From 172f4535ce8dcff79f626410af845f757c1563be Mon Sep 17 00:00:00 2001
From: Jerry Jiaheng Chen <80299683+CubeJerry@users.noreply.github.com>
Date: Thu, 12 Nov 2026 11:36:33 +0800
Subject: [PATCH] Lower the current selected quadratic Gaussian phase to a
 fixed tape table with complete owner return

---
 .../BINDING-OPTIMIZED.json                    |  17 +
 .../BINDING.json                              |  17 +
 .../ENVIRONMENT.json                          |   9 +
 .../literal-native-quadratic-phase/NOTICE     |  11 +
 .../PR-DESCRIPTION.md                         |  76 ++++
 .../PR-TITLE.txt                              |   1 +
 .../literal-native-quadratic-phase/PROOF.md   | 336 +++++++++++++++++
 .../literal-native-quadratic-phase/README.md  | 103 ++++++
 .../RECIPE.json                               | 130 +++++++
 .../RESULT-OPTIMIZED.json                     |  19 +
 .../RESULT.json                               |  19 +
 .../SOURCES.json                              |  71 ++++
 .../literal-native-quadratic-phase/TASK.json  |  40 ++
 .../VERIFY.json                               |  86 +++++
 .../bind_selected.py                          |  76 ++++
 .../independent/README.md                     |  38 ++
 .../independent/dense_check.py                | 144 ++++++++
 .../native-phase-dense-result.json            |  33 ++
 .../native-phase-source-bindings.json         |  71 ++++
 .../independent/peer/RESULT.json              | 262 +++++++++++++
 .../independent/peer/REVIEW.md                | 212 +++++++++++
 .../independent/peer/check.py                 | 322 ++++++++++++++++
 .../inputs/controller_reference.py            | 112 ++++++
 .../literal-native-quadratic-phase/phase.py   | 335 +++++++++++++++++
 .../reference/clifford_exact.py               | 149 ++++++++
 .../reference/native_verify.py                | 277 ++++++++++++++
 .../reference/outer-phase.py                  | 343 ++++++++++++++++++
 .../reference/selected-native-catalogue.py    |  97 +++++
 .../reference/selected-native.json.gz         | Bin 0 -> 350722 bytes
 .../reference/sequential-phase-PROOF.md       | 167 +++++++++
 .../reference/sequential-phase.py             | 125 +++++++
 .../literal-native-quadratic-phase/verify.py  | 201 ++++++++++
 32 files changed, 3899 insertions(+)
 create mode 100644 research/literal-native-quadratic-phase/BINDING-OPTIMIZED.json
 create mode 100644 research/literal-native-quadratic-phase/BINDING.json
 create mode 100644 research/literal-native-quadratic-phase/ENVIRONMENT.json
 create mode 100644 research/literal-native-quadratic-phase/NOTICE
 create mode 100644 research/literal-native-quadratic-phase/PR-DESCRIPTION.md
 create mode 100644 research/literal-native-quadratic-phase/PR-TITLE.txt
 create mode 100644 research/literal-native-quadratic-phase/PROOF.md
 create mode 100644 research/literal-native-quadratic-phase/README.md
 create mode 100644 research/literal-native-quadratic-phase/RECIPE.json
 create mode 100644 research/literal-native-quadratic-phase/RESULT-OPTIMIZED.json
 create mode 100644 research/literal-native-quadratic-phase/RESULT.json
 create mode 100644 research/literal-native-quadratic-phase/SOURCES.json
 create mode 100644 research/literal-native-quadratic-phase/TASK.json
 create mode 100644 research/literal-native-quadratic-phase/VERIFY.json
 create mode 100644 research/literal-native-quadratic-phase/bind_selected.py
 create mode 100644 research/literal-native-quadratic-phase/independent/README.md
 create mode 100644 research/literal-native-quadratic-phase/independent/dense_check.py
 create mode 100644 research/literal-native-quadratic-phase/independent/native-phase-dense-result.json
 create mode 100644 research/literal-native-quadratic-phase/independent/native-phase-source-bindings.json
 create mode 100644 research/literal-native-quadratic-phase/independent/peer/RESULT.json
 create mode 100644 research/literal-native-quadratic-phase/independent/peer/REVIEW.md
 create mode 100644 research/literal-native-quadratic-phase/independent/peer/check.py
 create mode 100644 research/literal-native-quadratic-phase/inputs/controller_reference.py
 create mode 100644 research/literal-native-quadratic-phase/phase.py
 create mode 100644 research/literal-native-quadratic-phase/reference/clifford_exact.py
 create mode 100644 research/literal-native-quadratic-phase/reference/native_verify.py
 create mode 100644 research/literal-native-quadratic-phase/reference/outer-phase.py
 create mode 100644 research/literal-native-quadratic-phase/reference/selected-native-catalogue.py
 create mode 100644 research/literal-native-quadratic-phase/reference/selected-native.json.gz
 create mode 100644 research/literal-native-quadratic-phase/reference/sequential-phase-PROOF.md
 create mode 100644 research/literal-native-quadratic-phase/reference/sequential-phase.py
 create mode 100644 research/literal-native-quadratic-phase/verify.py

diff --git a/research/literal-native-quadratic-phase/BINDING-OPTIMIZED.json b/research/literal-native-quadratic-phase/BINDING-OPTIMIZED.json
new file mode 100644
index 0000000000..203898de18
--- /dev/null
+++ b/research/literal-native-quadratic-phase/BINDING-OPTIMIZED.json
@@ -0,0 +1,17 @@
+{
+  "basis_states": 512,
+  "frames": [
+    3,
+    99
+  ],
+  "native_event_index": 24472,
+  "negative_controls": [
+    "changed absolute phase"
+  ],
+  "note": "Retained exact compiler and independent interpreter, plus new XOR-parity phase oracle. No full native catalogue replay.",
+  "owner": 415,
+  "rank": 1,
+  "side": "right",
+  "source_files": 8,
+  "status": "PASS_CURRENT_SELECTED_QUADRATIC_RECIPE_BINDING"
+}
diff --git a/research/literal-native-quadratic-phase/BINDING.json b/research/literal-native-quadratic-phase/BINDING.json
new file mode 100644
index 0000000000..203898de18
--- /dev/null
+++ b/research/literal-native-quadratic-phase/BINDING.json
@@ -0,0 +1,17 @@
+{
+  "basis_states": 512,
+  "frames": [
+    3,
+    99
+  ],
+  "native_event_index": 24472,
+  "negative_controls": [
+    "changed absolute phase"
+  ],
+  "note": "Retained exact compiler and independent interpreter, plus new XOR-parity phase oracle. No full native catalogue replay.",
+  "owner": 415,
+  "rank": 1,
+  "side": "right",
+  "source_files": 8,
+  "status": "PASS_CURRENT_SELECTED_QUADRATIC_RECIPE_BINDING"
+}
diff --git a/research/literal-native-quadratic-phase/ENVIRONMENT.json b/research/literal-native-quadratic-phase/ENVIRONMENT.json
new file mode 100644
index 0000000000..3ece38706a
--- /dev/null
+++ b/research/literal-native-quadratic-phase/ENVIRONMENT.json
@@ -0,0 +1,9 @@
+{
+  "python": "3.12.14 (main, Aug 25 2026, 14:00:49) [Clang 22.1.3 ]",
+  "platform": "linux",
+  "dependencies": "Python standard library only",
+  "normal_optimized_author_receipts_identical": true,
+  "relocated_independent_dense": "PASS, exact receipt equality",
+  "relocated_peer": "PASS_B3_INDEPENDENT_STRUCTURE",
+  "human_or_kernel_review": false
+}
diff --git a/research/literal-native-quadratic-phase/NOTICE b/research/literal-native-quadratic-phase/NOTICE
new file mode 100644
index 0000000000..7a1f1da9c5
--- /dev/null
+++ b/research/literal-native-quadratic-phase/NOTICE
@@ -0,0 +1,11 @@
+Prepared for Jerry Chen / CubeJerry with substantial OpenAI assistance.
+Apache-2.0. New contribution: literal local lowering, scoped proof and package.
+Ali Romman/#5375 supplies the incremental phase and repeated constant.
+The retained reference files preserve original source text.
+#7000 and its native/Gaussian lineage supply the selected source and engines;
+credit DreamingOfClouds, Ali Romman, Cinna Davis, zxv, unpaidmoon, icekylinx,
+CubeJerry and earlier authors named by the retained source.
+C2 literal-layer-phase supplies Gamma and signed staging/return. B1 literal-
+layer-egress supplies owner staging, and #7349 the retained literal machine.
+Exact original/copy source pins are in SOURCES.json. No predecessor identity,
+compiler, source construction or general monomial theorem is claimed as new.
diff --git a/research/literal-native-quadratic-phase/PR-DESCRIPTION.md b/research/literal-native-quadratic-phase/PR-DESCRIPTION.md
new file mode 100644
index 0000000000..1775e78a85
--- /dev/null
+++ b/research/literal-native-quadratic-phase/PR-DESCRIPTION.md
@@ -0,0 +1,76 @@
+## Result and purpose
+
+This package lowers the diagonal phase of an actual current #7000 native
+monomial to a fixed local tape table. The source is MOVE index 24472, owner 415,
+frames 3→99, rank 1, right factor, at main
+`e54523517d0f03e58352e647c403a5af3e525a57`. Its nonzero constant, linear and
+quadratic terms are retained, including the constant once per column.
+
+The graph has **8,211 locations, 23 tapes and four symbols**, using only
+local reads, one-cell writes, unit moves, fixed jumps and halt. For every
+positive supplied five-Gamma `(f,r,P,S,w)` descriptor and complete packed
+`[P] × [2^f]^9 × [S]` payload, it phases every Gaussian field and commits to
+the original owner. Success restores every head and private window. Exact
+signed range rejection preserves the public payload; failure cleanup and
+reuse are not claimed.
+
+With `Q=2^(9f)`, `N=PQS r`, `V=2Nw` and descriptor length E, the proved bill
+is `65536[E+10f+r+P+S+w+100PQ+V+1]`, hence O(V) on this supplied interface.
+A separate historical bound retains prior backing K0.
+
+## Novelty and predecessors
+
+#5375 already proves the multicounter update and repeated constant phase.
+Structured physical Gaussian monomials and selected record lists remain
+predecessors. The new contribution is the current recipe's literal table
+and complete owner contract, not a phase identity or full monomial theorem.
+
+The [outer phase package](https://github.com/CubeJerry/integer-mult-bounds/tree/8b549ca6557ccbb3017b104b27ea35051f990416/research/literal-layer-phase)
+supplies Gamma and signed staging/return; the
+[egress package](https://github.com/CubeJerry/integer-mult-bounds/tree/51ab0539059214fcfa044938c36a0b6017232be5/research/literal-layer-egress)
+supplies whole-owner staging, and #7349 supplies the machine. Exact source
+pins and copied original notices are retained.
+
+## Reproduction and checks
+
+Python 3.11+ and standard library only, from the repository root:
+
+```sh
+python3 -I -W error -B research/literal-native-quadratic-phase/bind_selected.py
+python3 -I -W error -O -B research/literal-native-quadratic-phase/bind_selected.py
+python3 -I -W error -B research/literal-native-quadratic-phase/verify.py
+python3 -I -W error -O -B research/literal-native-quadratic-phase/verify.py
+```
+
+Normal/optimized receipts agree. Source binding checks the actual event,
+exact lowering, retained independent kernel, all 512 monomial basis inputs
+and a phase fault. The separate arithmetic/address oracle checks 376
+complete packets, 11,991,714 transitions, all 336 width 1–3 endpoint pairs,
+53 exact range rejections and four semantic controls. Four packets use
+actual m=9,f=1; smaller fixed dimensions cover multiple columns. The author
+replay does not claim a complete actual f=2 packet.
+
+## Independent verification
+
+The lead's separate dense CPU/XOR-popcount oracle passes 16 cases and
+6,428,358 transitions, including explicitly incomplete selected f=2,3,4
+prefixes; eight exact Git source bindings pass. A complementary reviewer
+accepts the universal proof and complete bill. Their separate interpreter
+checks 11 complete successes, 9,362 records and 6,786,145 transitions, plus
+three mutations. A functionally correct full-width rewind is rejected by
+an independently derived exact movement census. Copied sources, reviews
+and receipts are in `independent/`; the relocated copies also pass.
+
+## Limitations and kappa
+
+**No new kappa.** Descriptor/layout production, gapped-layout conversion,
+affine routing, exterior scalars, recursive C and joined lifetimes remain
+open. No complete native word, selected layer or multiplier is admitted.
+All five foundational assumptions remain in the full conditional theorem.
+The next obligations are the actual descriptor/layout bridge, affine/child
+composition, and complete caller grid/headroom admission.
+
+Prepared for **@CubeJerry (Jerry Chen)** with substantial OpenAI assistance.
+Credit Ali Romman/#5375, #7000's retained native/Gaussian authors and the
+B1/C2/#7349 predecessors. Independent AI checks are not human or formal
+kernel acceptance. Apache-2.0.
diff --git a/research/literal-native-quadratic-phase/PR-TITLE.txt b/research/literal-native-quadratic-phase/PR-TITLE.txt
new file mode 100644
index 0000000000..8af5d76e24
--- /dev/null
+++ b/research/literal-native-quadratic-phase/PR-TITLE.txt
@@ -0,0 +1 @@
+Lower the current selected quadratic Gaussian phase to a fixed tape table with complete owner return
diff --git a/research/literal-native-quadratic-phase/PROOF.md b/research/literal-native-quadratic-phase/PROOF.md
new file mode 100644
index 0000000000..46a43bf214
--- /dev/null
+++ b/research/literal-native-quadratic-phase/PROOF.md
@@ -0,0 +1,336 @@
+# A literal quadratic phase for a current selected native recipe
+
+**Scoped enabling lemma; no new kappa.** Prepared for Jerry Chen / @CubeJerry
+with substantial OpenAI assistance. Canonical base:
+`e54523517d0f03e58352e647c403a5af3e525a57`.
+
+## 1. Exact claim, source and novelty
+
+Fix the nine-coordinate right monomial in the lowering of the actual retained
+#7000 native `MOVE` at zero-based program index **24472**:
+`["MOVE",415,3,99,false]`. Its before/after frames are 3 and 99 and its
+proper-width central child has rank 1. The source is
+`research/shared-wire-split-composition/data/final.json.gz`, compressed SHA256
+`2b94953dc14a122e3ef089633f05b58ec689ec5156feb7c784ed3e137dc01883`,
+with canonical uncompressed JSON SHA256
+`91429a6b96729292a8764836e00c87a6f7cd3b76f05159033f4867f69a98d097`.
+This is a current selected source event, not the historical event8814 fixture.
+
+`RECIPE.json` retains the whole exact lowering, including both monomial
+recipes. The right factor has constant phase 1, linear coefficients
+`(3,3,3,3,0,3,3,2,0)`, and symmetric quadratic row masks
+`(111,111,111,111,0,111,111,0,0)`. Diagonal entries are already included in
+the linear coefficients; only strictly upper matrix entries contribute
+quadratic terms. With `J={0,1,2,3,5,6}`, its input-coordinate phase is
+
+\[
+q(x)=1+3\sum_{i\in J}x_i+2x_7+
+      2\sum_{i<j,\ i,j\in J}x_ix_j\pmod4.                 \tag{1}
+\]
+
+The retained `clifford_exact.monomial` constructs `diag; permutation;
+translation; global phase`. Hence (1) is evaluated **before** the affine
+address map. The global unit phase commutes with that map. Its matrix rows
+and offset 1 remain in `RECIPE.json`; this package does not implement them.
+The source normalization/exterior scalar coefficients are not replaced by
+this Gaussian unit phase.
+
+`bind_selected.py` authenticates exact source bytes, locates the actual
+event, recomputes its lowering, and checks the complete factored kernel with
+the retained independently written signed-Pauli/Gaussian interpreter. It
+also checks all 512 right-monomial basis inputs against the separate formula
+`q(x)=1-(sum_J x_i mod2)+2*x_7`. This follows from
+`3h+2*binom(h,2)=h^2+2h`, which is 0 for even h and 3 for odd h modulo4.
+These are source-binding checks, not a new review of the full catalogue.
+
+The algebraic tensor phase and amortized multicounter update are prior work
+of Ali Romman in [#5375](https://github.com/CrocSwap/integer-mult-bounds/pull/5375),
+retained in `reference/sequential-phase-PROOF.md`. In particular that work
+already includes a constant phase once per active column. The contribution
+here is a **fixed local transition graph** for this actual selected recipe,
+including binary header acquisition, implicit complete address enumeration,
+all Gaussian fields and repetitions, exact original-owner commit, restored
+heads/private windows, and complete time and historical-extent bounds.
+
+The predecessor `physical-gaussian-monomial/PROOF.md` explicitly supplies
+structured tape control without a complete table. The selected tape-lift
+work supplies record lists. `current-serial-monomial-leaf` makes its
+descriptor iterator literal but leaves its numerical service structured.
+The streaming tariff leaves encoded-control lowering open. We do not claim
+their phase identities, physical monomial theorem, or tariff as new.
+
+## 2. Supplied local ABI and theorem
+
+The following packed interface is supplied, rather than inferred from the
+native word. Supply five positive integers `f,r,P,S,w` in canonical Elias
+Gamma form on HEADER, in that order. Gamma(z) is `ell` zero bits followed by
+the `ell+1` ordinary binary digits of z, where `ell=floor(log2 z)`. Let E be
+the total length of these five codes. No source/native descriptor conversion
+is hidden in the theorem.
+
+At DATA's origin supply the complete array indexed lexicographically by
+
+\[
+[P]\times[2^f]^m\times[S],\qquad m=9.                 \tag{2}
+\]
+
+The last coordinate of `[2^f]^m` varies fastest. Each record contains r
+Gaussian coefficients; each coefficient contains first w real bits and then
+w imaginary bits, both MSB first in signed two's-complement notation. All
+coefficients share an arbitrary fixed caller grid, which this operation
+does not change. Write
+
+\[
+Q=2^{mf},\quad N=PQS r,\quad V=2Nw.
+\]
+
+DATA contains exactly V binary cells and then a blank. It may have opaque
+data beyond that boundary. HEADER may have opaque data after the five codes.
+Both heads start at their respective origins. On each of the 21 private
+tapes, supply a blank work window starting one cell before its head's
+arbitrary origin and long enough for the extents below. All cells outside
+the declared work/input windows are opaque and must be preserved. Tape
+origins may differ and may be negative. Earlier allocated backing is not
+assumed to shrink on erasure.
+
+For coordinate integers `a=(a_0,...,a_(m-1))`, define column j by
+`x_(j,i)=(a_i >> j)&1`. The desired mode is
+
+\[
+\Phi(a)=\sum_{j=0}^{f-1}q(x_j)\pmod4.                \tag{3}
+\]
+
+**Theorem.** The fixed graph emitted by `compile_phase()` has 8,211
+locations, 23 tapes, alphabet `{blank,0,1,|}`, and only local `read`,
+one-cell `write`, unit `move`, fixed `goto`, and `halt` operations. It is
+independent of all five runtime integers and all payload data. On every
+legal complete packet, it accepts if and only if every requested rotated
+Gaussian component fits in the original w-bit signed range. On acceptance
+it replaces each coefficient z at address `(p,a,s)` by `i^Phi(a) z` at the
+same address and owner, preserves HEADER and all opaque cells, returns every
+head to its exact entry origin, and erases all private bodies and guards.
+
+On a named signed range failure it rejects before **any DATA write**, so
+the complete public payload and HEADER remain unchanged. Private work and
+head positions need not be restored on rejection. There is no rejected-call
+reuse, interruption recovery, or general malformed-input guarantee.
+
+The emitted graph can also be constructed for a fixed `1<=m<=9` recipe.
+Those smaller specializations are finite verification fixtures. No runtime
+recipe loader, input-sized graph, decoder oracle or extra address lookup is
+an operation of the selected graph. Python only constructs the fixed graph
+and simulates its local instructions. Observer integers such as the simulator
+step count do not affect any transition of the graph. The finite verifier
+step budget is external to the specified program.
+
+## 3. Header and initial state
+
+The retained C2 `gamma` helper represents each integer as a unary run on a
+fixed private tape. It counts leading zeroes on LENGTH, starts the target
+with one, and for each following binary bit b replaces unary v by `2v+b`.
+DOUBLE is erased after each such replacement. The code consumes exactly the
+number of tail bits recorded on LENGTH, without a decoded host value. HTRACE
+records one unary cell for each HEADER move and returns HEADER exactly to
+its origin; it is erased by that return.
+
+After the five decodes, COLUMNS, COEFF, OUTER, SPECTATOR and WIDTH contain
+`1^f,1^r,1^P,1^S,1^w`. All heads are at their body origins. Each private
+window has a unique local `|` at origin minus one. No boundary search can
+cross that guard.
+
+The initialization loop scans COLUMNS once. For each of its f ones it writes
+a zero on each of the m coordinate counters and advances those heads.
+The only numerical phase state is one of four control labels; each iteration
+adds the fixed constant q0=1 modulo4. On leaving the loop it therefore holds
+`f*q0 mod4`. It rewinds COLUMNS and all counters, giving the first address
+`a=(0,...,0)` and the correct mode. In particular replacing this initialization
+by a single q0, or by zero, would be wrong at general positive f.
+
+## 4. Synchronized counters and phase invariant
+
+Each counter holds its coordinate integer in exactly f cells, LSB first.
+At every `record` entry all m counter heads are at their origins. The
+integer values of the counters are the current address a; the four-state
+control is exactly (3). Neither P nor S nor the coefficient ordinal enters
+the phase. For each record, the program retains that control state while
+traversing S spectator copies and their r coefficients. COEFF is rewound
+after each r fields; SPECTATOR is rewound after each S records. Every
+header-supplied repetition is implemented by scanning these unary bodies.
+
+The next address is obtained by incrementing counter m-1. During a carry in
+counter i, **at each carry-bit read all m counter heads are aligned at the
+same column j**. The intervening fixed sequence of unit moves finishes
+before the next counter read. If the
+old bit is x, a flip changes (3) by
+
+\[
+(1-2x)\alpha_i+2\sum_{k\ne i}Q_{\min(i,k),\max(i,k)}x_{j,k}\pmod4. \tag{4}
+\]
+
+For the quadratic contribution the sign `1-2x` can be omitted because
+`-2=2 mod4`. The graph reads the fixed list of neighboring counters at
+their current cells, accumulates the sum in four-state control, flips the
+bit in counter i, and moves **all** m heads right by one. If x was zero,
+the carry ends, all heads rewind to their own guards, and the next record
+starts with the updated phase. If x was one, the carry continues.
+
+On the blank after f cells, the graph checks that all aligned heads see
+blank and rewinds them. Counter i has wrapped to zero and the control still
+equals (3), because every flip was paid by (4). It then carries into counter
+i-1. On overflow of counter 0 all m counters are zero. OUTER advances once;
+another outer row starts with the phase **f*q0**, not necessarily zero.
+The final row instead goes to the end-of-DATA check. No phase reset oracle
+is required: the invariant itself supplies the right wrapped value.
+
+Induction on individual flips proves the invariant. The standard binary
+odometer argument proves that the program visits every address in (2)
+exactly once in the prescribed order for each outer row. The proof covers
+arbitrary joint coefficient arrays; it assumes neither pure tensors nor
+uncorrelated columns. The tensor law merely specifies the diagonal action
+(3) on the basis coordinates of a complete array.
+
+## 5. Gaussian arithmetic, commit and exact failure domain
+
+The WIDTH run makes each component read consume exactly w bits, including
+leading zeroes and ones. REAL and IMAG are reused for one coefficient.
+Every DATA move writes one DTRACE cell; none writes DATA. The retained
+literal negation helper traverses a complete component from its LSB end,
+complements each bit and adds one with a one-bit carry. The ordinary carry
+invariant proves the modular result `-z mod 2^w`.
+
+At the sign cell the helper remembers both the original and resulting sign.
+Both are 1 exactly for the unique unrepresentable negation
+`z=-2^(w-1)`. It rejects that case. This argument also holds for w=1;
+no absolute-value operation on the original width is used. Copying or
+swapping a most-negative component is allowed when it is not negated.
+Thus the exact acceptance guards are:
+
+| Mode | Rotation | Components forbidden to equal `-2^(w-1)` |
+|---|---|---|
+| 0 | `(x,y)` | none |
+| 1 | `(-y,x)` | y |
+| 2 | `(-x,-y)` | x and y |
+| 3 | `(y,-x)` | x |
+
+Both components are transformed and staged in OUTPUT in the correct order.
+Their scratch bodies are erased before the next coefficient. A failure
+after earlier coefficients therefore can leave a partial private output,
+but the original complete DATA has not changed. All four operations are
+exact integer Gaussian unit multiplication on the unchanged caller grid.
+There is no rounding, truncation or intermediate scalar normalization.
+
+After all `P Q S r` coefficients the program requires DATA's next cell to
+be blank. DTRACE then returns DATA to its entry origin while erasing its
+own body. OUTPUT rewinds. Commit copies all V staged bits onto the **same
+DATA owner**, erasing each OUTPUT bit and rebuilding DTRACE. Its second
+return erases that trace and restores DATA. The program rewinds the now
+blank OUTPUT, clears every unary parameter and counter body, erases each
+local guard, and restores every private head to entry. No public marker is
+installed. All scans are bounded by bodies and private guards, so opaque
+cells outside the declared windows are never read or changed.
+
+## 6. Full literal time bound
+
+Each table instruction has unit cost. Put
+
+\[
+B=E+(m+1)f+r+P+S+w+(m+1)^2 P Q+V+1.
+\]
+
+For every accepted legal packet the explicit conservative bound is
+
+\[
+T\le65536 B.                                             \tag{5}
+\]
+
+The same bound holds up to rejection for a legal complete packet with a
+signed range failure. This is an intentionally loose uniform bound, not a
+claimed optimized streaming tariff. Its components are as follows.
+
+* Unary Gamma acquisition and return cost at most
+  `256(E+f+r+P+S+w+1)`: prefix updates `v->2v+b` cost at most `47v+35`
+  helper instructions plus the constant Gamma wrapper; the sum of earlier
+  binary prefix values of z is less than z. The five prefix/tail traces and
+  complete rewinds are included. This helper is copied from the frozen C2
+  program with only fixed tape identifiers changed.
+* Initialization and eventual counter/parameter cleanup cost
+  `O((m+1)f+r+P+S+w+1)` with an absolute constant below256. All zero bits,
+  guards, unused fixed ports, rewinds and erasures are included.
+* A complete mf-bit odometer cycle, including the final wrap, flips exactly
+  `F=2Q-2` bits. This remains true when its mf cells are split into m length-f
+  fields. Every flip uses at most m-1 neighbor reads, one write, m head
+  moves, and a bounded number of local reads/jumps. Returning the m heads
+  after a carry segment crosses exactly the cells just traversed; its work
+  is `O(m*segment_length+m)`. Each nonempty segment contains a flip, so all
+  rewinds, field-end tests, carries to preceding fields, and outer-row
+  advances fit `128(m+1)^2 P Q`. There is no f factor per address.
+* Each Gaussian pair uses at most `256w` instructions including two loads,
+  both possible negations, staging and component rewinds. The unary r/S
+  loops and their rewinds add O(N), bounded by another `32V` because
+  `r,S,w>=1`. They do not rerun phase arithmetic per coefficient.
+* Complete output/data return and commit cost `24V+13` with the retained
+  helper convention; end checking, parameter cleanup and fixed guards are
+  already covered. No retained history is rescanned.
+
+The sum is below (5), with substantial slack. These are instruction counts,
+not timings extrapolated from finite fixtures. Because m=9 is fixed and
+`r,S,w>=1`, `P Q<=N<=V/2`. Also `f,r,P,S,w<=V` and
+`E=O(f+r+P+S+w)`. Thus (5) gives **O(V)** on this supplied local ABI for
+every positive f. This absorption requires no long-record or RAM premise.
+The unreduced expression (5) exposes all header/repetition/counter terms.
+
+## 7. Historical storage and repeated invocation
+
+Let K0 include all previously backed cells and initial input/header storage,
+including opaque regions and current entry heads. Retiring or erasing cells
+does not subtract from K0. A sufficient additional backing bound is
+
+\[
+K_{\rm final}\le K_0+2V+4E+(m+4)f+4(r+P+S)+8w+256.       \tag{6}
+\]
+
+OUTPUT and DTRACE each reach V cells plus an end/guard boundary. HTRACE and
+LENGTH are bounded by E. DOUBLE never exceeds the largest decoded value.
+The five unary parameters have their declared lengths. REAL/IMAG reach w
+cells; the m counters each reach f cells. Fixed guards and end probes fit
+the constant in (6). The work regions can therefore be chosen in advance
+from those lengths, although the graph itself only discovers their ends
+by local scans. It creates no input-dependent number of tapes or objects.
+
+On a successful longer-then-shorter reuse, the active bodies are blank and
+heads restored; prior backing remains part of K0. The next execution touches
+only the next active lengths, so (5) does not depend on old high-water marks.
+This conclusion follows from local guard-delimited scans; it is not a claim
+that the sparse Python cell dictionary models deallocation. No analogous
+reuse result is claimed after a rejecting call.
+
+## 8. Composition limits and open next questions
+
+The exact source recipe binding does not prove that the native compiler
+produces the five-Gamma descriptor or the packed geometry (2) at a physical
+invocation. In particular a field layout with inactive gaps or interleaved
+spectators needs its own paid bridge or a more general literal mask program.
+Affirming those interfaces from a finite source index would be a hidden
+assumption. They remain supplied here.
+
+The affine map, its routing/Compact work, source access, caller scalar
+arithmetic, recursive rank-one C call, lifecycle composition and descriptor
+production remain outside this component. The local table and fresh output
+buffer do not establish global stock, tape, precision or recursive supplier
+admission. All five selected foundational assumptions remain in the full
+theorem, and the selected complete kappa is unchanged.
+
+**Open next questions:** (1) Produce this exact packed descriptor from one
+actual native invocation, or lower the existing arbitrary-active-mask update
+while pricing the required physical layout; (2) join the corresponding
+affine route and rank-one child with complete entry/return/lifecycle costs;
+(3) establish the caller's common grid and signed headroom at that joined
+occurrence. Each requires its own source-bound evidence, rather than a new
+claim about this standalone diagonal pass.
+
+Credit #5375 for incremental quadratic phases and tensor constants; the
+retained #7000 source/compiler and earlier Gaussian/Clifford contributors
+for the recipe; C2 `literal-layer-phase` for Gamma, signed staging and
+return helpers; B1 `literal-layer-egress` for the original-owner staging
+method; and #7349 for the retained literal machine. Independent AI review,
+when recorded, is not human or proof-kernel verification.
diff --git a/research/literal-native-quadratic-phase/README.md b/research/literal-native-quadratic-phase/README.md
new file mode 100644
index 0000000000..f36ff6105f
--- /dev/null
+++ b/research/literal-native-quadratic-phase/README.md
@@ -0,0 +1,103 @@
+# A fixed tape table for the current selected quadratic Gaussian phase
+
+**Scoped enabling lemma; no new kappa.** This package lowers the diagonal
+part of one actual current #7000 monomial to a fixed local tape program:
+native MOVE index 24472, owner 415, frames 3→99, rank 1, right factor. It has a nonzero
+constant, seven nonzero linear coefficients and fifteen quadratic pairs.
+`RECIPE.json` records the exact source lowering.
+
+The graph has **8,211 locations, 23 tapes and four symbols**. Runtime control
+uses only scanned symbols, one-cell writes, unit moves, fixed jumps and
+halt. It receives a supplied five-Gamma descriptor `(f,r,P,S,w)` and the
+complete packed payload `[P] × [2^f]^9 × [S]`, with r signed Gaussian fields
+per record. It applies the tensor phase, including the constant `i^f`, and
+commits the exact result to the original owner. Successful calls restore
+every head and private window. A signed range failure leaves the complete
+public payload unchanged, without promising failure cleanup or reuse.
+
+For `Q=2^(9f)`, `N=PQS r`, `V=2Nw` and descriptor length E, the complete
+bound is `65536[E+10f+r+P+S+w+100PQ+V+1]`, hence **O(V)** for every positive
+f on this supplied interface. It includes header acquisition, synchronized
+counters, repetitions, signed staging, commit and erasure. The proof also
+retains historical backing relative to prior history K0.
+
+## Novelty and limits
+
+[#5375](https://github.com/CrocSwap/integer-mult-bounds/pull/5375) already
+proves the incremental multicounter phase algorithm and repeated constant.
+The structured physical Gaussian monomial and selected record-list work
+are predecessors. The new result is this current recipe's literal table,
+complete original-owner contract and bill; see [PROOF.md](PROOF.md).
+
+The native descriptor producer, bridge from a gapped/interleaved layout,
+affine routing, source scalar arithmetic, recursive C call and caller
+lifecycle remain open. This is not a complete monomial, all-column supplier,
+selected layer or multiplier. All five foundational assumptions remain in
+the complete theorem and its conditional kappa is unchanged.
+
+## Evidence and reproduction
+
+Python 3.11+ and standard library only, from the repository root:
+
+```sh
+python3 -I -W error -B research/literal-native-quadratic-phase/bind_selected.py
+python3 -I -W error -O -B research/literal-native-quadratic-phase/bind_selected.py
+python3 -I -W error -B research/literal-native-quadratic-phase/verify.py
+python3 -I -W error -O -B research/literal-native-quadratic-phase/verify.py
+```
+
+Expected markers: `PASS_CURRENT_SELECTED_QUADRATIC_RECIPE_BINDING` and
+`PASS_LITERAL_NATIVE_QUADRATIC_PHASE_NO_NEW_KAPPA`. Normal/optimized outputs
+match the BINDING and RESULT receipts. These are standalone commands;
+the package does not claim a registered research-package schema.
+VERIFY.json records these direct commands and the two independent replays.
+
+Source binding authenticates eight unchanged references, reconstructs the
+actual recipe, checks its full kernel with the retained independent
+signed-Pauli/Gaussian interpreter, and checks all 512 right-monomial basis
+inputs against a separate parity formula. This is not a new catalogue review.
+
+The packet replay covers **376 complete packets, 11,991,714 transitions and
+all 336 width 1–3 Gaussian endpoint pairs**. It rejects 53 exact range cases
+without a public write. Four complete packets use actual m=9,f=1, including
+outer/spectator/coefficient repetitions and widths 1,2,5,17; smaller fixed
+dimensions cover f=1,2,3. Its direct address/polynomial/integer oracle does
+not use incremental phase updates. Four controls detect an omitted repeated
+phase, deleted actual quadratic pair, wrong linear sign and corrupt commit.
+No complete selected m=9,f=2 packet is claimed by this author replay.
+Universal claims rest on the written invariant and scan proofs.
+
+## Independent review
+
+The lead's separate dense CPU and XOR/popcount oracle pass 16 cases and
+6,428,358 transitions, including exact public preservation and successful
+longer-to-shorter reuse. Its selected f=2,3,4 inputs deliberately stop after
+300 coefficients: correct staged prefixes are checked, not full tensor
+execution. All eight exact Git source bindings were independently verified.
+
+A second reviewer accepts the universal phase/geometry, complete cost,
+retirement and historical-storage proof. A separate interpreter checks
+11 complete successes, 9,362 record entries and 6,786,145 transitions.
+It derives and verifies an exact counter-movement census. One cost control
+is functionally correct but unnecessarily scans the full counter width;
+the census detects that error. Review, sources, receipts and portable
+commands are in [independent/README.md](independent/README.md).
+
+## Open next questions
+
+1. Produce this descriptor and packed layout at a selected native occurrence,
+   or lower the existing arbitrary-active-mask version with physical costs.
+2. Join its affine route and rank-one child with complete source access,
+   current-chart, private-lifetime and return accounting.
+3. Establish the common grid and signed headroom at that joined caller.
+
+## Credit
+
+Prepared for **@CubeJerry (Jerry Chen)** with substantial OpenAI assistance
+in mathematics, implementation, checking and prose. Apache-2.0. Credit
+Ali Romman/#5375; #7000's DreamingOfClouds, Ali Romman, Cinna Davis, zxv,
+unpaidmoon, icekylinx and earlier native/Gaussian authors; C2's
+`literal-layer-phase` for Gamma and signed staging/return; B1's
+`literal-layer-egress` for owner staging; and #7349 for the retained literal
+machine. Exact pins are in [SOURCES.json](SOURCES.json). Independent AI
+review is not human or proof-kernel acceptance.
diff --git a/research/literal-native-quadratic-phase/RECIPE.json b/research/literal-native-quadratic-phase/RECIPE.json
new file mode 100644
index 0000000000..ea2bc6909e
--- /dev/null
+++ b/research/literal-native-quadratic-phase/RECIPE.json
@@ -0,0 +1,130 @@
+{
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "native_event_index": 24472,
+  "event": [
+    "MOVE",
+    415,
+    3,
+    99,
+    false
+  ],
+  "side": "right",
+  "rank": 1,
+  "recipe": {
+    "matrix_rows": [
+      239,
+      129,
+      130,
+      132,
+      136,
+      16,
+      160,
+      192,
+      256
+    ],
+    "offset": 1,
+    "quadratic_rows": [
+      111,
+      111,
+      111,
+      111,
+      0,
+      111,
+      111,
+      0,
+      0
+    ],
+    "linear_Z4": [
+      3,
+      3,
+      3,
+      3,
+      0,
+      3,
+      3,
+      2,
+      0
+    ],
+    "phase": 1
+  },
+  "lowered_recipe": {
+    "rank": 1,
+    "left": {
+      "matrix_rows": [
+        3,
+        5,
+        9,
+        17,
+        32,
+        65,
+        129,
+        1,
+        256
+      ],
+      "offset": 0,
+      "quadratic_rows": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "linear_Z4": [
+        2,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "phase": 0
+    },
+    "right": {
+      "matrix_rows": [
+        239,
+        129,
+        130,
+        132,
+        136,
+        16,
+        160,
+        192,
+        256
+      ],
+      "offset": 1,
+      "quadratic_rows": [
+        111,
+        111,
+        111,
+        111,
+        0,
+        111,
+        111,
+        0,
+        0
+      ],
+      "linear_Z4": [
+        3,
+        3,
+        3,
+        3,
+        0,
+        3,
+        3,
+        2,
+        0
+      ],
+      "phase": 1
+    }
+  },
+  "catalogue_pairs_examined": 1,
+  "native_compressed_sha256": "2b94953dc14a122e3ef089633f05b58ec689ec5156feb7c784ed3e137dc01883",
+  "native_canonical_sha256": "91429a6b96729292a8764836e00c87a6f7cd3b76f05159033f4867f69a98d097"
+}
diff --git a/research/literal-native-quadratic-phase/RESULT-OPTIMIZED.json b/research/literal-native-quadratic-phase/RESULT-OPTIMIZED.json
new file mode 100644
index 0000000000..a50e2c52de
--- /dev/null
+++ b/research/literal-native-quadratic-phase/RESULT-OPTIMIZED.json
@@ -0,0 +1,19 @@
+{
+  "complete_packets": 376,
+  "literal_transitions": 11991714,
+  "negative_controls": [
+    "drop_repeated_constant",
+    "drop_actual_quadratic_pair",
+    "wrong_actual_linear_sign",
+    "commit_bit_corruption"
+  ],
+  "range_rejections": 53,
+  "scope": "Finite evidence; universal f, payload, head and cost statements are in PROOF.md.",
+  "selected_complete_packets": 4,
+  "signed_endpoint_pairs": 336,
+  "source_sha256": "2be149aa1be504509bb480f85396f0be302082d638e81422f747daee5a3c1ce7",
+  "status": "PASS_LITERAL_NATIVE_QUADRATIC_PHASE_NO_NEW_KAPPA",
+  "table_locations": 8211,
+  "table_sha256": "e10a0070af3331d23963fb97ebcd4bb7e81314d94a43d41ce00d4c84c5747638",
+  "tapes": 23
+}
diff --git a/research/literal-native-quadratic-phase/RESULT.json b/research/literal-native-quadratic-phase/RESULT.json
new file mode 100644
index 0000000000..a50e2c52de
--- /dev/null
+++ b/research/literal-native-quadratic-phase/RESULT.json
@@ -0,0 +1,19 @@
+{
+  "complete_packets": 376,
+  "literal_transitions": 11991714,
+  "negative_controls": [
+    "drop_repeated_constant",
+    "drop_actual_quadratic_pair",
+    "wrong_actual_linear_sign",
+    "commit_bit_corruption"
+  ],
+  "range_rejections": 53,
+  "scope": "Finite evidence; universal f, payload, head and cost statements are in PROOF.md.",
+  "selected_complete_packets": 4,
+  "signed_endpoint_pairs": 336,
+  "source_sha256": "2be149aa1be504509bb480f85396f0be302082d638e81422f747daee5a3c1ce7",
+  "status": "PASS_LITERAL_NATIVE_QUADRATIC_PHASE_NO_NEW_KAPPA",
+  "table_locations": 8211,
+  "table_sha256": "e10a0070af3331d23963fb97ebcd4bb7e81314d94a43d41ce00d4c84c5747638",
+  "tapes": 23
+}
diff --git a/research/literal-native-quadratic-phase/SOURCES.json b/research/literal-native-quadratic-phase/SOURCES.json
new file mode 100644
index 0000000000..f4e1e676c9
--- /dev/null
+++ b/research/literal-native-quadratic-phase/SOURCES.json
@@ -0,0 +1,71 @@
+{
+  "schema": "standalone-provenance-v1",
+  "canonical_base": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "claim": "literal local phase only; no new kappa",
+  "files": [
+    {
+      "local_path": "reference/clifford_exact.py",
+      "path": "research/shared-wire-split-composition/frozen/clifford_exact.py",
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962"
+    },
+    {
+      "local_path": "reference/native_verify.py",
+      "path": "research/shared-wire-split-composition/frozen/native_verify.py",
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "git_blob": "2c209dfa9cfade3bc20ac5fc9d876b433a147f8c",
+      "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38"
+    },
+    {
+      "local_path": "reference/selected-native.json.gz",
+      "path": "research/shared-wire-split-composition/data/final.json.gz",
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "git_blob": "74563a7867d9d762b6588f65362369668841bb3e",
+      "sha256": "2b94953dc14a122e3ef089633f05b58ec689ec5156feb7c784ed3e137dc01883"
+    },
+    {
+      "local_path": "reference/selected-native-catalogue.py",
+      "path": "docs/research/pr-intake-evidence/pr7000-application/check_native_catalogue.py",
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "git_blob": "98b35c89c8bd52637633aa78d506244bca8ec9e1",
+      "sha256": "6120e1ea1e07eeba757694d209437a38f121b0a377cc466889a30fa52c561472"
+    },
+    {
+      "local_path": "reference/sequential-phase.py",
+      "path": "research/sequential-monomial-phase/phase.py",
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "git_blob": "26126045289a3dceeb8d096c984d536613e5345f",
+      "sha256": "256aa42f38e496bf91347078d8e2c731ee35b03fc0528f536a946f215f79fcd7"
+    },
+    {
+      "local_path": "reference/sequential-phase-PROOF.md",
+      "path": "research/sequential-monomial-phase/PROOF.md",
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "git_blob": "42ffd4007c1416eb4a45e47d181194d3e87742bb",
+      "sha256": "aed63dc425a410627a290b2e0251b27bc1befce85614cefbb8a82f199a1d2764"
+    },
+    {
+      "local_path": "reference/outer-phase.py",
+      "path": "research/literal-layer-phase/phase.py",
+      "repository": "CubeJerry/integer-mult-bounds",
+      "commit": "8b549ca6557ccbb3017b104b27ea35051f990416",
+      "status": "verified remote commit reported by lead publication gate",
+      "sha256": "9a360817b80e869398927f38a4f110de0b5a1f9620eeb23b19e166f112b83f23"
+    },
+    {
+      "local_path": "inputs/controller_reference.py",
+      "path": "research/literal-layer-egress/inputs/controller_reference.py",
+      "repository": "CubeJerry/integer-mult-bounds",
+      "commit": "51ab0539059214fcfa044938c36a0b6017232be5",
+      "sha256": "e664a748e6b6797d4b60f60a7a7dd68b59a5608acf56b9c5d8b466f363755f73",
+      "note": "Exact 112-line retained literal machine reference inherited through B1/C2; origin PR7349, head9dfe6747b8553fd11b408f2b045c5954ef1e6eaa, source blob3ab6d0361f35038ffcdf0cf81a157f7a17b40e72."
+    }
+  ]
+}
diff --git a/research/literal-native-quadratic-phase/TASK.json b/research/literal-native-quadratic-phase/TASK.json
new file mode 100644
index 0000000000..83af63786d
--- /dev/null
+++ b/research/literal-native-quadratic-phase/TASK.json
@@ -0,0 +1,40 @@
+{
+  "schema": "standalone-task-provenance-v1",
+  "outcome": "completed",
+  "review_status": "Author normal/-O checks, independent root dense arithmetic/source authentication, and complementary universal/cost peer review accepted.",
+  "worker": "literal_layer_boundary",
+  "wave": 3,
+  "track_seeds": [
+    20,
+    60,
+    75
+  ],
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "claim": "Actual MOVE24472 right quadratic phase: fixed graph for supplied packed five-Gamma ABI, every positive f, exact range rejection and complete O(V) owner return.",
+  "method": "Literal table; separate direct address/polynomial/Gaussian oracle; retained independent source kernel verification.",
+  "stopping_rule": "Stop if no exact current recipe or prior complete literal table; otherwise freeze after proof, source, cost and endpoint checks.",
+  "overlap": [
+    "sequential-monomial-phase",
+    "physical-gaussian-monomial",
+    "selected-monomial-tape-lift",
+    "selected-monomial-coordinate-refinement",
+    "streaming-gaussian-phase-tariff",
+    "current-serial-monomial-leaf",
+    "derived-phase-stream",
+    "encoded-phase-bank"
+  ],
+  "nonclaims": [
+    "Complete descriptor/layout bridge",
+    "Affine routing",
+    "Recursive C",
+    "Whole supplier",
+    "Foundational assumption removal",
+    "New kappa"
+  ],
+  "open_next_questions": [
+    "Produce actual native packed descriptor/layout with all conversion costs.",
+    "Join affine route and rank-one child with source/lifecycle costs.",
+    "Prove complete caller common grid and signed headroom."
+  ],
+  "kappa_status": "No new kappa; all five full-theorem assumptions remain."
+}
diff --git a/research/literal-native-quadratic-phase/VERIFY.json b/research/literal-native-quadratic-phase/VERIFY.json
new file mode 100644
index 0000000000..f6206bdfba
--- /dev/null
+++ b/research/literal-native-quadratic-phase/VERIFY.json
@@ -0,0 +1,86 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "research/literal-native-quadratic-phase/bind_selected.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Recorded standalone source binding and literal program checks; normal and optimized receipts agree."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "-O",
+        "research/literal-native-quadratic-phase/bind_selected.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Recorded standalone source binding and literal program checks; normal and optimized receipts agree."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "research/literal-native-quadratic-phase/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Recorded standalone source binding and literal program checks; normal and optimized receipts agree."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "-O",
+        "research/literal-native-quadratic-phase/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Recorded standalone source binding and literal program checks; normal and optimized receipts agree."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "research/literal-native-quadratic-phase/independent/dense_check.py",
+        "research/literal-native-quadratic-phase/phase.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Independent dense CPU and selected XOR/popcount oracle, including explicitly incomplete larger-column packets."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "research/literal-native-quadratic-phase/independent/peer/check.py",
+        "research/literal-native-quadratic-phase"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Independent proof/structural review replay and exact counter-movement controls."
+    }
+  ],
+  "inputs": [],
+  "note": "Direct reproduction manifest; TASK/SOURCES use their documented standalone schemas, not registered-package admission."
+}
diff --git a/research/literal-native-quadratic-phase/bind_selected.py b/research/literal-native-quadratic-phase/bind_selected.py
new file mode 100644
index 0000000000..583f4087c4
--- /dev/null
+++ b/research/literal-native-quadratic-phase/bind_selected.py
@@ -0,0 +1,76 @@
+"""Exact current-source recipe binding; retained independent Clifford check."""
+from pathlib import Path
+import copy
+import gzip
+import hashlib
+import importlib.util
+import json
+import sys
+
+HERE = Path(__file__).resolve().parent
+
+
+def need(value, message):
+    if not value:
+        raise ValueError(message)
+
+
+def module(name, path):
+    spec = importlib.util.spec_from_file_location(name, path)
+    result = importlib.util.module_from_spec(spec)
+    sys.modules[name] = result
+    spec.loader.exec_module(result)
+    return result
+
+
+def main():
+    pins = json.loads((HERE/'SOURCES.json').read_text())
+    for record in pins['files']:
+        need(hashlib.sha256((HERE/record['local_path']).read_bytes()).hexdigest() == record['sha256'],
+             'source bytes '+record['local_path'])
+    C = module('binding_compiler', HERE/'reference/clifford_exact.py')
+    V = module('binding_independent', HERE/'reference/native_verify.py')
+    raw = (HERE/'reference/selected-native.json.gz').read_bytes()
+    word = json.loads(gzip.decompress(raw))
+    record = json.loads((HERE/'RECIPE.json').read_text())
+    need(hashlib.sha256(raw).hexdigest() == record['native_compressed_sha256'], 'compressed selected word')
+    need(hashlib.sha256(json.dumps(word,separators=(',',':'),sort_keys=True).encode()).hexdigest() == record['native_canonical_sha256'], 'canonical selected word')
+    event = word['program'][record['native_event_index']]
+    need(event == ['MOVE',415,3,99,False] == record['event'], 'actual selected event')
+    before, after = event[2:4]
+    a = C.canonical_frame(9,tuple(word['frames'][before]))
+    b = C.canonical_frame(9,tuple(word['frames'][after]))
+    recipe, _ = C.lower_child(a.adjoint().then(b))
+    recipe = json.loads(json.dumps(recipe))
+    need(recipe == record['lowered_recipe'], 'exact source lowering')
+    need(recipe['rank'] == 1 and recipe[record['side']] == record['recipe'], 'selected right factor')
+    # Retained independent signed-Pauli/global-Gaussian method. This is not
+    # a new independent review of the complete selected native catalogue.
+    target = V.adjoint(V.frame(word['frames'][before],9)) + V.frame(word['frames'][after],9)
+    V.equal(V.recipe(recipe),target,'selected MOVE complete kernel')
+    selected = record['recipe']
+    J = (0,1,2,3,5,6)
+    for x in range(512):
+        h = sum((x >> i) & 1 for i in J)
+        mode = (1-(h % 2)+2*((x >> 7) & 1)) % 4
+        y = sum(((row & x).bit_count() % 2) << i for i,row in enumerate(selected['matrix_rows'])) ^ selected['offset']
+        expected = {y: ((1,0),(0,1),(-1,0),(0,-1))[mode]}
+        value, denominator = V.state(V.monomial(selected),x)
+        need(denominator == 0 and value == expected, 'input-coordinate absolute phase')
+    wrong = copy.deepcopy(recipe)
+    wrong['right']['phase'] = (wrong['right']['phase']+1) % 4
+    try:
+        V.equal(V.recipe(wrong),target,'changed absolute phase')
+    except ValueError:
+        pass
+    else:
+        raise ValueError('global phase control missed')
+    print(json.dumps({'status':'PASS_CURRENT_SELECTED_QUADRATIC_RECIPE_BINDING',
+          'native_event_index':record['native_event_index'],'frames':[before,after],
+          'owner':415,'side':'right','rank':1,'basis_states':512,
+          'source_files':len(pins['files']),'negative_controls':['changed absolute phase'],
+          'note':'Retained exact compiler and independent interpreter, plus new XOR-parity phase oracle. No full native catalogue replay.'},indent=2,sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/literal-native-quadratic-phase/independent/README.md b/research/literal-native-quadratic-phase/independent/README.md
new file mode 100644
index 0000000000..0819f7a66e
--- /dev/null
+++ b/research/literal-native-quadratic-phase/independent/README.md
@@ -0,0 +1,38 @@
+# Independent review and replay
+
+The lead independently wrote dense_check.py. It executes the exported graph
+without the author CPU or packet oracle, and derives the current phase from
+f-popcount(a0^a1^a2^a3^a5^a6)+2*popcount(a7) modulo4. Its source is copied
+unchanged; SHA256 163b9042622825c3d97153508336fc4fd285ac0bd01910186ee6eafb7d9e777a.
+
+The lead ran normal and optimized modes; receipts agree. The package-local
+copy was additionally run once and matched the committed result exactly.
+From the repository root:
+
+```sh
+python3 -I -W error -B research/literal-native-quadratic-phase/independent/dense_check.py research/literal-native-quadratic-phase/phase.py
+python3 -I -W error -O -B research/literal-native-quadratic-phase/independent/dense_check.py research/literal-native-quadratic-phase/phase.py
+```
+
+Result: 16 cases, 7 accepts, 11,776 coefficients in complete fixture inputs,
+6,428,358 transitions and three detected program mutations. The f=2,3,4
+cases each deliberately supply only 300 coefficients: they validate staged
+prefix arithmetic and public preservation on rejection, not full tensor
+execution. Successful longer-to-shorter reuse preserves the physical dense
+tape arrays and restores their heads/private cells.
+
+native-phase-source-bindings.json records independent authentication of all
+eight exact Git commit:path source bindings, including the published outer
+phase predecessor. This authenticates source bytes, not mathematical scope.
+
+The complementary peer accepts the universal phase/geometry, complete cost,
+retirement and historical-storage theorem. Its unchanged files are under
+peer/. It independently interprets the graph, derives exact head-movement
+counts and detects a functionally correct but more expensive full-width
+rewind. Program and final proof hashes are pinned in peer/REVIEW.md.
+The package-local checker also passed after relocation:
+
+```sh
+python3 -I -W error -B research/literal-native-quadratic-phase/independent/peer/check.py research/literal-native-quadratic-phase
+```
+Independent AI review is not human or formal-kernel acceptance.
diff --git a/research/literal-native-quadratic-phase/independent/dense_check.py b/research/literal-native-quadratic-phase/independent/dense_check.py
new file mode 100644
index 0000000000..6e3fd1e7ae
--- /dev/null
+++ b/research/literal-native-quadratic-phase/independent/dense_check.py
@@ -0,0 +1,144 @@
+"""Independent dense CPU and selected XOR/popcount phase oracle; Apache-2.0.
+Prepared for CubeJerry with OpenAI assistance. No author CPU/tests execute.
+"""
+from pathlib import Path
+from itertools import product
+import hashlib,importlib.util,json,sys
+class Rejected(Exception):pass
+
+def load(path):
+    spec=importlib.util.spec_from_file_location('phase_subject',path)
+    m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m);return m
+
+def gamma(n):
+    s=format(n,'b');return '0'*(len(s)-1)+s
+
+def bits(n,w):
+    if not -(1<<(w-1))<=n<(1<<(w-1)):raise OverflowError('signed range')
+    return format(n%(1<<w),f'0{w}b')
+
+class CPU:
+    def __init__(self,header,data,capacity=None):
+        self.origin=[31+3*t for t in range(23)];self.head=self.origin.copy()
+        size=capacity or max(2048,4*(len(data)+len(header)+100))
+        self.rows=[['_']*size for _ in range(23)];self.steps=0
+        for t,o in enumerate(self.origin):self.rows[t][o-5:o-1]=list('1|01')
+        self.supply(header,data)
+    def supply(self,header,data):
+        if self.head!=self.origin:raise AssertionError('not retired')
+        for t,s in ((0,header),(1,data)):
+            o=self.origin[t];self.rows[t][o:]=['_']*(len(self.rows[t])-o)
+            self.rows[t][o:o+len(s)]=list(s);self.rows[t][o+len(s)+1:o+len(s)+5]=list('1|01')
+        self.before=[r.copy() for r in self.rows]
+    def run(self,code):
+        pc=0;start=self.steps
+        while self.steps-start<25_000_000:
+            op,*args=code[pc];pc+=1;self.steps+=1
+            if op=='read':
+                t,edges=args;s=self.rows[t][self.head[t]]
+                if s not in edges:raise Rejected('undefined local branch')
+                pc=edges[s]
+            elif op=='write':
+                t,s=args
+                if s not in '_01|':raise AssertionError('alphabet')
+                self.rows[t][self.head[t]]=s
+            elif op=='move':
+                t,d=args
+                if d not in (-1,1):raise AssertionError('nonlocal move')
+                self.head[t]+=d
+                if not 0<=self.head[t]<len(self.rows[t]):raise AssertionError('capacity')
+            elif op=='goto':pc=args[0]
+            elif op=='halt':return
+            else:raise AssertionError('opcode')
+        raise AssertionError('CPU budget exhausted, not a successful rejection')
+    def check(self,expected):
+        desired=[r.copy() for r in self.before];o=self.origin[1];desired[1][o:o+len(expected)]=list(expected)
+        if self.rows!=desired:raise AssertionError('payload/header/exterior/cleanup')
+        if self.head!=self.origin:raise AssertionError('head return')
+
+def expected_bits(phases,pairs,w):
+    output=''
+    for q,(x,y) in zip(phases,pairs):
+        for _ in range(q):x,y=-y,x
+        output+=bits(x,w)+bits(y,w)
+    return output
+
+
+def require(ok,message):
+    if not ok:raise AssertionError(message)
+
+def phase(f,address):
+    a=[0]*9
+    for j in range(8,-1,-1):address,a[j]=divmod(address,1<<f)
+    require(address==0,'coordinate range')
+    parity=a[0]^a[1]^a[2]^a[3]^a[5]^a[6]
+    return (f-parity.bit_count()+2*a[7].bit_count())%4
+
+def exercise(code,params,pairs=None,alter=None,cpu=None):
+    f,r,P,S,w=params;Q=1<<(9*f);count=300 if alter=='prefix' else P*Q*S*r
+    phases=[phase(f,(j//(S*r))%Q) for j in range(count)]
+    if pairs is None:
+        lim=min(12,(1<<(w-1))-1)
+        pairs=[((7*j)%(2*lim+1)-lim,(11*j)%(2*lim+1)-lim) for j in range(count)]
+    header=''.join(gamma(n) for n in params)
+    data=''.join(bits(x,w)+bits(y,w) for x,y in pairs)
+    try:expected=expected_bits(phases,pairs,w)
+    except OverflowError:expected=None
+    if alter=='extra':data+='1';expected=None
+    if alter=='short':data=data[:-1];expected=None
+    if cpu is None:cpu=CPU(header,data)
+    else:cpu.supply(header,data)
+    start=cpu.steps
+    try:cpu.run(code)
+    except Rejected:
+        require(expected is None or alter=='prefix','unexpected literal rejection')
+        require(cpu.rows[0]==cpu.before[0] and cpu.rows[1]==cpu.before[1],'public rejection mutation')
+        if alter=='prefix':
+            o=cpu.origin[2]
+            require(cpu.rows[2][o:o+len(expected)]==list(expected),'incomplete f>1 staged prefix arithmetic')
+            require(cpu.head[2]==o+len(expected),'staged prefix length')
+        return cpu,cpu.steps-start,False,count
+    require(expected is not None and alter!='prefix','missed range/cardinality guard')
+    cpu.check(expected)
+    return cpu,cpu.steps-start,True,count
+
+def main():
+    path=Path(sys.argv[1]).resolve();m=load(path);code,alphabet=m.compile_phase()
+    require(set(alphabet)==set('_01|'),'alphabet')
+    require(m.SELECTED_LINEAR==(3,3,3,3,0,3,3,2,0),'linear source pin')
+    require(m.SELECTED_ROWS==(111,111,111,111,0,111,111,0,0),'quadratic source pin')
+    require(m.SELECTED_CONSTANT==1,'absolute phase source pin')
+    cases=steps=accepted=coefficients=0
+    cpu=CPU('','',capacity=100000);identities=tuple(map(id,cpu.rows))
+    for params in [(1,2,2,2,5),(1,1,1,1,3),(1,2,1,3,6)]:
+        cpu,n,ok,count=exercise(code,params,cpu=cpu)
+        cases+=1;steps+=n;accepted+=ok;coefficients+=count
+        require(tuple(map(id,cpu.rows))==identities,'retained tape identities')
+    phases=[phase(1,a) for a in range(512)]
+    for q in range(4):
+        where=phases.index(q)
+        for component in (0,1):
+            pairs=[(0,0)]*512;xy=[0,0];xy[component]=-2;pairs[where]=tuple(xy)
+            _,n,ok,count=exercise(code,(1,1,1,1,2),pairs)
+            cases+=1;steps+=n;accepted+=ok;coefficients+=count
+    for alter in ('extra','short'):
+        _,n,ok,count=exercise(code,(1,1,1,1,4),alter=alter)
+        cases+=1;steps+=n;accepted+=ok
+    prefixes=[]
+    for f in (2,3,4):
+        _,n,ok,count=exercise(code,(f,1,1,1,5),alter='prefix')
+        cases+=1;steps+=n
+        prefixes.append({'f':f,'staged_coefficients':count,'complete_packet':False})
+    variants=[
+        ('omit absolute tensor phase',m.compile_phase(constant=0)[0]),
+        ('omit quadratic cross terms',m.compile_phase(rows=tuple((1<<i) if x&1 else 0 for i,x in enumerate(m.SELECTED_LINEAR)))[0]),
+        ('wrong original-owner commit',[('write',1,'1') if x==('write',1,'0') else x for x in code]),
+    ]
+    controls=[]
+    for name,mutant in variants:
+        require(mutant!=code,'effective mutation')
+        try:exercise(mutant,(1,1,1,1,5))
+        except AssertionError:controls.append(name)
+        else:raise AssertionError('undetected '+name)
+    print(json.dumps({'status':'PASS_INDEPENDENT_NATIVE_PHASE','cases':cases,'accepted':accepted,'full_fixture_coefficients':coefficients,'transitions':steps,'locations':len(code),'tapes':23,'controls':controls,'incomplete_prefixes':prefixes,'phase_sha256':hashlib.sha256(path.read_bytes()).hexdigest(),'limitations':'All-f theorem rests on proof; f=2,3,4 checks are deliberately incomplete packets, with verified staged prefixes.'},sort_keys=True))
+if __name__=='__main__':main()
diff --git a/research/literal-native-quadratic-phase/independent/native-phase-dense-result.json b/research/literal-native-quadratic-phase/independent/native-phase-dense-result.json
new file mode 100644
index 0000000000..4f577b7e91
--- /dev/null
+++ b/research/literal-native-quadratic-phase/independent/native-phase-dense-result.json
@@ -0,0 +1,33 @@
+{
+  "accepted": 7,
+  "cases": 16,
+  "controls": [
+    "omit absolute tensor phase",
+    "omit quadratic cross terms",
+    "wrong original-owner commit"
+  ],
+  "full_fixture_coefficients": 11776,
+  "incomplete_prefixes": [
+    {
+      "complete_packet": false,
+      "f": 2,
+      "staged_coefficients": 300
+    },
+    {
+      "complete_packet": false,
+      "f": 3,
+      "staged_coefficients": 300
+    },
+    {
+      "complete_packet": false,
+      "f": 4,
+      "staged_coefficients": 300
+    }
+  ],
+  "limitations": "All-f theorem rests on proof; f=2,3,4 checks are deliberately incomplete packets, with verified staged prefixes.",
+  "locations": 8211,
+  "phase_sha256": "2be149aa1be504509bb480f85396f0be302082d638e81422f747daee5a3c1ce7",
+  "status": "PASS_INDEPENDENT_NATIVE_PHASE",
+  "tapes": 23,
+  "transitions": 6428358
+}
diff --git a/research/literal-native-quadratic-phase/independent/native-phase-source-bindings.json b/research/literal-native-quadratic-phase/independent/native-phase-source-bindings.json
new file mode 100644
index 0000000000..7f23bae82c
--- /dev/null
+++ b/research/literal-native-quadratic-phase/independent/native-phase-source-bindings.json
@@ -0,0 +1,71 @@
+{
+  "status": "PASS",
+  "package": "literal-native-quadratic-phase",
+  "method": "Exact Git commit:path bytes, SHA256, Git blob, and frozen local copy equality",
+  "rows": [
+    {
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "path": "research/shared-wire-split-composition/frozen/clifford_exact.py",
+      "bytes": 6824,
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+      "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2",
+      "pass": true
+    },
+    {
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "path": "research/shared-wire-split-composition/frozen/native_verify.py",
+      "bytes": 12323,
+      "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38",
+      "git_blob": "2c209dfa9cfade3bc20ac5fc9d876b433a147f8c",
+      "pass": true
+    },
+    {
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "path": "research/shared-wire-split-composition/data/final.json.gz",
+      "bytes": 350722,
+      "sha256": "2b94953dc14a122e3ef089633f05b58ec689ec5156feb7c784ed3e137dc01883",
+      "git_blob": "74563a7867d9d762b6588f65362369668841bb3e",
+      "pass": true
+    },
+    {
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "path": "docs/research/pr-intake-evidence/pr7000-application/check_native_catalogue.py",
+      "bytes": 5071,
+      "sha256": "6120e1ea1e07eeba757694d209437a38f121b0a377cc466889a30fa52c561472",
+      "git_blob": "98b35c89c8bd52637633aa78d506244bca8ec9e1",
+      "pass": true
+    },
+    {
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "path": "research/sequential-monomial-phase/phase.py",
+      "bytes": 5710,
+      "sha256": "256aa42f38e496bf91347078d8e2c731ee35b03fc0528f536a946f215f79fcd7",
+      "git_blob": "26126045289a3dceeb8d096c984d536613e5345f",
+      "pass": true
+    },
+    {
+      "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "path": "research/sequential-monomial-phase/PROOF.md",
+      "bytes": 8719,
+      "sha256": "aed63dc425a410627a290b2e0251b27bc1befce85614cefbb8a82f199a1d2764",
+      "git_blob": "42ffd4007c1416eb4a45e47d181194d3e87742bb",
+      "pass": true
+    },
+    {
+      "commit": "8b549ca6557ccbb3017b104b27ea35051f990416",
+      "path": "research/literal-layer-phase/phase.py",
+      "bytes": 11572,
+      "sha256": "9a360817b80e869398927f38a4f110de0b5a1f9620eeb23b19e166f112b83f23",
+      "git_blob": "c6e87695340a12d76d8f4fc8fb9ae069d3ec6a47",
+      "pass": true
+    },
+    {
+      "commit": "51ab0539059214fcfa044938c36a0b6017232be5",
+      "path": "research/literal-layer-egress/inputs/controller_reference.py",
+      "bytes": 5938,
+      "sha256": "e664a748e6b6797d4b60f60a7a7dd68b59a5608acf56b9c5d8b466f363755f73",
+      "git_blob": "3ab6d0361f35038ffcdf0cf81a157f7a17b40e72",
+      "pass": true
+    }
+  ]
+}
diff --git a/research/literal-native-quadratic-phase/independent/peer/RESULT.json b/research/literal-native-quadratic-phase/independent/peer/RESULT.json
new file mode 100644
index 0000000000..0324bd0810
--- /dev/null
+++ b/research/literal-native-quadratic-phase/independent/peer/RESULT.json
@@ -0,0 +1,262 @@
+{
+  "all_success_records": 9362,
+  "all_success_steps": 6786145,
+  "cardinality_rejections": [
+    {
+      "rejected": true,
+      "steps": 128670
+    },
+    {
+      "rejected": true,
+      "steps": 127858
+    }
+  ],
+  "complete_selected_dimensions": {
+    "f": [
+      1
+    ],
+    "m": 9
+  },
+  "functional_full_width_mutant_steps": 71445,
+  "identical_C2_helpers": [
+    "fresh",
+    "clear",
+    "unary_copy",
+    "header_step",
+    "gamma",
+    "trace_return",
+    "load_component",
+    "negate",
+    "stage_component"
+  ],
+  "locations": 8211,
+  "mutation_checks": [
+    {
+      "diagnostic": "absolute quadratic phase invariant",
+      "mutation": "missing tensor constant"
+    },
+    {
+      "diagnostic": "synchronized counter column",
+      "mutation": "unsynchronized counters"
+    },
+    {
+      "diagnostic": "exact carry/return move census",
+      "mutation": "functionally correct full-width return"
+    }
+  ],
+  "opcodes": {
+    "goto": 1084,
+    "halt": 1,
+    "move": 4559,
+    "read": 1960,
+    "write": 607
+  },
+  "proof_sha256": "55f0a575f618bf467d0e25a25802053455d701cee1f23fb8811a671a1bd59ff3",
+  "runs": [
+    {
+      "budget_base": 52244,
+      "counter_moves": 4100,
+      "dimension": 9,
+      "flips": 1022,
+      "history_after": 2150,
+      "history_before": 105,
+      "parameters": [
+        1,
+        1,
+        1,
+        1,
+        1
+      ],
+      "payload_bits": 1024,
+      "records": 512,
+      "steps": 153482
+    },
+    {
+      "budget_base": 127009,
+      "counter_moves": 8188,
+      "dimension": 9,
+      "flips": 2044,
+      "history_after": 49264,
+      "history_before": 105,
+      "parameters": [
+        1,
+        2,
+        2,
+        2,
+        3
+      ],
+      "payload_bits": 24576,
+      "records": 2048,
+      "steps": 1578734
+    },
+    {
+      "budget_base": 4133,
+      "counter_moves": 3112,
+      "dimension": 1,
+      "flips": 1022,
+      "history_after": 4220,
+      "history_before": 105,
+      "parameters": [
+        9,
+        1,
+        1,
+        1,
+        2
+      ],
+      "payload_bits": 2048,
+      "records": 512,
+      "steps": 141044
+    },
+    {
+      "budget_base": 4258,
+      "counter_moves": 812,
+      "dimension": 2,
+      "flips": 252,
+      "history_after": 6261,
+      "history_before": 105,
+      "parameters": [
+        3,
+        2,
+        2,
+        2,
+        3
+      ],
+      "payload_bits": 3072,
+      "records": 256,
+      "steps": 183560
+    },
+    {
+      "budget_base": 6178,
+      "counter_moves": 1276,
+      "dimension": 3,
+      "flips": 378,
+      "history_after": 6260,
+      "history_before": 105,
+      "parameters": [
+        2,
+        1,
+        3,
+        2,
+        4
+      ],
+      "payload_bits": 3072,
+      "records": 384,
+      "steps": 185849
+    },
+    {
+      "budget_base": 20510,
+      "counter_moves": 6444,
+      "dimension": 3,
+      "flips": 2044,
+      "history_after": 8305,
+      "history_before": 105,
+      "parameters": [
+        3,
+        1,
+        2,
+        1,
+        2
+      ],
+      "payload_bits": 4096,
+      "records": 1024,
+      "steps": 307271
+    },
+    {
+      "budget_base": 10527,
+      "counter_moves": 1716,
+      "dimension": 4,
+      "flips": 510,
+      "history_after": 8304,
+      "history_before": 105,
+      "parameters": [
+        2,
+        2,
+        1,
+        2,
+        2
+      ],
+      "payload_bits": 4096,
+      "records": 512,
+      "steps": 268512
+    },
+    {
+      "budget_base": 7904,
+      "counter_moves": 36,
+      "dimension": 1,
+      "flips": 6,
+      "history_after": 15809,
+      "history_before": 105,
+      "parameters": [
+        1,
+        31,
+        3,
+        3,
+        7
+      ],
+      "payload_bits": 7812,
+      "records": 18,
+      "steps": 408528
+    }
+  ],
+  "same_owner_reuse": [
+    {
+      "budget_base": 127009,
+      "counter_moves": 8188,
+      "dimension": 9,
+      "flips": 2044,
+      "history_after": 49264,
+      "history_before": 105,
+      "parameters": [
+        1,
+        2,
+        2,
+        2,
+        3
+      ],
+      "payload_bits": 24576,
+      "records": 2048,
+      "steps": 1578734
+    },
+    {
+      "budget_base": 52244,
+      "counter_moves": 4100,
+      "dimension": 9,
+      "flips": 1022,
+      "history_after": 49264,
+      "history_before": 49264,
+      "parameters": [
+        1,
+        1,
+        1,
+        1,
+        1
+      ],
+      "payload_bits": 1024,
+      "records": 512,
+      "steps": 153482
+    },
+    {
+      "budget_base": 181280,
+      "counter_moves": 12276,
+      "dimension": 9,
+      "flips": 3066,
+      "history_after": 55410,
+      "history_before": 49264,
+      "parameters": [
+        1,
+        3,
+        3,
+        1,
+        3
+      ],
+      "payload_bits": 27648,
+      "records": 1536,
+      "steps": 1826949
+    }
+  ],
+  "scope": "Zero-valued structural fixtures; no claim of complete selected f>1 enumeration.",
+  "source_sha256": "2be149aa1be504509bb480f85396f0be302082d638e81422f747daee5a3c1ce7",
+  "status": "PASS_B3_INDEPENDENT_STRUCTURE",
+  "table_sha256": "e10a0070af3331d23963fb97ebcd4bb7e81314d94a43d41ce00d4c84c5747638",
+  "tapes": 23
+}
diff --git a/research/literal-native-quadratic-phase/independent/peer/REVIEW.md b/research/literal-native-quadratic-phase/independent/peer/REVIEW.md
new file mode 100644
index 0000000000..38d6621391
--- /dev/null
+++ b/research/literal-native-quadratic-phase/independent/peer/REVIEW.md
@@ -0,0 +1,212 @@
+# Independent review of the literal selected native quadratic phase
+
+**Disposition: accept the scoped component theorem. No new kappa or removal
+of a complete selected foundational assumption.** This is an independent
+AI-assisted review for Jerry Chen / @CubeJerry, not human or proof-kernel
+verification. The author program was not edited. The source-specific local
+phase, supplied packed ABI, successful-call lifecycle, and displayed complete
+cost bounds are consistent with the frozen graph.
+
+## Exact objects and reproduction
+
+Canonical base: `e54523517d0f03e58352e647c403a5af3e525a57`.
+Package: `research/literal-native-quadratic-phase` in the B3 worktree.
+
+| Object | SHA256 |
+|---|---|
+| `phase.py` | `2be149aa1be504509bb480f85396f0be302082d638e81422f747daee5a3c1ce7` |
+| `PROOF.md` reviewed revision | `55f0a575f618bf467d0e25a25802053455d701cee1f23fb8811a671a1bd59ff3` |
+| Canonical generated graph JSON | `e10a0070af3331d23963fb97ebcd4bb7e81314d94a43d41ce00d4c84c5747638` |
+| Retained C2 helper source | `9a360817b80e869398927f38a4f110de0b5a1f9620eeb23b19e166f112b83f23` |
+
+The author made two proof-only precision edits before this final receipt:
+external simulation budgets do not belong to the program, and counter heads
+align at carry-bit reads after the intervening sequential moves finish.
+These clarifications are correct. They do not change the graph or theorem.
+
+Run this independently written interpreter with an absolute or relative
+path to the reviewed package:
+
+```sh
+python -I -W error -B check.py /path/to/research/literal-native-quadratic-phase
+python -I -O -W error -B check.py /path/to/research/literal-native-quadratic-phase
+```
+
+Both executions must emit the same JSON receipt, headed by
+`PASS_B3_INDEPENDENT_STRUCTURE`. The checker imports only the graph builder;
+it does not call the author's `Machine`, `Tape`, numerical oracle or
+verification program. Its own tape objects record every local instruction,
+head motion, visited extent, and write. It checks the complete instruction
+graph for fixed tape numbers, alphabet, destinations and unit moves.
+
+## Source phase and universal invariant
+
+The source selection is the actual retained native event 24472,
+`["MOVE",415,3,99,false]`, with frames 3 and 99. The retained source binding
+and complete monomial decomposition are distinct from this review's local
+ABI theorem. This package implements only the right factor's diagonal
+action before its affine map; it does not implement the whole event.
+
+The right recipe has `q0=1`, linear entries 3 on
+`J={0,1,2,3,5,6}`, entry 2 at coordinate 7, and the complete quadratic graph
+on J. Its diagonal entries are already included in the linear term. For
+`h=sum_{i in J} x_i`, the phase contribution is
+
+`1 + 3h + 2*binom(h,2) + 2*x7 = 1 + h*h + 2h + 2*x7 mod 4`.
+
+The middle two terms are 0 for even h and 3 for odd h. Thus the alternative
+source formula `1 - parity_J(x) + 2*x7` is correct. For f tensor columns
+the constant is **f*q0**, and not a single q0. The graph increments its
+constant control state once for every unary COLUMNS cell, so initialization
+has exactly this phase.
+
+For the independent execution checks I use the full integer quadratic form
+
+`f*q0 + sum_i alpha_i*popcount(a_i) + 2*sum_{i<j,Qij=1} popcount(a_i & a_j)`
+
+modulo 4. This is a separate expression from root's selected XOR/popcount
+oracle and from the author's incremental update. It is evaluated at every
+record entry while independently deriving the mixed-radix address from the
+record ordinal. The oracle also verifies the outer and spectator positions,
+coefficient reset, and exact DATA offset. Zero payloads deliberately remove
+numerical values from this structural check; root supplies the independent
+nonzero Gaussian tests.
+
+Universally, flipping bit x of coordinate i changes the linear phase by
+`(1-2x)*alpha_i`; a cross term changes by
+`2*(1-2x)*x_j = 2*x_j mod 4`. The graph reads exactly the fixed neighbors
+at their common column before changing its own bit. All active counter
+heads then move once before another carry-bit read. Returning all of them
+to their own guards preserves the relative column geometry. A full field
+wrap propagates the carry to the preceding coordinate. A full m-field wrap
+returns all counters to zero with phase f*q0, by the same bit-flip invariant.
+The absence of an explicit phase reset is therefore correct.
+
+This reasoning proves the phase invariant for every positive f and arbitrary
+joint arrays; it does not extrapolate that quantifier from finite tests.
+
+## Exact independently derived movement census
+
+Write `b=2^f` and `Q=b^m`. In one complete address cycle, coordinate i,
+indexed from 0 at the slow end, receives `b^(i+1)` nonempty carry segments.
+It flips `(2b-2)*b^i` bits. Hence
+
+`F = sum_i (2b-2)*b^i = 2Q-2`
+
+and the total number of segments is
+
+`C = sum_{j=1}^m b^j = b*(Q-1)/(b-1) < 2Q`.
+
+Every flip moves every active counter head forward once. A segment with t
+flips rewinds each active head by t+2 unit moves: t back across the body,
+one onto its guard, and one back to the origin. Thus each active counter
+has exactly `2F+2C` ripple/return moves in a complete cycle. Initialization,
+final body clearing and guards add exactly `4f+8`. For P repetitions the
+full accepted-call census is therefore
+
+**`4f + 8 + P*(2F+2C)` moves on every active counter**.
+
+Every unused fixed counter has exactly four guard-install/removal moves.
+The checker verifies these identities, the separate per-coordinate flip
+counts, and synchronization at each bit replacement. These are exact
+checks, not measurements fitted to an asymptotic curve.
+
+There is also a direct instruction bound. A flip at coordinate i costs
+`m+n_i+4 <= 2m+3` instructions, where n_i is its fixed number of neighbors.
+A nonwrapping segment's return costs `m*(3t+3)+1`. A wrapping segment's
+blank checks, return and carry cost at most `3mt+4m+3`. Consequently all
+segments cost at most
+
+`(5m+3)*F + (4m+3)*C < (18m+12)*Q`
+
+per full outer cycle. This is well below the proof's deliberately loose
+`128*(m+1)^2*P*Q` allowance. In particular, no full f-cell scan is charged
+per ordinary address.
+
+## Headers, arithmetic, complete cost and history
+
+An AST comparison confirms identity of all nine retained C2 helper
+functions: fresh labels, clearing, unary copying, header stepping, Gamma
+decoding, trace return, component load, signed negate, and component stage.
+Only the fixed global tape numbers differ. Their current loops were also
+inspected in this graph. The previous independent C2 review's direct loop
+counts remain applicable: a unary prefix update costs at most `47v+35`
+before its bounded wrapper, and the sum of previous binary prefixes is less
+than the final positive integer. Thus the five header decodes cost linear
+time in `E+f+r+P+S+w`. Unary expansion is paid.
+
+The complete two-component load/possible negations/stage cost is at most
+`62w+32 <= 94w` for w>=1, below the proof's 256w allowance. The signed
+minimum rejection test remains correct even at w=1: both old and new sign
+bits equal 1 only when modular negation tries to represent the forbidden
+positive endpoint. Required rotations negate exactly the components listed
+in the proof. No DATA write precedes successful processing of all fields
+and the exact final blank check. The subsequent commit targets the same
+DATA tape and exact cells that were read.
+
+Coefficient and spectator loops are linear in N, and the two returns plus
+commit cost `24V+13` under the retained helper convention. The explicit
+guard, initialization, parameter cleanup, header, odometer and Gaussian
+allowances sum far below
+
+`65536*(E+(m+1)f+r+P+S+w+(m+1)^2*P*Q+V+1)`.
+
+Because f>=1 and m is fixed, the advertised O(V) absorption on the supplied
+packed ABI is valid. This does not pay for obtaining that ABI from a
+native compiler invocation.
+
+Storage is bounded by active extents: OUTPUT and DTRACE use V each; HTRACE
+and LENGTH use at most E each; DOUBLE uses at most the largest parameter;
+the five unary parameters use their lengths; real/imaginary buffers use w
+each; and the m counters use f each. Each role has a fixed guard and end
+probe. These extents fit the stated additional bound
+
+`2V+4E+(m+4)f+4(r+P+S)+8w+256`.
+
+This bound counts new backed cells, not currently nonblank cells. The
+review's surfaces retain their historical extents after erasure. The three
+selected-graph calls on the same physical objects grow private historical
+extent `105 -> 49264 -> 49264 -> 55410`. The shorter middle call uses exactly
+the same number of instructions as a fresh call. All origins, opaque
+exterior values, guards and retired private bodies are checked. As the
+theorem states, rejection does not provide cleanup or later-call reuse.
+
+## Executed evidence and mutation detection
+
+The receipt contains 11 successful complete executions totaling **9,362
+record entries and 6,786,145 local transitions**. The selected m=9 graph has
+complete f=1 runs with differing r/P/S/width, including the reuse chain.
+Separate fixed recipes with m=1,2,3,4 exercise complete f=2,3,9 domains and
+nonzero constants, interactions and field carries. These smaller fixed
+graphs test the builder family; they are not advertised as complete
+selected m=9,f>1 enumeration. Root's separate checks cover selected higher-f
+prefixes and explicitly delimit that coverage.
+
+For every success the checker verifies HEADER has exactly E reads and 2E
+moves; DATA has exactly V+1 reads, V writes and 4V moves; all fields and
+repetitions have been consumed; public zero payload and exterior cells are
+unchanged; private bodies and guards are erased; and all 23 heads return.
+Extra and missing final DATA cells are separate rejection controls, each
+with zero public writes.
+
+Three independent source mutations are rejected:
+
+1. Removing the tensor constant update violates the absolute phase oracle.
+2. Advancing only the active counter head violates common-column geometry.
+3. Scanning to full counter width before every ordinary rewind preserves
+   all functional checks but fails the exact carry/return movement census.
+
+For the third mutant the checker first runs with only that census disabled
+and requires complete functional success. It then requires the movement
+diagnostic. This demonstrates a genuinely cost-sensitive negative control.
+
+## Retained limits
+
+Acceptance is limited to the stated diagonal component and supplied local
+geometry. Source binding to one native event does not create a descriptor
+emitter, affine route, recursive child interface, scalar/grid bridge or
+global allocation theorem. Earlier incremental phase mathematics remains
+credited to #5375; the helper/table/commit predecessors remain credited.
+All five selected foundational assumptions and the complete selected kappa
+remain unchanged. The exact packed-descriptor and affine/native joins in
+the author's open questions are substantive remaining obligations.
diff --git a/research/literal-native-quadratic-phase/independent/peer/check.py b/research/literal-native-quadratic-phase/independent/peer/check.py
new file mode 100644
index 0000000000..d3f0c44bc6
--- /dev/null
+++ b/research/literal-native-quadratic-phase/independent/peer/check.py
@@ -0,0 +1,322 @@
+"""Independent local-graph geometry, phase-state, cost and history review.
+
+Uses the supplied graph builder, never its Machine/Tape, oracle or verifier.
+Zero payloads isolate structural obligations; root separately checks arithmetic.
+Prepared for Jerry Chen / CubeJerry with OpenAI assistance. No new kappa.
+"""
+from collections import Counter
+import argparse
+import ast
+import hashlib
+import importlib.util
+import json
+from pathlib import Path
+import sys
+import types
+
+EXPECTED = '2be149aa1be504509bb480f85396f0be302082d638e81422f747daee5a3c1ce7'
+EXPECTED_TABLE = 'e10a0070af3331d23963fb97ebcd4bb7e81314d94a43d41ce00d4c84c5747638'
+EXPECTED_PROOF = '55f0a575f618bf467d0e25a25802053455d701cee1f23fb8811a671a1bd59ff3'
+BLANK = '_'
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def gamma(n):
+    word = bin(n)[2:]
+    return '0' * (len(word) - 1) + word
+
+
+def load_subject(package):
+    need(hashlib.sha256((package / 'phase.py').read_bytes()).hexdigest() == EXPECTED, 'source pin')
+    need(hashlib.sha256((package / 'PROOF.md').read_bytes()).hexdigest() == EXPECTED_PROOF, 'proof pin')
+    spec = importlib.util.spec_from_file_location('b3_peer_subject', package / 'phase.py')
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[spec.name] = module
+    spec.loader.exec_module(module)
+    return module
+
+
+class Surface:
+    def __init__(self, origin, index):
+        self.origin = origin
+        self.head = origin
+        self.low = origin - (1 if index >= 2 else 0)
+        self.high = origin + (index % 7 if index >= 2 else 0)
+        self.cells = {origin - 2: '|', origin + 1000000: '1'}
+
+    def get(self):
+        return self.cells.get(self.head, BLANK)
+
+    def move(self, amount):
+        self.head += amount
+        self.low = min(self.low, self.head)
+        self.high = max(self.high, self.head)
+
+
+def fresh_surfaces():
+    return [Surface(i * 19 - 113, i) for i in range(23)]
+
+
+def set_public(tape, word, header=False):
+    # Separately paid caller acquisition; preserve private histories and objects.
+    tape.cells = {tape.origin - 2: '|', tape.origin + 1000000: '1'}
+    tape.cells.update({tape.origin + i: bit for i, bit in enumerate(word)})
+    if header:
+        tape.cells[tape.origin + len(word)] = '|'
+
+
+def table_hash(code):
+    return hashlib.sha256(json.dumps(code, sort_keys=True, separators=(',', ':')).encode()).hexdigest()
+
+
+def validate_graph(code):
+    operations = Counter()
+    for instruction in code:
+        opcode, *args = instruction
+        need(opcode in ('read', 'write', 'move', 'goto', 'halt'), 'nonliteral opcode')
+        operations[opcode] += 1
+        if opcode in ('read', 'write', 'move'):
+            need(type(args[0]) is int and 0 <= args[0] < 23, 'unfixed tape selector')
+        if opcode == 'read':
+            need(set(args[1]) <= {'_', '0', '1', '|'}, 'unfixed read alphabet')
+            need(all(type(v) is int and 0 <= v < len(code) for v in args[1].values()), 'unfixed branch')
+        if opcode == 'write':
+            need(args[1] in ('_', '0', '1', '|'), 'unfixed write alphabet')
+        if opcode == 'move':
+            need(args[1] in (-1, 1), 'nonunit move')
+        if opcode == 'goto':
+            need(type(args[0]) is int and 0 <= args[0] < len(code), 'unfixed jump')
+    return dict(sorted(operations.items()))
+
+
+def execute(code, recipe, parameters, *, surfaces=None, extra=False, missing=False,
+            expect_reject=False, check_counter_bill=True):
+    linear, rows, constant = recipe
+    m = len(linear)
+    f, r, P, S, w = parameters
+    radix, Q = 2 ** f, 2 ** (m * f)
+    V = 2 * P * Q * S * r * w
+    header = ''.join(map(gamma, parameters))
+    packet = '0' * V + ('0' if extra else '')
+    if missing:
+        packet = packet[:-1]
+    tapes = fresh_surfaces() if surfaces is None else surfaces
+    need(all(t.head == t.origin for t in tapes), 'entry head')
+    private_before = [dict(t.cells) for t in tapes[2:]]
+    history_before = sum(t.high - t.low + 1 for t in tapes[2:])
+    set_public(tapes[0], header, True)
+    set_public(tapes[1], packet)
+    public_before = [dict(t.cells) for t in tapes[:2]]
+    records = [i for i, ins in enumerate(code)
+               if ins[0] == 'read' and ins[1] == 9 and set(ins[2]) == {'_', '1'}
+               and tuple(code[i + 1][:2]) == ('read', 7) and ins[2]['1'] == i + 1]
+    need(len(records) == 4, 'four phase-entry locations')
+    phase_at = dict(zip(records, range(4)))
+    counters = {'read': Counter(), 'write': Counter(), 'move': Counter()}
+    flips = Counter()
+    record_ordinal, pc, steps, rejected = 0, 0, 0, False
+    for _ in range(5000000):
+        if pc in phase_at and tapes[9].get() == '1':
+            pos = record_ordinal
+            values = []
+            for i in range(m):
+                tape = tapes[14 + i]
+                bits = [tape.cells.get(tape.origin + j, BLANK) for j in range(f)]
+                need(all(bit in ('0', '1') for bit in bits), 'counter shape')
+                need(tape.head == tape.origin, 'record counter origin')
+                values.append(sum((bit == '1') << j for j, bit in enumerate(bits)))
+            expected = [((pos // S) // radix ** (m - 1 - i)) % radix for i in range(m)]
+            need(values == expected, 'full-domain address order')
+            phase = constant * f + sum(a.bit_count() * alpha for a, alpha in zip(values, linear))
+            phase += 2 * sum((values[i] & values[j]).bit_count()
+                             for i in range(m) for j in range(i + 1, m) if rows[i] >> j & 1)
+            need(phase_at[pc] == phase % 4, 'absolute quadratic phase invariant')
+            need(tapes[8].head - tapes[8].origin == pos // (Q * S), 'P repetition')
+            need(tapes[9].head - tapes[9].origin == pos % S, 'S repetition')
+            need(tapes[7].head == tapes[7].origin, 'r repeat reset')
+            need(tapes[1].head - tapes[1].origin == 2 * w * r * pos, 'r complete coefficients')
+            record_ordinal += 1
+        opcode, *args = code[pc]
+        pc += 1
+        steps += 1
+        if opcode == 'halt':
+            break
+        if opcode == 'goto':
+            pc = args[0]
+            continue
+        tape = tapes[args[0]]
+        counters[opcode][args[0]] += 1
+        if opcode == 'read':
+            symbol = tape.get()
+            if symbol not in args[1]:
+                rejected = True
+                break
+            pc = args[1][symbol]
+        elif opcode == 'write':
+            old = tape.get()
+            if 14 <= args[0] < 14 + m and old in ('0', '1') and args[1] in ('0', '1'):
+                need(old != args[1], 'counter flip')
+                need(len({t.head - t.origin for t in tapes[14:14 + m]}) == 1,
+                     'synchronized counter column')
+                flips[args[0]] += 1
+            if args[1] == BLANK:
+                tape.cells.pop(tape.head, None)
+            else:
+                tape.cells[tape.head] = args[1]
+        else:
+            tape.move(args[1])
+    else:
+        raise ValueError('review execution cap')
+    need(rejected == expect_reject, 'rejection outcome')
+    need(counters['write'][0] == 0, 'header was written')
+    if rejected:
+        need(counters['write'][1] == 0 and [dict(t.cells) for t in tapes[:2]] == public_before,
+             'rejection modified public input')
+        return {'rejected': True, 'steps': steps}
+    need(record_ordinal == P * Q * S, 'exact full record cardinality')
+    need(counters['read'][1] == V + 1 and counters['write'][1] == V, 'payload read/write cardinality')
+    need(counters['read'][0] == len(header), 'header source-read extent')
+    need(counters['move'][1] == 4 * V and counters['move'][0] == 2 * len(header), 'paid public returns')
+    need([dict(t.cells) for t in tapes[:2]] == public_before, 'zero packet/public exterior changed')
+    need([dict(t.cells) for t in tapes[2:]] == private_before, 'private body or guard not retired')
+    need(all(t.head == t.origin for t in tapes), 'head not restored')
+    F = 2 * Q - 2
+    segments = sum(radix ** j for j in range(1, m + 1))
+    need(sum(flips.values()) == P * F, 'geometric ripple flip census')
+    need(all(flips[14 + i] == P * (2 * radix - 2) * radix ** i for i in range(m)),
+         'per-coordinate flip census')
+    if check_counter_bill:
+        expected_moves = 4 * f + 8 + P * (2 * F + 2 * segments)
+        need(all(counters['move'][14 + i] == expected_moves for i in range(m)),
+             'exact carry/return move census')
+        need(all(counters['move'][14 + i] == 4 for i in range(m, 9)), 'unused counter motion')
+    budget = len(header) + (m + 1) * f + r + P + S + w + (m + 1) ** 2 * P * Q + V + 1
+    need(steps <= 65536 * budget, 'complete literal time envelope')
+    history_after = sum(t.high - t.low + 1 for t in tapes[2:])
+    history_budget = 2 * V + 4 * len(header) + (m + 4) * f + 4 * (r + P + S) + 8 * w + 256
+    need(history_before <= history_after <= history_before + history_budget, 'retained history envelope')
+    return {'parameters': parameters, 'dimension': m, 'steps': steps, 'records': record_ordinal,
+            'flips': sum(flips.values()), 'counter_moves': counters['move'][14],
+            'history_before': history_before, 'history_after': history_after,
+            'budget_base': budget, 'payload_bits': V}
+
+
+def changed_graph(package, label, old, new, recipe):
+    source = (package / 'phase.py').read_text()
+    need(source.count(old) == 1, 'mutation anchor ' + label)
+    module = types.ModuleType('b3_peer_mutant_' + label)
+    module.__file__ = str(package / 'phase.py')
+    sys.modules[module.__name__] = module
+    exec(compile(source.replace(old, new), module.__file__, 'exec'), module.__dict__)
+    return module.compile_phase(*recipe)[0]
+
+
+def diagnostic(label, action, expected):
+    try:
+        action()
+    except ValueError as error:
+        need(expected in str(error), 'unexpected failure for ' + label + ': ' + str(error))
+        return {'mutation': label, 'diagnostic': str(error)}
+    raise ValueError('mutation escaped ' + label)
+
+
+def shared_helpers(package):
+    old = (package / 'reference' / 'outer-phase.py').read_bytes()
+    need(hashlib.sha256(old).hexdigest() == '9a360817b80e869398927f38a4f110de0b5a1f9620eeb23b19e166f112b83f23',
+         'C2 source pin')
+    before = {node.name: ast.dump(node, include_attributes=False) for node in ast.parse(old).body
+              if isinstance(node, ast.FunctionDef)}
+    after = {node.name: ast.dump(node, include_attributes=False)
+             for node in ast.parse((package / 'phase.py').read_text()).body
+             if isinstance(node, ast.FunctionDef)}
+    names = ('fresh', 'clear', 'unary_copy', 'header_step', 'gamma', 'trace_return',
+             'load_component', 'negate', 'stage_component')
+    need(all(before[name] == after[name] for name in names), 'retained helper AST identity')
+    return names
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument('package', type=Path)
+    args = parser.parse_args()
+    package = args.package.resolve()
+    subject = load_subject(package)
+    selected = (subject.SELECTED_LINEAR, subject.SELECTED_ROWS, subject.SELECTED_CONSTANT)
+    code, alphabet = subject.compile_phase()
+    need(len(code) == 8211 and table_hash(code) == EXPECTED_TABLE, 'frozen literal graph')
+    opcodes = validate_graph(code)
+    helpers = shared_helpers(package)
+    one = ((3,), (1,), 1)
+    two = ((1, 3), (3, 3), 2)
+    three = ((3, 2, 1), (7, 1, 5), 3)
+    four = ((0, 1, 2, 3), (14, 15, 11, 15), 1)
+    fixtures = [
+        (selected, (1, 1, 1, 1, 1)),
+        (selected, (1, 2, 2, 2, 3)),
+        (one, (9, 1, 1, 1, 2)),
+        (two, (3, 2, 2, 2, 3)),
+        (three, (2, 1, 3, 2, 4)),
+        (three, (3, 1, 2, 1, 2)),
+        (four, (2, 2, 1, 2, 2)),
+        (one, (1, 31, 3, 3, 7)),
+    ]
+    runs = []
+    for recipe, parameters in fixtures:
+        graph = subject.compile_phase(*recipe)[0]
+        validate_graph(graph)
+        runs.append(execute(graph, recipe, parameters))
+    tapes = fresh_surfaces()
+    identities = tuple(map(id, tapes))
+    reuse = []
+    for parameters in ((1, 2, 2, 2, 3), (1, 1, 1, 1, 1), (1, 3, 3, 1, 3)):
+        reuse.append(execute(code, selected, parameters, surfaces=tapes))
+        need(tuple(map(id, tapes)) == identities, 'physical role identity changed')
+    need(reuse[1]['steps'] == runs[0]['steps'], 'small call rescanned historical capacity')
+    rejection = [execute(code, selected, (1, 1, 1, 1, 1), extra=True, expect_reject=True),
+                 execute(code, selected, (1, 1, 1, 1, 1), missing=True, expect_reject=True)]
+    recipe, parameters = two, (3, 1, 2, 2, 2)
+    bad_constant = changed_graph(package, 'missing_tensor_constant', '(mode+constant) % 4',
+                                 '(mode+0) % 4', recipe)
+    mutants = [diagnostic('missing tensor constant', lambda: execute(bad_constant, recipe, parameters),
+                          'absolute quadratic phase invariant')]
+    bad_geometry = changed_graph(package, 'unsynchronized_counters',
+        "for tape in counters:\n                        a.op('move', tape, 1)",
+        "for tape in (counters[i],):\n                        a.op('move', tape, 1)", recipe)
+    mutants.append(diagnostic('unsynchronized counters', lambda: execute(bad_geometry, recipe, parameters),
+                              'synchronized counter column'))
+    long_scan = """a.mark('advanced'+str(mode))
+        a.mark('peer_scan'+str(mode))
+        a.op('read', counters[0], {'0': 'peer_step'+str(mode), ONE: 'peer_step'+str(mode),
+                                  BLANK: 'peer_done'+str(mode)})
+        a.mark('peer_step'+str(mode))
+        for tape in counters:
+            a.op('move', tape, 1)
+        a.op('goto', 'peer_scan'+str(mode))
+        a.mark('peer_done'+str(mode))"""
+    bad_cost = changed_graph(package, 'full_width_return', "a.mark('advanced'+str(mode))", long_scan, recipe)
+    # Establish that the third mutation passes functionality and only loses the
+    # independently derived exact amortized movement bill.
+    functional_mutant = execute(bad_cost, recipe, parameters, check_counter_bill=False)
+    mutants.append(diagnostic('functionally correct full-width return',
+        lambda: execute(bad_cost, recipe, parameters), 'exact carry/return move census'))
+    result = {
+        'status': 'PASS_B3_INDEPENDENT_STRUCTURE', 'source_sha256': EXPECTED,
+        'proof_sha256': EXPECTED_PROOF, 'table_sha256': EXPECTED_TABLE,
+        'locations': len(code), 'tapes': 23, 'opcodes': opcodes,
+        'identical_C2_helpers': list(helpers), 'runs': runs, 'same_owner_reuse': reuse,
+        'cardinality_rejections': rejection, 'mutation_checks': mutants,
+        'functional_full_width_mutant_steps': functional_mutant['steps'],
+        'complete_selected_dimensions': {'m': 9, 'f': [1]},
+        'all_success_steps': sum(run['steps'] for run in runs + reuse),
+        'all_success_records': sum(run['records'] for run in runs + reuse),
+        'scope': 'Zero-valued structural fixtures; no claim of complete selected f>1 enumeration.'
+    }
+    print(json.dumps(result, sort_keys=True, indent=2))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/literal-native-quadratic-phase/inputs/controller_reference.py b/research/literal-native-quadratic-phase/inputs/controller_reference.py
new file mode 100644
index 0000000000..3ab6d0361f
--- /dev/null
+++ b/research/literal-native-quadratic-phase/inputs/controller_reference.py
@@ -0,0 +1,112 @@
+"""Finite three-tape online rank-recursion control; Apache-2.0.
+Only PC and local symbols affect execution. Metrics/event logs are observers.
+"""
+from dataclasses import dataclass, field
+BLANK, ONE, MARK = '_', '1', '|'
+WIDTH, CHILD, STACK = range(3)
+
+class Builder:
+    def __init__(self): self.code, self.labels, self.serial = [], {}, 0
+    def mark(self,n):
+        if n in self.labels: raise ValueError('duplicate label')
+        self.labels[n]=len(self.code)
+    def op(self,*x): self.code.append(x)
+    def rewind(self,t,symbols):
+        self.serial+=1; loop=f'rw{self.serial}'; done=loop+'done'
+        self.op('move',t,-1); self.mark(loop)
+        self.op('read',t,{s:loop+'step' for s in symbols}|{MARK:done})
+        self.mark(loop+'step'); self.op('move',t,-1); self.op('goto',loop)
+        self.mark(done); self.op('move',t,1)
+    def finish(self,alphabet):
+        result=[]
+        for op,*a in self.code:
+            if op=='goto': a[0]=self.labels[a[0]]
+            elif op=='read': a[1]={s:self.labels[n] for s,n in a[1].items()}
+            result.append((op,*a))
+        return tuple(result),tuple(alphabet)
+
+def compile_controller(m,ranks):
+    """Parameters are compile-time constants, absent from run-time state."""
+    if type(m) is not int or m<2 or not ranks: raise ValueError('bad fixed profile')
+    if any(type(r) is not int or not 1<=r<m for r in ranks): raise ValueError('rank range')
+    ranks=tuple(ranks); tags=tuple(f't{i}' for i in range(len(ranks)))
+    a=Builder(); a.mark('enter'); a.op('read',WIDTH,{ONE:'probe0'})
+    for k in range(m):
+        a.mark(f'probe{k}'); a.op('read',WIDTH,{ONE:f'advance{k}',BLANK:f'leaf{k}'})
+        a.mark(f'advance{k}'); a.op('move',WIDTH,1)
+        a.op('goto',f'probe{k+1}' if k+1<m else 'internal_rewind')
+        a.mark(f'leaf{k}'); a.rewind(WIDTH,(ONE,)); a.op('emit','L'); a.op('goto','complete')
+    a.mark('internal_rewind'); a.rewind(WIDTH,(ONE,)); a.op('emit','E'); a.op('goto','child0')
+    for i,r in enumerate(ranks):
+        a.mark(f'child{i}'); a.op('goto',f'build{i}_0')
+        for rem in range(m):
+            a.mark(f'build{i}_{rem}')
+            a.op('read',WIDTH,{ONE:f'build{i}_{rem}bit',BLANK:f'built{i}'})
+            a.mark(f'build{i}_{rem}bit'); quotient,nxt=divmod(rem+r,m)
+            if quotient: a.op('write',CHILD,ONE); a.op('move',CHILD,1)
+            a.op('move',WIDTH,1); a.op('goto',f'build{i}_{nxt}')
+        a.mark(f'built{i}'); a.rewind(WIDTH,(ONE,)); a.rewind(CHILD,(ONE,))
+        a.mark(f'push{i}'); a.op('read',WIDTH,{ONE:f'push{i}bit',BLANK:f'pushed{i}'})
+        a.mark(f'push{i}bit'); a.op('write',STACK,ONE); a.op('write',WIDTH,BLANK)
+        a.op('move',WIDTH,1); a.op('move',STACK,1); a.op('goto',f'push{i}')
+        a.mark(f'pushed{i}'); a.op('write',STACK,tags[i]); a.op('move',STACK,1); a.rewind(WIDTH,(BLANK,))
+        a.mark(f'transfer{i}'); a.op('read',CHILD,{ONE:f'transfer{i}bit',BLANK:f'transferred{i}'})
+        a.mark(f'transfer{i}bit'); a.op('write',WIDTH,ONE); a.op('write',CHILD,BLANK)
+        a.op('move',WIDTH,1); a.op('move',CHILD,1); a.op('goto',f'transfer{i}')
+        a.mark(f'transferred{i}'); a.rewind(WIDTH,(ONE,)); a.rewind(CHILD,(BLANK,))
+        a.op('emit',f'D{i}'); a.op('goto','enter')
+    a.mark('complete'); a.op('emit','X'); a.op('move',STACK,-1)
+    a.op('read',STACK,{MARK:'root_done',**{t:f'return{i}' for i,t in enumerate(tags)}})
+    a.mark('root_done'); a.op('move',STACK,1); a.op('halt')
+    for i in range(len(ranks)):
+        a.mark(f'return{i}'); a.op('write',STACK,BLANK); a.op('move',STACK,-1)
+        a.mark(f'erase{i}'); a.op('read',WIDTH,{ONE:f'erase{i}bit',BLANK:f'erased{i}'})
+        a.mark(f'erase{i}bit'); a.op('write',WIDTH,BLANK); a.op('move',WIDTH,1); a.op('goto',f'erase{i}')
+        a.mark(f'erased{i}'); a.rewind(WIDTH,(BLANK,))
+        a.mark(f'pop{i}'); a.op('read',STACK,{ONE:f'pop{i}bit',MARK:f'popped{i}',**{t:f'popped{i}' for t in tags}})
+        a.mark(f'pop{i}bit'); a.op('write',WIDTH,ONE); a.op('write',STACK,BLANK)
+        a.op('move',WIDTH,1); a.op('move',STACK,-1); a.op('goto',f'pop{i}')
+        a.mark(f'popped{i}'); a.op('move',STACK,1); a.rewind(WIDTH,(ONE,)); a.op('emit',f'U{i}')
+        a.op('goto',f'child{i+1}' if i+1<len(ranks) else 'complete')
+    return a.finish((BLANK,'0',ONE,MARK,*tags))
+
+@dataclass
+class Tape:
+    cells:dict
+    head:int
+    low:int=field(init=False)
+    high:int=field(init=False)
+    def __post_init__(self): self.low=self.high=self.head
+    def read(self): return self.cells.get(self.head,BLANK)
+    def write(self,s):
+        if s==BLANK: self.cells.pop(self.head,None)
+        else: self.cells[self.head]=s
+    def move(self,d):
+        self.head+=d; self.low=min(self.low,self.head); self.high=max(self.high,self.head)
+
+class Machine:
+    def __init__(self,code,tapes):
+        self.code,self.tapes=code,tapes; self.pc=self.steps=0; self.events=[]; self.halted=False
+    def step(self):
+        op,*a=self.code[self.pc]; self.steps+=1; self.pc+=1
+        if op=='read':
+            s=self.tapes[a[0]].read()
+            if s not in a[1]: raise ValueError('invalid scanned symbol')
+            self.pc=a[1][s]
+        elif op=='write': self.tapes[a[0]].write(a[1])
+        elif op=='move': self.tapes[a[0]].move(a[1])
+        elif op=='goto': self.pc=a[0]
+        elif op=='emit': self.events.append(a[0])
+        elif op=='halt': self.halted=True
+        else: raise ValueError('invalid opcode')
+    def run(self,budget=10_000_000):
+        while not self.halted:
+            if self.steps>=budget: raise ValueError('finite verifier budget')
+            self.step()
+        return self
+
+def initial_tapes(width,origins=(0,0,0),ancestor=('0','1','t0','_','|','0')):
+    if width<1: raise ValueError('positive width')
+    u,c,p=origins
+    return [Tape({u-1:MARK,**{u+i:ONE for i in range(width)}},u),Tape({c-1:MARK},c),
+            Tape({p-1:MARK,**{p-2-i:s for i,s in enumerate(ancestor) if s!=BLANK}},p)]
diff --git a/research/literal-native-quadratic-phase/phase.py b/research/literal-native-quadratic-phase/phase.py
new file mode 100644
index 0000000000..d489781458
--- /dev/null
+++ b/research/literal-native-quadratic-phase/phase.py
@@ -0,0 +1,335 @@
+"""Literal selected native quadratic phase, with complete owner commit.
+
+Prepared for Jerry Chen / CubeJerry with substantial OpenAI assistance.
+Apache-2.0. The multi-counter phase algorithm is credited to #5375. Header,
+signed staging and return helpers are retained from the frozen C2 phase.
+Only fixed-table read/write/unit-move/goto/halt instructions execute.
+"""
+from pathlib import Path
+import hashlib
+import sys
+
+HERE = Path(__file__).resolve().parent
+MACHINE = HERE / 'inputs' / 'controller_reference.py'
+EXPECTED_MACHINE = 'e664a748e6b6797d4b60f60a7a7dd68b59a5608acf56b9c5d8b466f363755f73'
+if hashlib.sha256(MACHINE.read_bytes()).hexdigest() != EXPECTED_MACHINE:
+    raise ValueError('frozen literal machine source drift')
+sys.path.insert(0, str(MACHINE.parent))
+from controller_reference import Builder, Machine, Tape, BLANK, ONE, MARK
+
+(HEADER, DATA, OUTPUT, HTRACE, LENGTH, DOUBLE, COLUMNS, COEFF, OUTER,
+ SPECTATOR, WIDTH, REAL, IMAG, DTRACE) = range(14)
+COUNTERS = tuple(range(14, 23))
+PRIVATE = tuple(range(2, 23))
+BITS = ('0', ONE)
+SELECTED_LINEAR = (3, 3, 3, 3, 0, 3, 3, 2, 0)
+SELECTED_ROWS = (111, 111, 111, 111, 0, 111, 111, 0, 0)
+SELECTED_CONSTANT = 1
+
+
+def fresh(a, stem):
+    a.serial += 1
+    return stem + str(a.serial)
+
+
+def clear(a, tape, symbols=(ONE,)):
+    k = fresh(a, 'clear')
+    a.mark(k)
+    a.op('read', tape, {s: k+'bit' for s in symbols} | {BLANK: k+'done'})
+    a.mark(k+'bit')
+    a.op('write', tape, BLANK)
+    a.op('move', tape, 1)
+    a.op('goto', k)
+    a.mark(k+'done')
+    a.rewind(tape, (BLANK,))
+
+
+def unary_copy(a, source, target, *, repeat=1, extra=0, erase=False):
+    k = fresh(a, 'copy')
+    a.mark(k)
+    a.op('read', source, {ONE: k+'bit', BLANK: k+'done'})
+    a.mark(k+'bit')
+    for _ in range(repeat):
+        a.op('write', target, ONE)
+        a.op('move', target, 1)
+    if erase:
+        a.op('write', source, BLANK)
+    a.op('move', source, 1)
+    a.op('goto', k)
+    a.mark(k+'done')
+    for _ in range(extra):
+        a.op('write', target, ONE)
+        a.op('move', target, 1)
+    a.rewind(source, (BLANK,) if erase else (ONE,))
+    a.rewind(target, (ONE,))
+
+
+def header_step(a):
+    a.op('write', HTRACE, ONE)
+    a.op('move', HTRACE, 1)
+    a.op('move', HEADER, 1)
+
+
+def gamma(a, target):
+    """Actual positive Gamma code to unary, using only literal operations."""
+    k = fresh(a, 'gamma')
+    a.mark(k+'prefix')
+    a.op('read', HEADER, {'0': k+'zero', ONE: k+'one'})
+    a.mark(k+'zero')
+    a.op('write', LENGTH, ONE)
+    a.op('move', LENGTH, 1)
+    header_step(a)
+    a.op('goto', k+'prefix')
+    a.mark(k+'one')
+    header_step(a)
+    a.op('write', target, ONE)
+    a.op('move', LENGTH, -1)
+    a.mark(k+'tail')
+    a.op('read', LENGTH, {ONE: k+'tailbit', MARK: k+'done'})
+    a.mark(k+'tailbit')
+    a.op('write', LENGTH, BLANK)
+    a.op('read', HEADER, {str(b): k+'digit'+str(b) for b in (0, 1)})
+    for b in (0, 1):
+        a.mark(k+'digit'+str(b))
+        unary_copy(a, target, DOUBLE, repeat=2, extra=b)
+        clear(a, target)
+        unary_copy(a, DOUBLE, target, erase=True)
+        header_step(a)
+        a.op('move', LENGTH, -1)
+        a.op('goto', k+'tail')
+    a.mark(k+'done')
+    a.op('move', LENGTH, 1)
+
+
+def trace_return(a, tape, trace):
+    k = fresh(a, 'return')
+    a.op('move', trace, -1)
+    a.mark(k)
+    a.op('read', trace, {ONE: k+'bit', MARK: k+'done'})
+    a.mark(k+'bit')
+    a.op('write', trace, BLANK)
+    a.op('move', trace, -1)
+    a.op('move', tape, -1)
+    a.op('goto', k)
+    a.mark(k+'done')
+    a.op('move', trace, 1)
+
+
+def load_component(a, target):
+    k = fresh(a, 'load')
+    a.mark(k)
+    a.op('read', WIDTH, {ONE: k+'bit', BLANK: k+'done'})
+    a.mark(k+'bit')
+    a.op('read', DATA, {str(b): k+str(b) for b in (0, 1)})
+    for b in (0, 1):
+        a.mark(k+str(b))
+        a.op('write', target, str(b))
+        a.op('move', target, 1)
+        a.op('move', DATA, 1)
+        a.op('write', DTRACE, ONE)
+        a.op('move', DTRACE, 1)
+        a.op('move', WIDTH, 1)
+        a.op('goto', k)
+    a.mark(k+'done')
+    a.rewind(WIDTH, (ONE,))
+    a.rewind(target, BITS)
+
+
+def negate(a, tape):
+    """Exact w-bit integer negation; reject the unrepresentable minimum."""
+    k = fresh(a, 'neg')
+    a.mark(k+'seek')
+    a.op('read', tape, {'0': k+'step', ONE: k+'step', BLANK: k+'last'})
+    a.mark(k+'step')
+    a.op('move', tape, 1)
+    a.op('goto', k+'seek')
+    a.mark(k+'last')
+    a.op('move', tape, -1)
+    a.op('goto', k+'c1x0y0')
+    for carry in (0, 1):
+        for old_sign in (0, 1):
+            for new_sign in (0, 1):
+                name = k+f'c{carry}x{old_sign}y{new_sign}'
+                a.mark(name)
+                a.op('read', tape, {str(b): name+'b'+str(b) for b in (0, 1)} |
+                     {MARK: 'reject' if old_sign and new_sign else k+'done'})
+                for bit in (0, 1):
+                    total = 1-bit+carry
+                    digit, nxt = total % 2, total//2
+                    a.mark(name+'b'+str(bit))
+                    a.op('write', tape, str(digit))
+                    a.op('move', tape, -1)
+                    a.op('goto', k+f'c{nxt}x{bit}y{digit}')
+    a.mark(k+'done')
+    a.op('move', tape, 1)
+
+
+def stage_component(a, tape):
+    k = fresh(a, 'stage')
+    a.mark(k)
+    a.op('read', tape, {str(b): k+str(b) for b in (0, 1)} | {BLANK: k+'done'})
+    for b in (0, 1):
+        a.mark(k+str(b))
+        a.op('write', OUTPUT, str(b))
+        a.op('move', OUTPUT, 1)
+        a.op('write', tape, BLANK)
+        a.op('move', tape, 1)
+        a.op('goto', k)
+    a.mark(k+'done')
+    a.rewind(tape, (BLANK,))
+
+
+def initialize(a, counters, constant):
+    """Generate f zero bits on each counter and retain constant*f mod 4."""
+    a.op('goto', 'initialize0')
+    for mode in range(4):
+        name = 'initialize'+str(mode)
+        a.mark(name)
+        a.op('read', COLUMNS, {ONE: name+'bit', BLANK: name+'done'})
+        a.mark(name+'bit')
+        for tape in counters:
+            a.op('write', tape, '0')
+            a.op('move', tape, 1)
+        a.op('move', COLUMNS, 1)
+        a.op('goto', 'initialize'+str((mode+constant) % 4))
+        a.mark(name+'done')
+        a.rewind(COLUMNS, (ONE,))
+        for tape in counters:
+            a.rewind(tape, BITS)
+        a.op('goto', 'record'+str(mode))
+
+
+def advance(a, counters, linear, rows):
+    """Fixed finite lowering of the credited synchronized counter updater."""
+    m = len(counters)
+    for i in range(m):
+        neighbors = tuple(j for j in range(m) if j != i and
+                          (rows[min(i, j)] >> max(i, j)) & 1)
+        for mode in range(4):
+            name = f'advance{i}q{mode}'
+            a.mark(name)
+            a.op('read', counters[i], {str(b): name+'b'+str(b) for b in (0, 1)} |
+                 {BLANK: name+'overflow'})
+            for bit in (0, 1):
+                stem = name+'b'+str(bit)
+                a.mark(stem)
+                nxt = (mode+(1-2*bit)*linear[i]) % 4
+                a.op('goto', stem+f'n0q{nxt}')
+                for k, j in enumerate(neighbors):
+                    for value in range(4):
+                        label = stem+f'n{k}q{value}'
+                        a.mark(label)
+                        a.op('read', counters[j], {
+                            str(b): stem+f'n{k+1}q{(value+2*b) % 4}'
+                            for b in (0, 1)})
+                for value in range(4):
+                    a.mark(stem+f'n{len(neighbors)}q{value}')
+                    a.op('write', counters[i], str(1-bit))
+                    for tape in counters:
+                        a.op('move', tape, 1)
+                    a.op('goto', f'advance{i}q{value}' if bit else
+                         'advanced'+str(value))
+            a.mark(name+'overflow')
+            for tape in counters:
+                target = fresh(a, 'all_end')
+                a.op('read', tape, {BLANK: target})
+                a.mark(target)
+            for tape in counters:
+                a.rewind(tape, BITS)
+            if i:
+                a.op('goto', f'advance{i-1}q{mode}')
+            else:
+                a.op('move', OUTER, 1)
+                a.op('read', OUTER, {ONE: 'record'+str(mode), BLANK: 'finish'})
+    for mode in range(4):
+        a.mark('advanced'+str(mode))
+        for tape in counters:
+            a.rewind(tape, BITS)
+        a.op('goto', 'record'+str(mode))
+
+
+def compile_phase(linear=SELECTED_LINEAR, rows=SELECTED_ROWS,
+                  constant=SELECTED_CONSTANT):
+    """Recipe is finite construction-time data; no runtime size is an argument."""
+    m = len(linear)
+    if not 1 <= m <= len(COUNTERS) or len(rows) != m:
+        raise ValueError('fixed recipe dimension')
+    if type(constant) is not int or not 0 <= constant < 4:
+        raise ValueError('fixed constant phase')
+    if any(type(x) is not int or not 0 <= x < 4 for x in linear):
+        raise ValueError('fixed linear phase')
+    if any(type(x) is not int or not 0 <= x < 1 << m for x in rows):
+        raise ValueError('fixed quadratic rows')
+    if any(((rows[i] >> j) ^ (rows[j] >> i)) & 1
+           for i in range(m) for j in range(m)):
+        raise ValueError('symmetric source quadratic matrix')
+    if any((linear[i] ^ (rows[i] >> i)) & 1 for i in range(m)):
+        raise ValueError('source diagonal parity')
+    counters = COUNTERS[:m]
+    a = Builder()
+    for tape in PRIVATE:
+        a.op('move', tape, -1)
+        a.op('write', tape, MARK)
+        a.op('move', tape, 1)
+    # Supplied local ABI: Gamma(f), Gamma(r), Gamma(P), Gamma(S), Gamma(w).
+    for target in (COLUMNS, COEFF, OUTER, SPECTATOR, WIDTH):
+        gamma(a, target)
+    trace_return(a, HEADER, HTRACE)
+    initialize(a, counters, constant)
+    for mode in range(4):
+        a.mark('record'+str(mode))
+        a.op('read', SPECTATOR, {ONE: 'coefficient'+str(mode), BLANK: 'record_end'+str(mode)})
+        a.mark('coefficient'+str(mode))
+        a.op('read', COEFF, {ONE: 'payload'+str(mode), BLANK: 'coefficient_end'+str(mode)})
+        a.mark('payload'+str(mode))
+        load_component(a, REAL)
+        load_component(a, IMAG)
+        if mode in (2, 3):
+            negate(a, REAL)
+        if mode in (1, 2):
+            negate(a, IMAG)
+        order = (IMAG, REAL) if mode % 2 else (REAL, IMAG)
+        for tape in order:
+            stage_component(a, tape)
+        a.op('move', COEFF, 1)
+        a.op('goto', 'coefficient'+str(mode))
+        a.mark('coefficient_end'+str(mode))
+        a.rewind(COEFF, (ONE,))
+        a.op('move', SPECTATOR, 1)
+        a.op('goto', 'record'+str(mode))
+        a.mark('record_end'+str(mode))
+        a.rewind(SPECTATOR, (ONE,))
+        a.op('goto', f'advance{m-1}q{mode}')
+    advance(a, counters, linear, rows)
+    a.mark('finish')
+    a.op('read', DATA, {BLANK: 'verified', '0': 'reject', ONE: 'reject'})
+    a.mark('verified')
+    trace_return(a, DATA, DTRACE)
+    a.rewind(OUTPUT, BITS)
+    a.mark('commit')
+    a.op('read', OUTPUT, {str(b): 'commit'+str(b) for b in (0, 1)} | {BLANK: 'committed'})
+    for bit in ('0', ONE):
+        a.mark('commit'+bit)
+        a.op('write', DATA, bit)
+        a.op('write', OUTPUT, BLANK)
+        a.op('move', OUTPUT, 1)
+        a.op('move', DATA, 1)
+        a.op('write', DTRACE, ONE)
+        a.op('move', DTRACE, 1)
+        a.op('goto', 'commit')
+    a.mark('committed')
+    trace_return(a, DATA, DTRACE)
+    a.rewind(OUTPUT, (BLANK,))
+    a.rewind(OUTER, (ONE,))
+    for tape in (COLUMNS, OUTER, COEFF, SPECTATOR, WIDTH):
+        clear(a, tape)
+    for tape in counters:
+        clear(a, tape, BITS)
+    for tape in PRIVATE:
+        a.op('move', tape, -1)
+        a.op('write', tape, BLANK)
+        a.op('move', tape, 1)
+    a.op('halt')
+    a.mark('reject')
+    a.op('read', HEADER, {})
+    return a.finish((BLANK, '0', ONE, MARK))
diff --git a/research/literal-native-quadratic-phase/reference/clifford_exact.py b/research/literal-native-quadratic-phase/reference/clifford_exact.py
new file mode 100644
index 0000000000..166841a037
--- /dev/null
+++ b/research/literal-native-quadratic-phase/reference/clifford_exact.py
@@ -0,0 +1,149 @@
+"""Small exact Gaussian-dyadic Clifford compiler, independent of gx internals.
+Paulis use i^p X^x Z^z. State vectors use integer Gaussian numerators / 2^d.
+The tableau is supplemented by a literal circuit, so global phases are kept.
+"""
+from dataclasses import dataclass
+
+def parity(x):return x.bit_count()&1
+def rank(rows):
+ p={}
+ for x in rows:
+  while x:
+   b=x.bit_length()-1
+   if b not in p:p[b]=x;break
+   x^=p[b]
+ return len(p)
+def transpose(rows,n):return tuple(sum(((rows[j]>>i)&1)<<j for j in range(n)) for i in range(n))
+def mv(rows,x):return sum(parity(r&x)<<i for i,r in enumerate(rows))
+def inv(rows,n):
+ a=[r|(1<<(n+i)) for i,r in enumerate(rows)]
+ for j in range(n):
+  k=next(i for i in range(j,n) if a[i]>>j&1);a[j],a[k]=a[k],a[j]
+  for i in range(n):
+   if i!=j and a[i]>>j&1:a[i]^=a[j]
+ assert [r&((1<<n)-1) for r in a]==[1<<i for i in range(n)]
+ return tuple(r>>n for r in a)
+def basis_extend(cols,n):
+ out=[]
+ for x in list(cols)+[1<<i for i in range(n)]:
+  if rank(out+[x])>len(out):out.append(x)
+ assert len(out)==n
+ return tuple(out)
+def nullspace(rows,n):
+ a=list(rows);p=[];k=0
+ for j in range(n):
+  t=next((i for i in range(k,len(a)) if a[i]>>j&1),None)
+  if t is None:continue
+  a[t],a[k]=a[k],a[t]
+  for i in range(len(a)):
+   if i!=k and a[i]>>j&1:a[i]^=a[k]
+  p.append(j);k+=1
+ return tuple((1<<j)|sum(((a[i]>>j)&1)<<v for i,v in enumerate(p)) for j in range(n) if j not in p)
+def pmul(a,b):
+ x,z,p=a;y,w,q=b
+ return x^y,z^w,(p+q+2*parity(z&y))%4
+def identity(n):return tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,0) for i in range(n)])
+def apply(T,p):
+ n=len(T)//2;x,z,ph=p;out=(0,0,ph)
+ for i in range(n):
+  if x>>i&1:out=pmul(out,T[i])
+ for i in range(n):
+  if z>>i&1:out=pmul(out,T[n+i])
+ return out
+def compose(A,B):return tuple(apply(A,p) for p in B)
+def inverse(T):
+ n=len(T)//2;cols=[x|(z<<n) for x,z,p in T];rows=transpose(cols,2*n);cols_inv=transpose(inv(rows,2*n),2*n);out=[]
+ for j,w in enumerate(cols_inv):
+  x=w&((1<<n)-1);z=w>>n;u,v,p=apply(T,(x,z,0));assert (u,v)==identity(n)[j][:2];out.append((x,z,(-p)%4))
+ return tuple(out)
+def qvalue(Q,alpha,x,n):
+ return (sum(alpha[i] for i in range(n) if x>>i&1)+2*sum(parity((Q[i]>>(i+1))&(x>>(i+1))) for i in range(n) if x>>i&1))%4
+def mul_i(a,p):
+ x,y=a;p%=4
+ return [(x,y),(-y,x),(-x,-y),(y,-x)][p]
+@dataclass
+class Op:
+ n:int
+ tab:tuple
+ circuit:list
+ def then(self,b):
+  assert self.n==b.n
+  return Op(self.n,compose(b.tab,self.tab),self.circuit+b.circuit)
+ def adjoint(self):
+  circ=[]
+  for e in reversed(self.circuit):
+   if e[0]=='P':circ.append(('P',inv(e[1],self.n)))
+   elif e[0]=='D':circ.append(('D',e[1],tuple((-x)%4 for x in e[2])))
+   elif e[0]=='C':circ.append(('C',e[1],-e[2]))
+   elif e[0]=='phase':circ.append(('phase',(-e[1])%4))
+   else:circ.append(e)
+  return Op(self.n,inverse(self.tab),circ)
+ def zero_state(self,basis=0):
+  N=1<<self.n;vec=[(0,0)]*N;vec[basis]=(1,0);den=0
+  for e in self.circuit:
+   if e[0]=='P':
+    v=[(0,0)]*N
+    for x,a in enumerate(vec):v[mv(e[1],x)]=a
+    vec=v
+   elif e[0]=='X':vec=[vec[x^e[1]] for x in range(N)]
+   elif e[0]=='D':vec=[mul_i(a,qvalue(e[1],e[2],x,self.n)) for x,a in enumerate(vec)]
+   elif e[0]=='phase':vec=[mul_i(a,e[1]) for a in vec]
+   elif e[0]=='C':
+    q,sgn=e[1:];v=[]
+    for x,(a,b) in enumerate(vec):
+     c,d=vec[x^q]
+     # C=(1+i)I/2+(1-i)X/2; inverse conjugates both coefficients.
+     v.append((a-sgn*b+c+sgn*d,b+sgn*a+d-sgn*c))
+    vec=v;den+=1
+   else:raise AssertionError(e)
+   while den and all((a&1)==(b&1)==0 for a,b in vec):vec=[(a//2,b//2) for a,b in vec];den-=1
+  return vec,den
+def ident(n):return Op(n,identity(n),[])
+def perm(n,G):
+ cols=transpose(G,n);gi=inv(G,n)
+ return Op(n,tuple([(a,0,0) for a in cols]+[(0,a,0) for a in gi]),[('P',tuple(G))])
+def diag(n,Q,alpha=None):
+ Q=tuple(Q);assert transpose(Q,n)==Q
+ alpha=tuple(alpha) if alpha is not None else tuple((Q[i]>>i)&1 for i in range(n))
+ assert all(alpha[i]%2==(Q[i]>>i)&1 for i in range(n))
+ return Op(n,tuple([(1<<i,Q[i],alpha[i]) for i in range(n)]+[(0,1<<i,0) for i in range(n)]),[('D',Q,alpha)])
+def cgate(n,q,sgn=1):
+ return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(q if q>>i&1 else 0,1<<i,(3*sgn)%4 if q>>i&1 else 0) for i in range(n)]),[('C',q,sgn)])
+def xgate(n,b):return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,2*((b>>i)&1)) for i in range(n)]),[('X',b)])
+def fourier(n,axes=None):
+ out=ident(n)
+ for i in range(n) if axes is None else axes:out=out.then(cgate(n,1<<i))
+ return out
+def hadamard(n,axes):
+ axes=list(axes);Q=tuple(1<<i if i in axes else 0 for i in range(n));S=diag(n,Q)
+ return S.then(fourier(n,axes)).then(S)
+def canonical_frame(n,U):
+ r=len(U)
+ if r==0:return ident(n)
+ if r==n:return fourier(n)
+ G=transpose(basis_extend(U,n),n);gi=inv(G,n);Q=tuple((1<<i)^sum(parity(gi_i&gi_j)<<j for j,gi_j in enumerate(transpose(gi,n))) for i,gi_i in enumerate(transpose(gi,n)))
+ K=perm(n,G).then(diag(n,Q))
+ return K.adjoint().then(fourier(n,range(r))).then(K)
+def monomial(op):
+ n=op.n;T=op.tab;assert all(x==0 and p%2==0 for x,z,p in T[n:]);G=transpose([x for x,z,p in T[:n]],n);b=mv(G,sum((p//2)<<i for i,(x,z,p) in enumerate(T[n:])))
+ gt=transpose(G,n);cols=[mv(gt,z) for x,z,p in T[:n]];Q=transpose(cols,n);alpha=[(p-2*parity(b&z))%4 for x,z,p in T[:n]]
+ out=diag(n,Q,alpha).then(perm(n,G)).then(xgate(n,b));assert out.tab==T
+ vec,den=op.zero_state();assert den==0 and all(a==(0,0) for i,a in enumerate(vec) if i!=b);phase=[(1,0),(0,1),(-1,0),(0,-1)].index(vec[b]);out=out.then(Op(n,identity(n),[('phase',phase)]));assert out.zero_state()==op.zero_state()
+ return out,dict(matrix_rows=G,offset=b,quadratic_rows=Q,linear_Z4=alpha,phase=phase)
+def lower_child(op):
+ n=op.n;Bcols=[x for x,z,p in op.tab[n:]];r=rank(Bcols)
+ if not r:
+  mon,rec=monomial(op);return dict(rank=0,left=rec,right=None),mon
+ G=transpose(basis_extend(Bcols,n),n);A=inv(G,n);PA=perm(n,A);part=op.then(PA);Bcols=[x for x,z,p in part.tab[n:]];Dcols=[z for x,z,p in part.tab[n:]]
+ picks=[];selected=[]
+ for j,x in enumerate(Bcols):
+  if rank(selected+[x])>len(selected):picks.append(j);selected.append(x)
+ assert len(picks)==r and all(x>>r==0 for x in Bcols)
+ Br=transpose(selected,r);Bi=inv(Br,r);Dr=transpose([Dcols[j]&((1<<r)-1) for j in picks],r)
+ Qsmall=tuple(sum(parity(row&col)<<j for j,col in enumerate(transpose(Bi,r))) for row in Dr);Q=Qsmall+(0,)*(n-r);DQ=diag(n,Q)
+ V=part.then(DQ).then(hadamard(n,range(r)));assert all(x==0 for x,z,p in V.tab[n:]);Vmon,Vrec=monomial(V)
+ S=diag(n,tuple(1<<i if i<r else 0 for i in range(n)))
+ left=S.adjoint().then(DQ.adjoint()).then(PA.adjoint());right=Vmon.then(S.adjoint()).then(xgate(n,(1<<r)-1))
+ lmon,lrec=monomial(left);rmon,rrec=monomial(right);compiled=rmon.then(fourier(n,range(r))).then(lmon)
+ assert compiled.tab==op.tab and compiled.zero_state()==op.zero_state()
+ return dict(rank=r,left=lrec,right=rrec),compiled
diff --git a/research/literal-native-quadratic-phase/reference/native_verify.py b/research/literal-native-quadratic-phase/reference/native_verify.py
new file mode 100644
index 0000000000..2c209dfa9c
--- /dev/null
+++ b/research/literal-native-quadratic-phase/reference/native_verify.py
@@ -0,0 +1,277 @@
+"""Check native operators with exact Gaussian dyadic arithmetic.
+
+The interpreter is adapted from the independent campaign native review.
+It does not import the source-code producer or the Clifford compiler.
+Research used OpenAI Codex under the user's direction.
+"""
+
+from functools import lru_cache
+
+
+def require(condition, label):
+    if not condition:
+        raise ValueError(label)
+
+
+def parity(value):
+    return value.bit_count() % 2
+
+
+def mul(matrix, vector):
+    return sum(parity(row & vector) << i for i, row in enumerate(matrix))
+
+
+def transpose(matrix):
+    return tuple(sum(((row >> j) & 1) << i for i, row in enumerate(matrix)) for j in range(len(matrix)))
+
+
+def inverse(matrix):
+    n = len(matrix)
+    rows = [row | (1 << (n+i)) for i, row in enumerate(matrix)]
+    for j in range(n):
+        pivot = next((i for i in range(j, n) if rows[i] >> j & 1), None)
+        require(pivot is not None, 'singular binary chart')
+        rows[j], rows[pivot] = rows[pivot], rows[j]
+        for i in range(n):
+            if i != j and rows[i] >> j & 1:
+                rows[i] ^= rows[j]
+    require([row & ((1 << n)-1) for row in rows] == [1 << i for i in range(n)], 'binary inverse')
+    return tuple(row >> n for row in rows)
+
+
+def extend(vectors, n):
+    columns, pivots = [], {}
+    for value in tuple(vectors) + tuple(1 << i for i in range(n)):
+        reduced = value
+        while reduced:
+            j = reduced.bit_length()-1
+            if j not in pivots:
+                pivots[j] = reduced
+                columns.append(value)
+                break
+            reduced ^= pivots[j]
+    require(len(columns) == n, 'frame basis extension')
+    return tuple(columns)
+
+
+def adjoint(circuit):
+    out = []
+    for gate in reversed(circuit):
+        if gate[0] == 'P':
+            out.append(('P', inverse(gate[1])))
+        elif gate[0] == 'D':
+            out.append(('D', gate[1], tuple(-a % 4 for a in gate[2])))
+        elif gate[0] == 'C':
+            out.append(('C', gate[1], -gate[2]))
+        elif gate[0] == 'phase':
+            out.append(('phase', -gate[1] % 4))
+        else:
+            require(gate[0] == 'X', 'adjoint gate type')
+            out.append(gate)
+    return tuple(out)
+
+
+def fourier(rank):
+    return tuple(('C', 1 << i, 1) for i in range(rank))
+
+
+def frame(vectors, n):
+    rank = len(vectors)
+    if rank == 0:
+        return ()
+    if rank == n:
+        return fourier(n)
+    matrix = transpose(extend(vectors, n))
+    columns = transpose(inverse(matrix))
+    quadratic = tuple((1 << i) ^ sum(parity(a & b) << j for j, b in enumerate(columns)) for i, a in enumerate(columns))
+    diagonal = tuple((row >> i) & 1 for i, row in enumerate(quadratic))
+    chart = (('P', matrix), ('D', quadratic, diagonal))
+    return adjoint(chart) + fourier(rank) + chart
+
+
+def monomial(record):
+    matrix = tuple(record['matrix_rows'])
+    quadratic = tuple(record['quadratic_rows'])
+    linear = tuple(record['linear_Z4'])
+    require(transpose(quadratic) == quadratic, 'quadratic symmetry')
+    require(len(matrix) == len(quadratic) == len(linear) == 9, 'monomial dimension')
+    require(all(a % 2 == (quadratic[i] >> i) & 1 for i, a in enumerate(linear)), 'diagonal parity')
+    inverse(matrix)
+    return (('D', quadratic, linear), ('P', matrix), ('X', record['offset']), ('phase', record['phase']))
+
+
+def recipe(record):
+    rank = record['rank']
+    require(isinstance(rank, int) and 0 <= rank <= 9, 'child rank')
+    if rank == 0:
+        require(record['right'] is None, 'rank zero right factor')
+        return monomial(record['left'])
+    return monomial(record['right']) + fourier(rank) + monomial(record['left'])
+
+
+def tableau(circuit, n=9):
+    rows = [(1 << i, 0, 0) for i in range(n)] + [(0, 1 << i, 0) for i in range(n)]
+    for gate in circuit:
+        tag = gate[0]
+        out = []
+        inverse_transpose = transpose(inverse(gate[1])) if tag == 'P' else None
+        for x, z, phase in rows:
+            if tag == 'P':
+                x, z = mul(gate[1], x), mul(inverse_transpose, z)
+            elif tag == 'X':
+                phase += 2 * parity(z & gate[1])
+            elif tag == 'C':
+                if parity(z & gate[1]):
+                    x ^= gate[1]
+                    phase += 3 * gate[2]
+            elif tag == 'D':
+                quadratic, linear = gate[1:]
+                phase += sum(linear[i] for i in range(n) if x >> i & 1)
+                phase += 2 * sum(parity((quadratic[i] >> (i+1)) & (x >> (i+1))) for i in range(n) if x >> i & 1)
+                z ^= mul(quadratic, x)
+            else:
+                require(tag == 'phase', 'tableau gate type')
+            out.append((x, z, phase % 4))
+        rows = out
+    return tuple(rows)
+
+
+@lru_cache(maxsize=512)
+def action(gate):
+    if gate[0] == 'P':
+        return tuple(mul(gate[1], x) for x in range(512))
+    require(gate[0] == 'D', 'state action type')
+    quadratic, linear = gate[1:]
+    return tuple((sum(linear[i] for i in range(9) if x >> i & 1)
+                  + 2 * sum(parity((quadratic[i] >> (i+1)) & (x >> (i+1))) for i in range(9) if x >> i & 1)) % 4 for x in range(512))
+
+
+def rotate(pair, power):
+    a, b = pair
+    return ((a, b), (-b, a), (-a, -b), (b, -a))[power % 4]
+
+
+def state(circuit, basis=0):
+    value, denominator = {basis: (1, 0)}, 0
+    for gate in circuit:
+        tag = gate[0]
+        result = {}
+        if tag == "C":
+            direction, sign = gate[1:]
+            for address, (a, b) in value.items():
+                for target, pair in ((address, (a - sign*b, b + sign*a)),
+                                     (address ^ direction, (a + sign*b, b - sign*a))):
+                    old = result.get(target, (0, 0))
+                    result[target] = (old[0] + pair[0], old[1] + pair[1])
+            denominator += 1
+        else:
+            for address, pair in value.items():
+                if tag == "P":
+                    target = mul(gate[1], address)
+                elif tag == "X":
+                    target = address ^ gate[1]
+                else:
+                    target = address
+                    if tag == "D":
+                        pair = rotate(pair, action(gate)[address])
+                    else:
+                        require(tag == "phase", "Unknown Gaussian state gate")
+                        pair = rotate(pair, gate[1])
+                require(target not in result, "A monomial address map is not injective")
+                result[target] = pair
+        value = {address: pair for address, pair in result.items() if pair != (0, 0)}
+    while denominator and all((a | b) % 2 == 0 for a, b in value.values()):
+        denominator -= 1
+        value = {address: (a//2, b//2) for address, (a, b) in value.items()}
+    return value, denominator
+
+
+def equal(actual, expected, label):
+    require(tableau(actual) == tableau(expected), label + ": signed Pauli action")
+    require(state(actual) == state(expected), label + ": exact global phase")
+
+
+def verify(data, certificate):
+    word = data["word"]
+    original, clean = data["programs"], data["clean_programs"]
+    source_recipes = data["recipes"]
+    residuals = data["residuals"]["recipes"]
+    native_tags = {"climb", "adapter", "residual_adapter"}
+    for orientation in ("forward", "backward"):
+        native = [event for event in clean[orientation] if event[0] in native_tags]
+        require(native == [event for event in original[orientation] if event[0] in native_tags],
+                "A retained native event changed")
+        require(len(native) == 5056, "The retained native event count changed")
+    frames = [frame(tuple(vectors), 9) for vectors in word["frames"]]
+    used = {(event[3], event[4]) for program in clean.values()
+            for event in program if event[0] == "climb"}
+    children = {(record["old"], record["new"]): record for record in source_recipes["children"]}
+    require(len(children) == len(source_recipes["children"]) and used <= set(children),
+            "The retained child recipe coverage is incomplete")
+    for old, new in sorted(used):
+        record = children[old, new]
+        require(record["rank"] == len(word["frames"][new]) - len(word["frames"][old]),
+                "A retained child has the wrong rank")
+        equal(recipe(record), adjoint(frames[old]) + frames[new], "Retained child")
+    endpoints = {(record["role"], record["port"]): record
+                 for record in source_recipes["endpoint_adapters"]}
+    require(len(endpoints) == len(source_recipes["endpoint_adapters"])
+            and set(endpoints) == {(role, p) for role in ("source", "target") for p in range(112)},
+            "The endpoint adapter coverage is incomplete")
+    for (role, port), record in endpoints.items():
+        direction = word["ports"][port]
+        conventional = (("C", direction, 1),) if role == "source" else (
+            ("C", direction, -1),) + fourier(9)
+        index = word["start"][port] if role == "source" else word["final"][112 + port]
+        require(record["frame"] == index and record["rank"] == 0, "An endpoint frame changed")
+        equal(recipe(record), adjoint(conventional) + frames[index], "Endpoint adapter")
+    used_residuals = set()
+    for program in clean.values():
+        for event in program:
+            if event[0] != "residual_adapter":
+                continue
+            _, position, register, entry, *tail = event
+            terminal = tail[0] if tail else 1
+            key = str(entry) if terminal == 1 else str(entry) + ":" + str(terminal)
+            require(key in residuals and position in ("pre", "post"), "A residual binding changed")
+            require(entry == word["start"][register] and terminal == word["final"][register],
+                    "A residual endpoint changed")
+            used_residuals.add(key)
+    for key in sorted(used_residuals):
+        record = residuals[key]
+        equal(recipe(record), adjoint(frames[record["frame"]]) + frames[record.get("terminal", 1)],
+              "Residual adapter")
+    records = certificate["input_recipes"]
+    ports = certificate["compensation_ports"]
+    require([record["port"] for record in records] == ports and len(records) == 8,
+            "The eight input recipes changed")
+    columns = 0
+    for item in records:
+        port, label = item["port"], item["label"]
+        require(label == word["ports"][port] and item["canonical_start_frame_id"] == word["start"][port],
+                "An input recipe has the wrong supplier label")
+        record = item["recipe"]
+        require(record["rank"] == 1, "An input copy has the wrong child rank")
+        for side in ("left", "right"):
+            factor = record[side]
+            require(factor["quadratic_rows"] == [0] * 9 and factor["linear_Z4"] == [0] * 9
+                    and factor["offset"] == factor["phase"] == 0,
+                    "An input factor is not an address permutation")
+        actual, expected = recipe(record), (("C", label, 1),)
+        equal(actual, expected, "Positive inherited input line")
+        for basis in range(512):
+            require(state(actual, basis) == state(expected, basis), "A complete input Gaussian column differs")
+            columns += 1
+        negative = monomial(record["right"]) + (("C", 1, -1),) + monomial(record["left"])
+        require(tableau(negative) != tableau(expected) or state(negative) != state(expected),
+                "An inverse input child was not detected")
+        phased = actual + (("phase", 1),)
+        require(tableau(phased) == tableau(actual) and state(phased) != state(expected),
+                "An input phase-only error was not detected")
+    return {"status": "PASS_EXACT_INPUT_NATIVE_FRAMES", "retained_native_events_per_orientation": 5056,
+            "retained_child_recipes": len(used), "endpoint_adapter_recipes": len(endpoints),
+            "residual_adapter_recipes": len(used_residuals), "new_input_recipes": len(records),
+            "new_exact_gaussian_columns": columns, "inverse_input_child_controls": len(records),
+            "phase_only_controls": len(records), "address_permutation_factors": 2 * len(records),
+            "method": "Independent signed Pauli and exact Gaussian interpretation; no compiler import",
+            "scope": "Positive inherited input frames before every retained adapter. No recursive hosting claim."}
diff --git a/research/literal-native-quadratic-phase/reference/outer-phase.py b/research/literal-native-quadratic-phase/reference/outer-phase.py
new file mode 100644
index 0000000000..c6e8769534
--- /dev/null
+++ b/research/literal-native-quadratic-phase/reference/outer-phase.py
@@ -0,0 +1,343 @@
+"""Header-derived literal S_D phase sweep on implicit layer addresses.
+
+Prepared for Jerry Chen / CubeJerry with substantial OpenAI assistance.
+Apache-2.0. Gaussian phases and counter amortization are credited prior work;
+see PROOF.md. Python constructs one fixed graph, never an input-sized graph.
+Only scanned symbols, fixed jumps, one-cell writes and unit moves execute.
+"""
+from pathlib import Path
+import hashlib
+import sys
+
+HERE = Path(__file__).resolve().parent
+MACHINE = HERE / 'inputs' / 'controller_reference.py'
+EXPECTED_MACHINE = 'e664a748e6b6797d4b60f60a7a7dd68b59a5608acf56b9c5d8b466f363755f73'
+if hashlib.sha256(MACHINE.read_bytes()).hexdigest() != EXPECTED_MACHINE:
+    raise ValueError('frozen literal machine source drift')
+sys.path.insert(0, str(MACHINE.parent))
+from controller_reference import Builder, Machine, Tape, BLANK, ONE, MARK
+
+(HEADER, DATA, OUTPUT, HTRACE, LENGTH, DOUBLE, PREC, GLOBAL, COEFF, DIM,
+ OUTER, SPECTATOR, BLOCK, SELECT, WIDTH, MASK, COUNT, REAL, IMAG, DTRACE) = range(20)
+PRIVATE = tuple(range(2, 20))
+BITS = ('0', ONE)
+
+
+def fresh(a, stem):
+    a.serial += 1
+    return stem + str(a.serial)
+
+
+def clear(a, tape, symbols=(ONE,)):
+    k = fresh(a, 'clear')
+    a.mark(k)
+    a.op('read', tape, {s: k+'bit' for s in symbols} | {BLANK: k+'done'})
+    a.mark(k+'bit')
+    a.op('write', tape, BLANK)
+    a.op('move', tape, 1)
+    a.op('goto', k)
+    a.mark(k+'done')
+    a.rewind(tape, (BLANK,))
+
+
+def unary_copy(a, source, target, *, repeat=1, extra=0, erase=False):
+    k = fresh(a, 'copy')
+    a.mark(k)
+    a.op('read', source, {ONE: k+'bit', BLANK: k+'done'})
+    a.mark(k+'bit')
+    for _ in range(repeat):
+        a.op('write', target, ONE)
+        a.op('move', target, 1)
+    if erase:
+        a.op('write', source, BLANK)
+    a.op('move', source, 1)
+    a.op('goto', k)
+    a.mark(k+'done')
+    for _ in range(extra):
+        a.op('write', target, ONE)
+        a.op('move', target, 1)
+    a.rewind(source, (BLANK,) if erase else (ONE,))
+    a.rewind(target, (ONE,))
+
+
+def header_step(a):
+    a.op('write', HTRACE, ONE)
+    a.op('move', HTRACE, 1)
+    a.op('move', HEADER, 1)
+
+
+def gamma(a, target):
+    """Actual positive Gamma code to unary, using only literal operations."""
+    k = fresh(a, 'gamma')
+    a.mark(k+'prefix')
+    a.op('read', HEADER, {'0': k+'zero', ONE: k+'one'})
+    a.mark(k+'zero')
+    a.op('write', LENGTH, ONE)
+    a.op('move', LENGTH, 1)
+    header_step(a)
+    a.op('goto', k+'prefix')
+    a.mark(k+'one')
+    header_step(a)
+    a.op('write', target, ONE)
+    a.op('move', LENGTH, -1)
+    a.mark(k+'tail')
+    a.op('read', LENGTH, {ONE: k+'tailbit', MARK: k+'done'})
+    a.mark(k+'tailbit')
+    a.op('write', LENGTH, BLANK)
+    a.op('read', HEADER, {str(b): k+'digit'+str(b) for b in (0, 1)})
+    for b in (0, 1):
+        a.mark(k+'digit'+str(b))
+        unary_copy(a, target, DOUBLE, repeat=2, extra=b)
+        clear(a, target)
+        unary_copy(a, DOUBLE, target, erase=True)
+        header_step(a)
+        a.op('move', LENGTH, -1)
+        a.op('goto', k+'tail')
+    a.mark(k+'done')
+    a.op('move', LENGTH, 1)
+
+
+def trace_return(a, tape, trace):
+    k = fresh(a, 'return')
+    a.op('move', trace, -1)
+    a.mark(k)
+    a.op('read', trace, {ONE: k+'bit', MARK: k+'done'})
+    a.mark(k+'bit')
+    a.op('write', trace, BLANK)
+    a.op('move', trace, -1)
+    a.op('move', tape, -1)
+    a.op('goto', k)
+    a.mark(k+'done')
+    a.op('move', trace, 1)
+
+
+def make_geometry(a):
+    """Generate D K-bit chunks with active position rho, all LSB first."""
+    a.mark('build_chunk')
+    a.op('read', DIM, {ONE: 'build_before', BLANK: 'geometry_done'})
+    a.mark('build_before')
+    a.op('read', BLOCK, {ONE: 'build_select', BLANK: 'reject'})
+    a.mark('build_select')
+    a.op('read', SELECT, {ONE: 'select_step'})
+    a.mark('select_step')
+    a.op('move', SELECT, 1)
+    a.op('read', SELECT, {ONE: 'mask_before', BLANK: 'mask_active'})
+    a.mark('build_after')
+    a.op('read', BLOCK, {ONE: 'mask_after', BLANK: 'chunk_done'})
+    for name, bit, nxt in (('mask_before', '0', 'build_before'),
+                           ('mask_active', ONE, 'build_after'),
+                           ('mask_after', '0', 'build_after')):
+        a.mark(name)
+        a.op('write', MASK, bit)
+        a.op('move', MASK, 1)
+        a.op('write', COUNT, '0')
+        a.op('move', COUNT, 1)
+        a.op('move', BLOCK, 1)
+        a.op('goto', nxt)
+    a.mark('chunk_done')
+    a.rewind(BLOCK, (ONE,))
+    a.rewind(SELECT, (ONE,))
+    a.op('move', DIM, 1)
+    a.op('goto', 'build_chunk')
+    a.mark('geometry_done')
+    a.rewind(DIM, (ONE,))
+    a.rewind(MASK, BITS)
+    a.rewind(COUNT, BITS)
+
+
+def load_component(a, target):
+    k = fresh(a, 'load')
+    a.mark(k)
+    a.op('read', WIDTH, {ONE: k+'bit', BLANK: k+'done'})
+    a.mark(k+'bit')
+    a.op('read', DATA, {str(b): k+str(b) for b in (0, 1)})
+    for b in (0, 1):
+        a.mark(k+str(b))
+        a.op('write', target, str(b))
+        a.op('move', target, 1)
+        a.op('move', DATA, 1)
+        a.op('write', DTRACE, ONE)
+        a.op('move', DTRACE, 1)
+        a.op('move', WIDTH, 1)
+        a.op('goto', k)
+    a.mark(k+'done')
+    a.rewind(WIDTH, (ONE,))
+    a.rewind(target, BITS)
+
+
+def negate(a, tape):
+    """Exact w-bit integer negation; reject the unrepresentable minimum."""
+    k = fresh(a, 'neg')
+    a.mark(k+'seek')
+    a.op('read', tape, {'0': k+'step', ONE: k+'step', BLANK: k+'last'})
+    a.mark(k+'step')
+    a.op('move', tape, 1)
+    a.op('goto', k+'seek')
+    a.mark(k+'last')
+    a.op('move', tape, -1)
+    a.op('goto', k+'c1x0y0')
+    for carry in (0, 1):
+        for old_sign in (0, 1):
+            for new_sign in (0, 1):
+                name = k+f'c{carry}x{old_sign}y{new_sign}'
+                a.mark(name)
+                a.op('read', tape, {str(b): name+'b'+str(b) for b in (0, 1)} |
+                     {MARK: 'reject' if old_sign and new_sign else k+'done'})
+                for bit in (0, 1):
+                    total = 1-bit+carry
+                    digit, nxt = total % 2, total//2
+                    a.mark(name+'b'+str(bit))
+                    a.op('write', tape, str(digit))
+                    a.op('move', tape, -1)
+                    a.op('goto', k+f'c{nxt}x{bit}y{digit}')
+    a.mark(k+'done')
+    a.op('move', tape, 1)
+
+
+def stage_component(a, tape):
+    k = fresh(a, 'stage')
+    a.mark(k)
+    a.op('read', tape, {str(b): k+str(b) for b in (0, 1)} | {BLANK: k+'done'})
+    for b in (0, 1):
+        a.mark(k+str(b))
+        a.op('write', OUTPUT, str(b))
+        a.op('move', OUTPUT, 1)
+        a.op('write', tape, BLANK)
+        a.op('move', tape, 1)
+        a.op('goto', k)
+    a.mark(k+'done')
+    a.rewind(tape, (BLANK,))
+
+
+def advance(a):
+    """Full binary increment; phase is always fixed four-state control."""
+    for mode in range(4):
+        name = 'advance'+str(mode)
+        a.mark(name)
+        a.op('read', COUNT, {str(b): name+'b'+str(b) for b in (0, 1)} |
+             {BLANK: 'overflow'+str(mode)})
+        for bit in (0, 1):
+            a.mark(name+'b'+str(bit))
+            a.op('read', MASK, {str(active): name+f'b{bit}m{active}' for active in (0, 1)})
+            for active in (0, 1):
+                a.mark(name+f'b{bit}m{active}')
+                nxt = (mode + (1-2*bit)*active) % 4
+                a.op('write', COUNT, str(1-bit))
+                a.op('move', COUNT, 1)
+                a.op('move', MASK, 1)
+                a.op('goto', 'advance'+str(nxt) if bit else 'advanced'+str(nxt))
+    for mode in range(4):
+        a.mark('advanced'+str(mode))
+        a.rewind(COUNT, BITS)
+        a.rewind(MASK, BITS)
+        a.op('goto', 'record'+str(mode))
+        a.mark('overflow'+str(mode))
+        a.op('read', MASK, {BLANK: 'wrapped'+str(mode)})
+        a.mark('wrapped'+str(mode))
+        if mode:
+            a.op('goto', 'reject')
+        else:
+            a.rewind(COUNT, BITS)
+            a.rewind(MASK, BITS)
+            a.op('move', OUTER, 1)
+            a.op('read', OUTER, {ONE: 'record0', BLANK: 'finish'})
+
+
+def compile_phase():
+    a = Builder()
+    for tape in PRIVATE:
+        a.op('move', tape, -1)
+        a.op('write', tape, MARK)
+        a.op('move', tape, 1)
+    # Actual header: p,d,r,D,P,S,K,rho+1,w. Runtime values stay on tapes.
+    for target in (PREC, GLOBAL, COEFF, DIM, OUTER, SPECTATOR, BLOCK, SELECT, WIDTH):
+        gamma(a, target)
+    trace_return(a, HEADER, HTRACE)
+    a.mark('dimension_check')
+    a.op('read', DIM, {ONE: 'dimension_global', BLANK: 'dimension_ok'})
+    a.mark('dimension_global')
+    a.op('read', GLOBAL, {ONE: 'dimension_step', BLANK: 'reject'})
+    a.mark('dimension_step')
+    a.op('move', DIM, 1)
+    a.op('move', GLOBAL, 1)
+    a.op('goto', 'dimension_check')
+    a.mark('dimension_ok')
+    a.rewind(DIM, (ONE,))
+    a.rewind(GLOBAL, (ONE,))
+    a.mark('width_check')
+    a.op('read', PREC, {ONE: 'width_precision', BLANK: 'width_extra0'})
+    a.mark('width_precision')
+    a.op('read', WIDTH, {ONE: 'width_step', BLANK: 'reject'})
+    a.mark('width_step')
+    a.op('move', PREC, 1)
+    a.op('move', WIDTH, 1)
+    a.op('goto', 'width_check')
+    for i in range(2):
+        a.mark('width_extra'+str(i))
+        a.op('read', WIDTH, {ONE: 'width_more'+str(i), BLANK: 'reject'})
+        a.mark('width_more'+str(i))
+        a.op('move', WIDTH, 1)
+    a.rewind(PREC, (ONE,))
+    a.rewind(WIDTH, (ONE,))
+    make_geometry(a)
+    for tape in (PREC, GLOBAL, DIM, BLOCK, SELECT):
+        clear(a, tape)
+    a.op('goto', 'record0')
+    for mode in range(4):
+        a.mark('record'+str(mode))
+        a.op('read', SPECTATOR, {ONE: 'coefficient'+str(mode), BLANK: 'record_end'+str(mode)})
+        a.mark('coefficient'+str(mode))
+        a.op('read', COEFF, {ONE: 'payload'+str(mode), BLANK: 'coefficient_end'+str(mode)})
+        a.mark('payload'+str(mode))
+        load_component(a, REAL)
+        load_component(a, IMAG)
+        if mode in (2, 3):
+            negate(a, REAL)
+        if mode in (1, 2):
+            negate(a, IMAG)
+        order = (IMAG, REAL) if mode % 2 else (REAL, IMAG)
+        for tape in order:
+            stage_component(a, tape)
+        a.op('move', COEFF, 1)
+        a.op('goto', 'coefficient'+str(mode))
+        a.mark('coefficient_end'+str(mode))
+        a.rewind(COEFF, (ONE,))
+        a.op('move', SPECTATOR, 1)
+        a.op('goto', 'record'+str(mode))
+        a.mark('record_end'+str(mode))
+        a.rewind(SPECTATOR, (ONE,))
+        a.op('goto', 'advance'+str(mode))
+    advance(a)
+    a.mark('finish')
+    # Header-derived geometry determines cardinality, not a host oracle.
+    a.op('read', DATA, {BLANK: 'verified', '0': 'reject', ONE: 'reject'})
+    a.mark('verified')
+    trace_return(a, DATA, DTRACE)
+    a.rewind(OUTPUT, BITS)
+    a.mark('commit')
+    a.op('read', OUTPUT, {str(b): 'commit'+str(b) for b in (0, 1)} | {BLANK: 'committed'})
+    for bit in ('0', ONE):
+        a.mark('commit'+bit)
+        a.op('write', DATA, bit)
+        a.op('write', OUTPUT, BLANK)
+        a.op('move', OUTPUT, 1)
+        a.op('move', DATA, 1)
+        a.op('write', DTRACE, ONE)
+        a.op('move', DTRACE, 1)
+        a.op('goto', 'commit')
+    a.mark('committed')
+    trace_return(a, DATA, DTRACE)
+    a.rewind(OUTPUT, (BLANK,))
+    a.rewind(OUTER, (ONE,))
+    for tape in (OUTER, COEFF, SPECTATOR, WIDTH):
+        clear(a, tape)
+    for tape in (MASK, COUNT):
+        clear(a, tape, BITS)
+    for tape in PRIVATE:
+        a.op('move', tape, -1)
+        a.op('write', tape, BLANK)
+        a.op('move', tape, 1)
+    a.op('halt')
+    a.mark('reject')
+    a.op('read', HEADER, {})
+    return a.finish((BLANK, '0', ONE, MARK))
diff --git a/research/literal-native-quadratic-phase/reference/selected-native-catalogue.py b/research/literal-native-quadratic-phase/reference/selected-native-catalogue.py
new file mode 100644
index 0000000000..98b35c89c8
--- /dev/null
+++ b/research/literal-native-quadratic-phase/reference/selected-native-catalogue.py
@@ -0,0 +1,97 @@
+#!/usr/bin/env python3
+"""Complete #7000 paid/helper chart catalogue, including inherited maps.
+
+Reuses the pinned producer's Clifford compiler and distinct exact signed-Pauli
+and Gaussian phase interpreter; it does not call the old delta-only checker.
+Credit the retained compiler and independent-native authors. OpenAI Codex
+assistance for Douglas Colkitt; Apache-2.0.
+"""
+from pathlib import Path
+from hashlib import sha256
+import copy
+import gzip
+import importlib.util
+import json
+import sys
+import argparse
+
+HERE=Path(__file__).resolve().parent
+ROOT=HERE.parents[3]
+PACKAGE=ROOT/'research/shared-wire-split-composition'
+
+
+def need(ok,message):
+    if not ok:raise ValueError(message)
+
+
+def module(relative,name):
+    path=PACKAGE/relative
+    pins=json.loads((HERE/'SOURCES.json').read_text())['files']
+    expected=next(p['sha256'] for p in pins if p['path']==path.relative_to(ROOT).as_posix())
+    need(sha256(path.read_bytes()).hexdigest()==expected,'engine source hash')
+    spec=importlib.util.spec_from_file_location(name,path)
+    value=importlib.util.module_from_spec(spec);sys.modules[name]=value;spec.loader.exec_module(value)
+    return value
+
+
+def main():
+    parser=argparse.ArgumentParser()
+    parser.add_argument('--start',type=int,default=0)
+    parser.add_argument('--stop',type=int)
+    args=parser.parse_args()
+    need(not sys.flags.optimize,'retained compiler assertions must be enabled')
+    C=module('frozen/clifford_exact.py','complete_catalogue_compiler')
+    V=module('frozen/native_verify.py','complete_catalogue_interpreter')
+    raw=(PACKAGE/'data/final.json.gz').read_bytes()
+    pins=json.loads((HERE/'SOURCES.json').read_text())['files']
+    need(sha256(raw).hexdigest()==next(p['sha256'] for p in pins if p['path'].endswith('data/final.json.gz')),'word hash')
+    word=json.loads(gzip.decompress(raw))
+    main_fields=set(word['live'])|{x+112 for x in word['live']}|set(word['hosted']['X2'])|set(word['hosted']['Y2'])
+    paid={(e[2],e[3]) for e in word['program'] if e[0]=='MOVE'}
+    helpers={(word['initial'][r],word['final'][r]) for r in range(word['public']) if r not in main_fields}
+    charts={};independent={};records=[]
+    def chart(index):
+        if index not in charts:charts[index]=C.canonical_frame(9,tuple(word['frames'][index]))
+        return charts[index]
+    def frame(index):
+        if index not in independent:independent[index]=V.frame(word['frames'][index],9)
+        return independent[index]
+    catalogue=sorted(paid|helpers)
+    stop=len(catalogue) if args.stop is None else args.stop
+    need(0<=args.start<stop<=len(catalogue),'valid catalogue slice')
+    for i in range(args.start,stop):
+        a,b=catalogue[i]
+        recipe,_=C.lower_child(chart(a).adjoint().then(chart(b)))
+        need(recipe['rank']==abs(len(word['frames'][a])-len(word['frames'][b]))>0,'positive exact rank')
+        V.equal(V.recipe(recipe),V.adjoint(frame(a))+frame(b),'complete paid/helper map '+str((a,b)))
+        records.append([a,b,recipe])
+        if i%250==0:print('complete catalogue',i,flush=True)
+    main_records=[]
+    for x,q in (zip(word['live'],word['ports']) if stop==len(catalogue) else []):
+        a=word['initial'][x];b=word['final'][x+112]
+        left,_=C.lower_child(C.cgate(9,q,1).adjoint().then(chart(a)))
+        right,_=C.lower_child(chart(b).adjoint().then(C.cgate(9,q,-1).then(C.fourier(9))))
+        need(left['rank']==right['rank']==0,'main boundary adapters are monomial')
+        V.equal(V.recipe(left),(('C',q,-1),)+frame(a),'main input absolute phase')
+        V.equal(V.recipe(right),V.adjoint(frame(b))+(('C',q,-1),)+V.fourier(9),'main output absolute phase')
+        V.equal(frame(word['final'][x]),V.fourier(9),'main full Fourier endpoint')
+        main_records.append([q,left,right])
+    a,b,recipe=records[0]
+    wrong=copy.deepcopy(recipe);wrong['left']['phase']=(wrong['left']['phase']+1)%4
+    try:V.equal(V.recipe(wrong),V.adjoint(frame(a))+frame(b),'phase-only fault')
+    except ValueError:pass
+    else:raise ValueError('global phase control accepted')
+    result=dict(status='PASS_CURRENT_NATIVE_CATALOGUE_SLICE',
+        word_sha256=sha256(raw).hexdigest(),distinct_paid_maps=len(paid),distinct_helper_maps=len(helpers),
+        catalogue_size=len(catalogue),slice_start=args.start,slice_stop=stop,
+        catalogue_digest=sha256(json.dumps(catalogue,separators=(',',':')).encode()).hexdigest(),
+        checked_maps=len(records),main_boundary_pairs=len(main_records),
+        exact_recipe_digest=sha256(json.dumps(records,separators=(',',':'),sort_keys=True).encode()).hexdigest(),
+        main_boundary_digest=sha256(json.dumps(main_records,separators=(',',':'),sort_keys=True).encode()).hexdigest(),
+        phase_control_rejected=True,scope='Complete signed Pauli action plus exact Gaussian global phase; '
+        'for Clifford maps these determine every matrix column. Generic tensor-product and physical '
+        'grammar application is written in the review. No random-field substitution.')
+    print(json.dumps(result,indent=2))
+
+
+if __name__=='__main__':main()
diff --git a/research/literal-native-quadratic-phase/reference/selected-native.json.gz b/research/literal-native-quadratic-phase/reference/selected-native.json.gz
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diff --git a/research/literal-native-quadratic-phase/reference/sequential-phase-PROOF.md b/research/literal-native-quadratic-phase/reference/sequential-phase-PROOF.md
new file mode 100644
index 0000000000..42ffd4007c
--- /dev/null
+++ b/research/literal-native-quadratic-phase/reference/sequential-phase-PROOF.md
@@ -0,0 +1,167 @@
+# Sequential production of the actual monomial phase
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Source contract and result
+
+Jose Manuel's [PR #2250](https://github.com/CrocSwap/integer-mult-bounds/pull/2250)
+implements the Gaussian monomial with the retained
+[source phase formula](https://github.com/CrocSwap/integer-mult-bounds/blob/7c472563474e60185e6ca101a17622133d047e56/research/selected-complex-compact-adapters/monomial.py).
+Its phase_pass calls tensor_phase on decoded addresses. This package replaces
+that address oracle with sequential counter tapes. The retained algebra and
+Gaussian coefficient operations are not new results.
+
+Fix a compiled recipe of dimension m>=1. Its linear coefficients alpha_i are
+in Z/4, its quadratic bits Q_ij use only i<j, and its absolute phase is p0.
+The supplied physical mask has b>=1 binary symbols followed by END. Its
+ones identify active low-bit positions in each of m equal-width fields.
+For the source geometry, b=fK and the ones are rho+jK, 0<=j<f. The mask is
+read-only and starts at its origin. It is an explicit supplied input; this
+package does not manufacture it from encoded f,K,rho.
+
+Enumerate the m b-bit address fields lexicographically, with the last field
+fastest. Put R=2^(mb), and let x_ij be bit j of field i. The required phase is
+
+```text
+p(x) = f p0 + sum_(active j) [sum_i alpha_i x_ij
+                            + 2 sum_(i<k) Q_ik x_ij x_kj] mod 4.
+```
+
+The producer emits exactly this four-state phase at each record. It receives
+neither b, f, R nor a decoded address as runtime control data. It uses m owned
+counter tapes, one borrowed mask and fixed recipe tables. All heads start at
+their current segment origins after LEFT; arbitrary ancestor prefixes are
+preserved. At completion the counter segments are blank and all heads and the
+mask return exactly. No complete phase table is stored.
+
+The complete physical read/write/unit-move bill is at most
+
+```text
+32(m+1)(R+b+1).
+```
+
+For fixed recipe dimension, this is linear in the complete record count plus
+initial mask width. There is no per-record factor b or f. This is a concrete
+source-specific construction. Binary-counter amortization is classical;
+Cinna Davis's cached chirp construction and Ryanson Jonathan's
+[PR #4325](https://github.com/CrocSwap/integer-mult-bounds/pull/4325)
+are credited related predecessors. Their
+[pinned audit](https://github.com/CrocSwap/integer-mult-bounds/blob/7c472563474e60185e6ca101a17622133d047e56/research/pr4100-chirp-odometer-tariff/README.md)
+uses cached table products at a different port, not this quadratic bit phase.
+
+## Initialization and phase update
+
+Scan the physical mask once. For each bit, append zero to every counter.
+When the mask bit is one, add p0 to the phase state modulo four. Thus the
+initial phase is f p0 without decoding f. Append END to each counter and
+restore all heads. Each counter now holds the corresponding address field
+in least-significant-bit-first order.
+
+Advance the last field by a binary carry. At a changed bit j, all counter
+heads and the mask head stand at j. Read its old value x from counter i.
+If the mask is active, update the phase by
+
+```text
+delta = (1-2x) alpha_i + 2 sum_(k != i) Q_(min(i,k),max(i,k)) x_kj mod 4.
+```
+
+Then replace x by 1-x. The linear part changes by the displayed signed term.
+Each quadratic term changes by plus or minus 2 x_kj; these are equal modulo
+four. No diagonal or lower-triangular raw Q entry is used. Thus this update
+is exactly the difference of the retained source polynomial.
+
+If x was zero, the increment is complete. Rewind all heads to their current
+LEFT markers. If x was one, move every counter and mask one cell right and
+continue the carry. At mask END, check all counter END symbols, rewind, and
+carry into the preceding field. When all fields wrap, the complete domain
+has been consumed. The counters are zero again, and the phase returns to its
+initial value. Erase every counter bit and END, then restore its head.
+
+Induction over bit flips proves phase correctness. Standard binary carry
+induction proves lexicographic order, no omitted or repeated record, and exact
+termination after R outputs. All variable-sized progress is encoded in head
+positions and counter cells. Recipe indexes, current symbols, phase and carry
+range over fixed finite sets. The executable is a finite-symbol program; a
+separate native transition-table compiler or proof-kernel artifact is not supplied.
+
+## Complete action and space proof
+
+A rewind from offset h costs 2h+3 actions. Initialization, including checking
+empty counter owners, writing END and restoring the borrowed mask, costs
+
+```text
+(4m+4)b + 8m+4.
+```
+
+Consider one field increment which changes t>=1 bits. Each changed bit costs
+at most m+2 reads/writes for the mask, old bit, cross terms and update. Carry
+movement costs at most (m+1)t. The final rewinds cost at most
+(m+1)(2t+3). END checks cost at most m+1. The total is at most
+(4m+5)t+4(m+1), hence at most 9(m+1)t.
+
+Across all R increments, including the final wrap, the global mb-bit counter
+changes exactly 2R-2 bits. Grouping them by field does not change that sum.
+Therefore all advance operations cost less than 18(m+1)R. Counter cleanup
+costs m(5b+9), including each erased END, blank check and final rewind.
+Combining these quantities gives the stated conservative bound.
+
+The owned counter bodies use m(b+1) cells, plus their LEFT markers. The
+borrowed mask uses b+1 body cells plus its LEFT marker. Blank counter backing
+remains physical capacity after cleanup. Fixed recipe tables contain O(m^2)
+compiled bits. There is no R-entry phase cache. The caller consumes each
+four-state output before advancing; it must pay for any retained copy.
+
+## Actual Gaussian consumer
+
+apply_phase retains the source's real/imaginary coefficient loop, with two
+coefficient scratch tapes and one-bit negation carry. Its loop is driven by
+physical counter overflow, not layout.records or layout.decode. It consumes
+one Gaussian coefficient per record in the demonstrated ABI. Multiple packed
+coefficients require a separately bound repetition grammar.
+
+For coefficient width p>=2 and V=2pR payload bits, the retained coefficient
+path uses exactly 18V source tape actions when erases are counted as one
+operation per erased cell. This includes input consumption and erasure,
+coefficient scratch cleanup, and output head return. The new complete local
+phase/coefficient interval therefore costs at most
+
+```text
+18V + 32(m+1)(R+b+1).
+```
+
+The input is owned and consumed. A caller which must preserve it needs a paid
+copy. Borrowed mask and dirty counter ancestors are preserved. The output is
+separately owned; its V cells and the two p-cell coefficient scratch bodies
+remain charged. Coefficient primitives retain their source representation
+and precision assumptions. Equality is exact modulo 2^p; ordinary Gaussian
+integer interpretation also needs the source's symmetric headroom condition.
+
+For an inverse monomial, apply the negative phase after the affine inverse
+has recovered the input coordinates. This producer does not authorize phase
+evaluation on unrecovered output coordinates. Zero-coefficient equal-width
+spectator fields can be inserted into the compiled recipe; their counters
+still determine physical record order. Other mixed-width packing geometries
+remain a caller integration obligation.
+
+## Evidence and limits
+
+The actual recipe is the retained family 65689/event 8814 right recipe, with
+offset 426 and absolute phase 1. Its affine map is not executed by this package;
+only its phase is supplied here. The source binding and native scalar theorem
+remain inherited from the pinned package, not independently rederived.
+
+The verifier checks 224 generic complete domains with 9760 records. Three
+actual source domains contain 512,262144,262144 records. They cover f=1,K=1,
+f=2,K=1, and f=1,K=2,rho=1, including inactive bits and the nonzero tensor
+constant. Six 1024-record payload cases include an independent spectator bit,
+both signs and dirty ancestor lengths 16,257,4096. They match the unchanged
+source output, a separate modular Gaussian oracle and every coefficient
+meter field. Each adds 120506 physical phase actions. Six malformed-state
+controls reject. Failed ABI calls need not restore partial work.
+
+Mask manufacture, arbitrary mixed-width caller packing, affine routing,
+recursive C, physical owner allocation, production codebook, full monomial
+invoice, native/stock/precision and outer recovery remain separate. No full
+147-bit production payload was materialized. The complete kappa calculation
+and selected frontier are unchanged. Independent human and proof-kernel
+acceptance remain open.
diff --git a/research/literal-native-quadratic-phase/reference/sequential-phase.py b/research/literal-native-quadratic-phase/reference/sequential-phase.py
new file mode 100644
index 0000000000..2612604528
--- /dev/null
+++ b/research/literal-native-quadratic-phase/reference/sequential-phase.py
@@ -0,0 +1,125 @@
+"""Sequential counter production of source Gaussian monomial phases.
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+The tensor phase follows Jose Manuel's retained monomial implementation.
+Counter amortization retains Cinna Davis and the PR4325 audit's credit.
+"""
+ZERO,ONE,LEFT,END,BLANK=range(5)
+
+
+def need(ok,message):
+    if not ok:raise ValueError(message)
+
+
+class Meter:
+    def __init__(self):self.reads=self.writes=self.moves=0
+    @property
+    def total(self):return self.reads+self.writes+self.moves
+
+
+class Tape:
+    """Hardware model; only current-symbol operations are algorithm inputs."""
+    def __init__(self,prefix=(),meter=None):
+        self.cells=list(prefix)+[LEFT];self.head=len(self.cells);self.floor=self.head-1
+        self.meter=meter if meter is not None else Meter()
+    def read(self):
+        need(self.head>=self.floor,'ancestor read');self.meter.reads+=1
+        return self.cells[self.head] if self.head<len(self.cells) else BLANK
+    def write(self,value):
+        need(value in range(5) and self.head>=self.floor,'owned finite-symbol write');self.meter.writes+=1
+        while len(self.cells)<=self.head:self.cells.append(BLANK)
+        self.cells[self.head]=value
+    def move(self,direction):
+        need(direction in (-1,1),'unit move');self.head+=direction;self.meter.moves+=1
+        need(self.head>=self.floor,'ancestor crossed')
+
+
+def emit(tape,value):tape.write(value);tape.move(1)
+
+
+def rewind(tape):
+    tape.move(-1)
+    while tape.read()!=LEFT:tape.move(-1)
+    tape.move(1)
+
+
+class Phases:
+    """The recipe is compiled data; mask length and addresses stay on tapes."""
+    def __init__(self,recipe,mask,counters):
+        self.alpha=tuple(recipe['linear_Z4']);self.quadratic=tuple(recipe['quadratic_rows'])
+        self.counters=tuple(counters);self.mask=mask;self.phase=0;self.done=False
+        m=len(self.alpha)
+        need(m>=1 and len(self.quadratic)==len(self.counters)==m,'compiled recipe extent')
+        need(all(type(a) is int and 0<=a<4 for a in self.alpha),'compiled linear phases')
+        need(all(type(a) is int and 0<=a<1<<m for a in self.quadratic),'compiled quadratic rows')
+        need(type(recipe['phase']) is int and 0<=recipe['phase']<4,'compiled absolute phase')
+        for tape in self.counters:need(tape.read()==BLANK,'empty counter owner')
+        nonempty=False
+        while (active:=mask.read())!=END:
+            need(active in (ZERO,ONE),'binary active mask');nonempty=True
+            if active:self.phase=(self.phase+recipe['phase'])%4
+            for tape in self.counters:emit(tape,ZERO)
+            mask.move(1)
+        need(nonempty,'positive counter width')
+        for tape in self.counters:emit(tape,END);rewind(tape)
+        rewind(mask)
+    def restore_heads(self):
+        for tape in self.counters:rewind(tape)
+        rewind(self.mask)
+    def advance(self):
+        need(not self.done,'live complete phase domain')
+        for i in range(len(self.counters)-1,-1,-1):
+            while (active:=self.mask.read())!=END:
+                need(active in (ZERO,ONE),'binary active mask')
+                bit=self.counters[i].read();need(bit in (ZERO,ONE),'binary counter')
+                if active:
+                    delta=self.alpha[i] if bit==0 else -self.alpha[i]
+                    for j,tape in enumerate(self.counters):
+                        if j!=i and (self.quadratic[min(i,j)]>>max(i,j))&1:
+                            other=tape.read();need(other in (ZERO,ONE),'aligned binary counter')
+                            delta+=2*other
+                    self.phase=(self.phase+delta)%4
+                self.counters[i].write(1-bit)
+                if bit==0:self.restore_heads();return True
+                for tape in self.counters:tape.move(1)
+                self.mask.move(1)
+            for tape in self.counters:need(tape.read()==END,'equal counter extents')
+            self.restore_heads()
+        self.done=True
+        return False
+    def close(self):
+        need(self.done,'complete phase consumption')
+        for tape in self.counters:
+            while (symbol:=tape.read())!=END:
+                need(symbol in (ZERO,ONE),'counter cleanup');emit(tape,BLANK)
+            emit(tape,BLANK);need(tape.read()==BLANK,'counter end');rewind(tape)
+
+
+def apply_phase(source,producer,coefficient_bits,TapeClass,copy,inverse=False):
+    """Retained Gaussian coefficient loop, driven by physical phase overflow.
+
+    This is adapted from the pinned monomial.phase_pass. Sequential coefficient
+    primitives and their precision contract remain inherited.
+    """
+    need(coefficient_bits>=2,'Gaussian coefficient extent')
+    source.rewind();out=TapeClass(source.meter);real=TapeClass(source.meter);imag=TapeClass(source.meter)
+    def coefficient(tape,negative):
+        if negative:
+            carry=1
+            for _ in range(coefficient_bits):
+                value=(1-tape.read(1)[0])+carry
+                out.write(bytes((value&1,)));carry=value>>1
+        else:copy(tape,out,coefficient_bits)
+        tape.finish()
+    while True:
+        phase=(-producer.phase if inverse else producer.phase)%4
+        source.meter.control_evaluations+=1
+        copy(source,real,coefficient_bits);copy(source,imag,coefficient_bits)
+        real.rewind();imag.rewind()
+        if phase==0:coefficient(real,False);coefficient(imag,False)
+        elif phase==1:coefficient(imag,True);coefficient(real,False)
+        elif phase==2:coefficient(real,True);coefficient(imag,True)
+        else:coefficient(imag,False);coefficient(real,True)
+        real.erase();imag.erase()
+        if not producer.advance():break
+    source.finish();source.erase();out.rewind();producer.close();return out
diff --git a/research/literal-native-quadratic-phase/verify.py b/research/literal-native-quadratic-phase/verify.py
new file mode 100644
index 0000000000..fd30072812
--- /dev/null
+++ b/research/literal-native-quadratic-phase/verify.py
@@ -0,0 +1,201 @@
+"""Bounded complete-packet replay and direct polynomial/integer oracles.
+
+The oracle enumerates addresses independently; it never uses the table's
+incremental phase update. Prepared for CubeJerry with OpenAI assistance.
+"""
+from pathlib import Path
+import hashlib
+import importlib.util
+import itertools
+import json
+import sys
+
+HERE = Path(__file__).resolve().parent
+sys.path.insert(0, str(HERE))
+import phase as P
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def gamma(value):
+    bits = bin(value)[2:]
+    return '0'*(len(bits)-1)+bits
+
+
+def phases(m, f, linear, rows, constant):
+    radix = 1 << f
+    out = []
+    for address in itertools.product(range(radix), repeat=m):
+        total = 0
+        for column in range(f):
+            bits = [(address[i] >> column) & 1 for i in range(m)]
+            total += constant + sum(a*x for a, x in zip(linear, bits))
+            total += 2*sum(bits[i]*bits[j] for i in range(m)
+                           for j in range(i+1, m) if (rows[i] >> j) & 1)
+        out.append(total % 4)
+    return out
+
+
+def rotate(z, mode):
+    # Independent Gaussian-integer multiplication, not the table's case list.
+    a, b = z
+    c, d = ((1, 0), (0, 1), (-1, 0), (0, -1))[mode]
+    return a*c-b*d, a*d+b*c
+
+
+def encode(values, w):
+    return ''.join(format(x % (1 << w), '0'+str(w)+'b')
+                   for z in values for x in z)
+
+
+def packet(f, r, outer, spectator, w, linear=P.SELECTED_LINEAR,
+           rows=P.SELECTED_ROWS, constant=P.SELECTED_CONSTANT,
+           values=None, code=None, corrupt=None, origins=None):
+    m = len(linear)
+    modes = phases(m, f, linear, rows, constant)
+    modes = [q for _ in range(outer) for q in modes
+             for _ in range(spectator) for _ in range(r)]
+    if values is None:
+        cap = (1 << (w-1))-1
+        values = [((7*j+3) % (2*cap+1)-cap,
+                   (11*j+1) % (2*cap+1)-cap) for j in range(len(modes))]
+    need(len(values) == len(modes), 'complete payload extent')
+    expected = [rotate(z, q) for z, q in zip(values, modes)]
+    low, high = -(1 << (w-1)), (1 << (w-1))-1
+    fits = all(low <= x <= high for z in expected for x in z)
+    header = ''.join(gamma(x) for x in (f, r, outer, spectator, w))
+    data = encode(values, w)
+    V, Q, E = len(data), 1 << (m*f), len(header)
+    if origins is None:
+        origins = tuple(17-7*i for i in range(23))
+    tapes = []
+    for i, origin in enumerate(origins):
+        span = V+E+sum((f, r, outer, spectator, w))+20
+        cells = {origin-5: '1', origin-4: '|', origin+span: '0'}
+        if i == P.HEADER:
+            cells.update({origin+j: b for j, b in enumerate(header)})
+        elif i == P.DATA:
+            cells.update({origin+j: b for j, b in enumerate(data)})
+        tapes.append(P.Tape(cells, origin))
+    before = [dict(t.cells) for t in tapes]
+    table = P.compile_phase(linear, rows, constant)[0] if code is None else code
+    if corrupt:
+        table = corrupt(table)
+    machine = P.Machine(table, tapes)
+    budget = 65536*(E+(m+1)*f+r+outer+spectator+w+(m+1)**2*outer*Q+V+1)
+    rejected = False
+    try:
+        machine.run(budget)
+    except ValueError as error:
+        need(str(error) == 'invalid scanned symbol', 'unexpected rejection '+str(error))
+        rejected = True
+    need(rejected == (not fits), 'exact signed acceptance')
+    need(tapes[P.HEADER].cells == before[P.HEADER], 'header preserved')
+    if rejected:
+        need(tapes[P.DATA].cells == before[P.DATA], 'all public data preserved on rejection')
+    else:
+        goal = dict(before[P.DATA])
+        goal.update({origins[P.DATA]+i: bit for i, bit in enumerate(encode(expected, w))})
+        need(tapes[P.DATA].cells == goal, 'independent direct phase/integer endpoint')
+        need(all(t.head == origin for t, origin in zip(tapes, origins)), 'all heads restored')
+        need(all(t.cells == old for t, old in zip(tapes[2:], before[2:])), 'private blank and opaque cells restored')
+    delta_history = sum(t.high-t.low for t in tapes)
+    # Public bodies are initially backed; this intentionally generous check
+    # also includes their full scans rather than subtracting initial backing.
+    history_bound = 3*V+5*E+(m+4)*f+4*(r+outer+spectator)+8*w+256
+    need(delta_history <= history_bound, 'historical extent bound')
+    return {'steps': machine.steps, 'rejected': rejected, 'V': V,
+            'history': delta_history, 'modes': sorted(set(modes))}
+
+
+def fixed_table_check(code, alphabet):
+    need(tuple(alphabet) == ('_', '0', '1', '|'), 'fixed alphabet')
+    for op, *args in code:
+        need(op in ('read', 'write', 'move', 'goto', 'halt'), 'literal instruction')
+        if op in ('read', 'write', 'move'):
+            need(type(args[0]) is int and 0 <= args[0] < 23, 'fixed physical port')
+        if op == 'read':
+            need(set(args[1]) <= set(alphabet), 'read symbols')
+            need(all(type(k) is int and 0 <= k < len(code) for k in args[1].values()), 'fixed branch')
+        if op == 'write':
+            need(args[1] in alphabet, 'write symbol')
+        if op == 'move':
+            need(args[1] in (-1, 1), 'unit move')
+        if op == 'goto':
+            need(type(args[0]) is int and 0 <= args[0] < len(code), 'fixed goto')
+
+
+def main():
+    code, alphabet = P.compile_phase()
+    fixed_table_check(code, alphabet)
+    recipe = json.loads((HERE/'RECIPE.json').read_text())
+    need(tuple(recipe['recipe']['linear_Z4']) == P.SELECTED_LINEAR and
+         tuple(recipe['recipe']['quadratic_rows']) == P.SELECTED_ROWS and
+         recipe['recipe']['phase'] == P.SELECTED_CONSTANT, 'selected recipe constants')
+    results = []
+    # Genuine complete selected m=9 arrays at f=1; all 512 addresses.
+    for args in ((1,1,1,1,2), (1,2,2,2,5), (1,1,1,1,1), (1,1,1,1,17)):
+        results.append(packet(*args, code=code))
+    # More than one column in complete lower-dimensional specializations.
+    for m in (1,2,3):
+        rows = tuple((1 << m)-1 for _ in range(m))
+        linear = tuple(1+2*(i % 2) for i in range(m))
+        for f in (1,2,3):
+            for constant in range(4):
+                results.append(packet(f,2,2,2,3,linear,rows,constant))
+    # Every pair at widths 1..3, for every phase. Constant f=1 gives
+    # the same phase at both addresses and isolates exact endpoint handling.
+    endpoints = 0
+    for w in (1,2,3):
+        interval = range(-(1 << (w-1)), 1 << (w-1))
+        for mode in range(4):
+            table = P.compile_phase((0,), (0,), mode)[0]
+            for pair in itertools.product(interval, repeat=2):
+                results.append(packet(1,1,1,1,w,(0,),(0,),mode,[pair,pair],table))
+                endpoints += 1
+    controls = []
+    # Wrong constant per column; f=2 must retain i^(2*q0).
+    wrong = P.compile_phase((1,1), (3,3), 0)[0]
+    try:
+        packet(2,1,1,1,3,(1,1),(3,3),1,code=wrong)
+    except ValueError:
+        controls.append('drop_repeated_constant')
+    # Dropping one actual quadratic edge while retaining parity/symmetry.
+    rows = list(P.SELECTED_ROWS);rows[0] ^= 2;rows[1] ^= 1
+    wrong = P.compile_phase(P.SELECTED_LINEAR,tuple(rows),P.SELECTED_CONSTANT)[0]
+    try:
+        packet(1,1,1,1,3,code=wrong)
+    except ValueError:
+        controls.append('drop_actual_quadratic_pair')
+    # Changing one odd coefficient reverses its zero/one-flip contribution.
+    linear = list(P.SELECTED_LINEAR);linear[0] = 1
+    wrong = P.compile_phase(tuple(linear),P.SELECTED_ROWS,P.SELECTED_CONSTANT)[0]
+    try:
+        packet(1,1,1,1,3,code=wrong)
+    except ValueError:
+        controls.append('wrong_actual_linear_sign')
+    # Corrupt the actual commit instruction, keeping input and oracle intact.
+    wrong = list(code)
+    index = next(i for i, row in enumerate(wrong) if row == ('write', P.DATA, '0'))
+    wrong[index] = ('write', P.DATA, '1')
+    try:
+        packet(1,1,1,1,3,code=tuple(wrong))
+    except ValueError:
+        controls.append('commit_bit_corruption')
+    need(len(controls) == 4, 'all semantic mutants rejected')
+    out = {'status':'PASS_LITERAL_NATIVE_QUADRATIC_PHASE_NO_NEW_KAPPA',
+           'source_sha256':hashlib.sha256((HERE/'phase.py').read_bytes()).hexdigest(),
+           'table_locations':len(code), 'tapes':23,
+           'table_sha256':hashlib.sha256(json.dumps(code,separators=(',',':'),sort_keys=True).encode()).hexdigest(),
+           'complete_packets':len(results), 'selected_complete_packets':4,
+           'signed_endpoint_pairs':endpoints, 'range_rejections':sum(x['rejected'] for x in results),
+           'literal_transitions':sum(x['steps'] for x in results), 'negative_controls':controls,
+           'scope':'Finite evidence; universal f, payload, head and cost statements are in PROOF.md.'}
+    print(json.dumps(out,indent=2,sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()