Prove degree-881 payload involutions split a sector-3 joint space

research/c3-payload-sector3-split/PROOF.md

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+# Payload involutions that fix sector 3 and split its joint spaces
+
+This note answers the second open question of the c=3 sign-character
+certificate: whether a payload involution from the degree-881 one-slot
+certificate fixes sector 3 and splits a joint space of the explicit pair.
+The selected conditional kappa stays `781451/250000000`. Nothing here is a
+characteristic polynomial, a basis, or a census of every operator outside
+the signed-permutation group.
+
+The claim is a **proof**. The trace formula is an identity for commuting
+involutions. The port operators, the 512 addresses, and the 64-element
+group below are finite sets, and the verifier evaluates the identities on
+every element of each set. That is not a heuristic search, and it is not an
+exhaustive census of a larger matrix group.
+
+## Operators
+
+The input is the pinned 8-by-112 query, SHA256
+`5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3`.
+Its columns lie in `{-1,0,1}^8`, each has weight 4, and `R R^T = 56 I_8`.
+The 112 port labels are distinct odd-weight vectors in `F_2^9`.
+
+Bell block `c=3` is the 4096-dimensional space `C^8 ⊗ C^{512}`. Write
+`e=511` and
+
+    B_3 = sum_q (r_q r_q^T) ⊗ P_q(3),
+
+where the sum runs over the 112 ports and, on an address `a`,
+
+    P_q(3) e_a = e_{a+q}                 if q·3 = 1,
+    P_q(3) e_a = (-1)^{q·a} e_a          if q·3 = 0.
+
+The degree-881 certificate attaches two payload involutions to this block:
+
+    X_3 e_a = e_{a+3},
+    Z_{508} e_a = (-1)^{(3+e)·a} e_a.
+
+Both act as the identity on the eight coordinates. Here `3+e = 508` and
+`3·508 = 0`, so the two operators commute and their product is an
+involution. In the certificate's sector notation, the eigenvalue of `X_3`
+is `s` and the eigenvalue of `Z_{508}` is `(-1)^t`.
+
+The explicit pair from the c=3 certificate is
+
+| name | permutation | minus signs |
+|---|---|---|
+| `u` | `(0 2)(4 6)` | coordinates 0 and 2 |
+| `v` | `(0 4)(2 6)` | coordinates 0 and 4 |
+| `uv` | `(0 6)(2 4)` | coordinates 0 and 6 |
+
+`H` is the order-16 group of diagonal signs that are positive on the even
+coordinates `{0,2,4,6}` and arbitrary on the odd coordinates `{1,3,5,7}`.
+The 64 products of `{1,u,v,uv}` with `H` are the operators used below. Each
+one is rebuilt from the query as a signed coordinate permutation whose
+induced port map `T` is linear and orthogonal on `F_2^9`.
+
+## They lie outside the signed-permutation representation
+
+A symmetry coming from a signed coordinate permutation acts as `S ⊗ U_T`,
+and `T` is linear, so `U_T` fixes the address `0`. Then
+
+    (S ⊗ U_T)(e_0 ⊗ e_0) = ε[σ(0)] e_{σ(0)} ⊗ e_0.
+
+`X_3` sends that same vector to `e_0 ⊗ e_3`. The address moved, so `X_3` is
+not equal to any such symmetry. `Z_{508}` fixes every address and multiplies
+`e_0` by `+1` but `e_4` by `-1`, because bit 2 of 508 is set. A coordinate
+sign does not depend on the address, so `Z_{508}` is not a signed
+permutation either. The same address translation shows that the product is
+not one. This is why the order-3072 stabilizer census does not decide these
+operators.
+
+The certificate defines `X_c` and `Z_{c+e}` as operators on block `c`. For
+`c ≠ 3` that block is a different summand of the Bell decomposition. Those
+operators are not endomorphisms of sector 3. The Walsh transform sends label
+3 to 508, so it does not fix the block either. The named payload involutions
+from the certificate that fix sector 3 are exactly `X_3` and `Z_{508}`.
+Their product is the remaining nontrivial word in those two operators. It
+fixes the same block, and the trace table below shows that it splits every
+joint space of this pair as well.
+
+## They commute with B_3 and with the joint projectors
+
+Every port has odd weight, so `q·e = 1` and `q·508 = q·3 + 1`. If `q·3 = 1`,
+then `q·508 = 0`: translation by `q` commutes with the diagonal character
+`Z_{508}`. If `q·3 = 0`, then `P_q(3)` is itself diagonal and commutes with
+`Z_{508}`. Translations commute with `X_3`, and a diagonal `P_q(3)` has the
+same sign at `a` and at `a+3` precisely because `q·3 = 0`. Thus both payload
+involutions commute with every `P_q(3)`, and therefore with `B_3`. The
+verifier also applies both sides to all 512 addresses for every port.
+
+Each of the 64 signed permutations was rebuilt from the query. For each one,
+`T(3) = 3`, `T(508) = 508`, and `T(511) = 511`. The transpose of `T` inverts
+`T` on every address. Consequently `T(a+3) = T(a)+3` and
+`508·T(a) = 508·a`, so `S ⊗ U_T` commutes with both payload involutions.
+
+The same rebuild gives the conjugation identity on addresses,
+
+    U_T P_q(3) U_T^{-1} = P_{Tq}(3),
+
+checked for every port and every address. On coordinates, the signed
+permutation sends each column to `±` a column, so it permutes the rank-one
+tensors `r_q r_q^T`. The sum `B_3` is therefore invariant. The spectral
+projections of `u`, `v`, and the elements of `H` are polynomials in those
+symmetries, so they commute with `B_3`. The payload involutions commute with
+those symmetries, so they preserve every joint eigenspace, and `B_3`
+preserves each intersection of a joint eigenspace with an eigenspace of
+`X_3` or `Z_{508}`.
+
+`X_3` is diagonalizable because it is a fixed-point-free involution of the
+addresses: each pair `{a, a+3}` contributes eigenvalues `±1`. `Z_{508}` is
+already diagonal in the address basis. Their product is an involution on
+each of those pairs because `3·508 = 0`, so it is diagonalizable as well.
+The minimal polynomial of each restriction divides `x^2-1`, which has
+distinct roots, so the restrictions to an invariant joint space remain
+diagonalizable. For an involution of trace `τ` on a space of dimension `d`,
+the eigenvalue dimensions are `(d+τ)/2` and `(d-τ)/2`.
+
+## Traces
+
+On a signed permutation `g = S ⊗ U_T`, the block trace of
+`I ⊗ (τ_b χ_f) ∘ g` factors as the product of the coordinate trace and the
+address trace. The coordinate trace is the signed count of fixed
+coordinates. The address trace counts solutions of `(T+I)a = b`, weighted
+by `(-1)^{f·T(a)}`. The verifier computes that count by enumerating all 512
+addresses and, separately, by solving the affine equation and summing the
+character on its kernel. The two counts agree on every element and every
+operator used here.
+
+Let `P_{χ,ε,δ}` be the projector onto the `χ`-isotypic component of `H` and
+the `(ε,δ)` joint eigenspace of `(u,v)`. Since `H`, `u`, and `v` commute and
+are involutions,
+
+    tr(W P_{χ,ε,δ})
+      = (1/64) sum_{h in H} χ(h)
+        [tr(Wh) + ε tr(Wuh) + δ tr(Wvh) + εδ tr(Wuvh)].
+
+The same formula with `W = I` reproduces the published joint dimensions,
+including `216^4` on the trivial character and `8^4` on the unique
+32-dimensional character. The character dimensions themselves are the
+published histogram
+
+    {864: 1, 432: 4, 192: 6, 80: 4, 32: 1}.
+
+Summing the four `(s,t)` intersections over all 16 characters and all four
+`(u,v)` eigenvalues rebuilds four blocks of dimension 1024, which are the
+degree-881 sectors of this Bell block. The `+1` space of `X_3` on the whole
+block has dimension 2048, matching its global trace 0.
+
+## The split
+
+`Z_{508}` and the product `X_3 Z_{508}` have trace 0 on every joint space of
+this pair. Every joint dimension below is positive and even, so each of
+those two involutions splits every joint space in half.
+
+`X_3` does not act by a scalar on any of them. Its traces depend only on the
+character weight and on the `(u,v)` eigenvalues:
+
+| weight | dimensions `(++,+-, -+, --)` | traces of `X_3` |
+|---:|---|---|
+| 0 | 216, 216, 216, 216 | -48, 16, 16, 16 |
+| 1 | 180, 84, 84, 84 | 36, -12, -12, -12 |
+| 2 | 48, 48, 48, 48 | -24, 8, 8, 8 |
+| 3 | 44, 12, 12, 12 | 12, -4, -4, -4 |
+| 4 | 8, 8, 8, 8 | 0, 0, 0, 0 |
+
+On the trivial character, the `(+1,+1)` space has dimension 216 and `X_3`
+has trace `-48`. The eigenvalue `+1` therefore has dimension **84** and the
+eigenvalue `-1` has dimension **132**. Both summands are `B_3`-invariant.
+`Z_{508}` splits the same space as **108 + 108**. The address trace of
+`Z_{508}`, and of the product, is already 0 on each of the 64 group
+elements, so the joint traces vanish before the character average. Because
+those traces are 0, `Z_{508}` halves each `X_3` eigenspace. The
+intersections with the four degree-881 sectors `(s,t)` are
+
+| `(s,t)` | dimension inside the 216-space |
+|---|---:|
+| `(+1, 0)` | 42 |
+| `(+1, 1)` | 42 |
+| `(-1, 0)` | 66 |
+| `(-1, 1)` | 66 |
+
+All four are positive, so this joint space meets every sector of the
+degree-881 splitting of block 3. The same four-way count is positive on
+every other joint space of this pair. In particular each 8-dimensional
+weight-4 joint space meets all four sectors in dimension 2. That is a
+dimension count. It is not the characteristic polynomial of an
+8-dimensional joint space, and it is not the characteristic polynomial of a
+16-dimensional sum of two of them.
+
+The other three trivial joint spaces have `X_3` trace 16, hence split as
+116 + 100, and their sector intersections are 58, 58, 50, and 50.
+
+## What is not claimed
+
+The argument uses this query, Bell block 3, the named operators `X_3` and
+`Z_{508}`, the order-16 group `H`, and the explicit pair `(u,v)`. It does
+not list a basis and does not compute a minimal or characteristic
+polynomial. It does not say that an arbitrary operator outside the
+signed-permutation group splits these spaces. It does not rerun the
+degree-881 coefficient certificate. A 42-dimensional invariant subspace
+does not replace that certificate. The selected conditional kappa is
+unchanged.
+
+The rejection controls are part of the verifier, not part of the theorem. A
+translation by 1 does not commute with `u`. The address character of the
+vector 1 does not commute with every port operator. Dropping one element of
+`H` does not reproduce dimension 864. The identity trace on the 216-space is
+216, not `-48`. The coordinate transposition `(0 1)` is not a symmetry of
+the query.

research/c3-payload-sector3-split/README.md

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+# Payload involutions split a sector-3 joint space
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.
+
+For the historical 8-by-112 centre query, the degree-881 payload involutions
+that fix Bell block `c=3` are `X_3` and `Z_{508}`. Both lie outside the
+signed-permutation group classified for this block. On the explicit
+commuting pair from that classification, `X_3` splits the trivial
+`(+1,+1)` joint space of dimension 216 into `B_3`-invariant subspaces of
+dimensions **84** (eigenvalue `+1`) and **132** (eigenvalue `-1`).
+`Z_{508}` splits the same space into dimensions **108 and 108**. The
+intersections of that 216-space with the four degree-881 sectors have
+dimensions **42, 42, 66, and 66**.
+
+The selected conditional kappa stays `781451/250000000`. [PROOF.md](PROOF.md)
+separates the trace-formula proof from the facts that are not claimed.
+
+## Result
+
+The live input is the query pinned in [SOURCES.json](SOURCES.json), SHA256
+`5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3`.
+`X_3` translates addresses by 3. `Z_{508}` is the diagonal character of
+`3 xor 511`. They commute with each other, with `B_3`, and with the
+order-64 group generated by the explicit pair
+
+    u = (0 2)(4 6) with minus signs on coordinates 0 and 2,
+    v = (0 4)(2 6) with minus signs on coordinates 0 and 4,
+
+together with the order-16 sign group that is positive on the even
+coordinates. The joint dimensions of that pair match the published c=3
+table, including `216^4` on the trivial character.
+
+`X_3` has trace `-48` on the trivial `(+1,+1)` space. Its `+1` eigenspace
+there has dimension 84 and its `-1` eigenspace has dimension 132. The same
+trace formula gives both eigenvalues on every other joint space of this
+pair. The address trace of `Z_{508}`, and of the product, is 0 on each of
+the 64 group elements, so every joint trace of those two operators is 0
+and each joint space is split in half. Inside the 216-space, the
+intersections with the four degree-881 sectors are 42, 42, 66, and 66, for
+`(s,t)` equal to `(+1,0)`, `(+1,1)`, `(-1,0)`, and `(-1,1)`. Summing those
+intersections over every character and every joint eigenvalue rebuilds the
+four 1024-dimensional sectors.
+
+This is a proof. The address counts are exhaustive on a set of 512
+elements, once by enumeration and once by an affine solve, and the two
+counts agree. It is not a stopped search and not a claim that every
+operator outside the signed-permutation group splits these spaces.
+
+## Scope
+
+Catalogue card `formal-interface-lemma` asks for a formal interface binding.
+The live question is open question 2 of
+[PR #6965](https://github.com/CrocSwap/integer-mult-bounds/pull/6965),
+recorded in discussion comment
+[19507308](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19507308).
+The predecessor classifies signed permutations only. This package starts
+from the payload involutions named in Krzysztof Wojt's degree-881
+certificate and tests the ones whose block is sector 3.
+
+It does not redo that signed-permutation census, and it does not compute a
+characteristic polynomial. The 16-dimensional weight-4 calculation left
+open by #6965 stays open; the trace table here only records that each
+8-dimensional weight-4 joint space meets the four sectors in dimension 2.
+The 216-dimensional `c=1` polynomial is a different block.
+
+Selected-baseline orientation returned `PASS_RESEARCH_BASELINE_BINDINGS_ONLY`
+for kappa `781451/250000000`.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+python3 -B -W error research/c3-payload-sector3-split/verify.py
+python3 -B -O research/c3-payload-sector3-split/verify.py
+python3 -B scripts/research_package.py --package research/c3-payload-sector3-split --check-only
+```
+
+The direct marker is `PASS_C3_PAYLOAD_SECTOR3_SPLIT`. Acceptance uses
+explicit checks, so `-O` does not skip them. The run rebuilds the 64
+symmetries from the query, checks commutation on every port and every
+address, and evaluates the joint trace formula. On the measured host that
+takes about five seconds. Rejection controls are a translation by 1, an
+address character other than 508, a truncated sign average, the identity
+trace on the same 216-space, and a coordinate transposition. No Lean,
+Docker, degree-881 coefficient replay, or selected cold replay was run.
+
+## Overlap
+
+`scripts/find_overlap.py` was refreshed on 2026-11-10 against upstream
+`17469fff78c22941280f04d39dc5a1faee09c840`. Coverage: local notes 10178 of
+10178 HEAD paths, complete; 317 open PRs, pagination complete, fetched
+2026-11-10T08:48:35Z; 6971 discussion records, pagination complete, fetched
+2026-11-10T08:49:37Z. Closed PRs were searched through local READMEs, task
+metadata and the ledger only.
+
+The phrase `payload involution` matches the degree-881 certificate, the
+#6965 finding that leaves this question open, and the two local notes that
+point at it. No open PR body reports a trace of `X_3` or `Z_{508}` on a
+sector-3 joint space. This search is not a novelty certificate.
+
+## Open next questions
+
+The structured list is in [TASK.json](TASK.json).
+
+- The characteristic polynomial of the 84-dimensional `X_3 = +1` subspace
+  of the trivial `(+1,+1)` joint space is still uncomputed. It would place
+  this split inside the known degree-881 polynomial. The trace formula
+  gives the dimension without a basis.
+- One other cross-coset pair, with a different odd sign, might move the
+  trace `-48` onto another `(u,v)` eigenvalue. This package fixes the
+  explicit masks from #6965.
+
+## Credit
+
+JoyboyBrian's handoff recorded the sign-character dimensions. The explicit
+pair, the joint dimensions, and the signed-permutation boundary are the
+certificate in upstream PR #6965. Ali Romman's PR4958 and PR4720 retain the
+query and the Bell action. Nick Petalas's PR5120 is the degree-410 lower
+bound. Krzysztof Wojt's exact one-slot package defines `X_c` and `Z_{c+e}`
+and is the degree-881 annihilator. The doubled representation comes from
+PR1719. New work is Apache-2.0.

research/c3-payload-sector3-split/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "17469fff78c22941280f04d39dc5a1faee09c840",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/c3-centre-sign-split/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "8da44a79e0e3f7d3cfcd7ebb8c46cc46146a9869",
+      "sha256": "6d9446760eb90201b96a73c78a04fdc0320c0c3973dc9d96f05a7fd135a45558"
+    },
+    {
+      "path": "research/centre-polynomial-cache-limits/frozen/work/centre-encoding-campaign-20261014/lanes/circuit-interface/INTERFACE.json",
+      "usage": "input",
+      "git_blob": "4201e9a04c12273b919e2cae60a6b6482c8150f3",
+      "sha256": "5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3"
+    },
+    {
+      "path": "research/exact-one-slot-gram/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "91854803c13d19d7962aafedf37aca0004827514",
+      "sha256": "30ccfa6d6623313a3c73c7107b7d9c3872eeffc07d2062e7e9b797162585e9b4"
+    }
+  ]
+}