Count intermediate carries of the frame-506 inverse elimination

research/frame506-inverse-row-carries/PROOF.md

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+# Intermediate carries of one frame-506 inverse elimination
+
+This is a proof for one fixed elimination word, together with three exhaustive
+finite censuses. It uses the carry formula of PR #4957, section 3, applied to
+each elementary factor. It does not replace the closed-form envelope of PR
+#7104, and it does not change the selected frontier.
+
+## Word
+
+The frozen frame 506 of helper 462 has rank 4. With image rows `B` and kernel
+rows `A`,
+
+```text
+S = [ B^T | A^T ].
+```
+
+The same integer matrix is recorded in
+`research/bit-completed-backend-api/science/RESULT.json`. Exact elimination,
+the algorithm in that package's `science/check.py`, produces 5 swaps, 5 scales
+and 14 shears. The published factor list is reproduced exactly, and the right
+product is `S`. Reversing the word and inverting each factor produces a word
+for `S^{-1}`: the same five swaps, with each scale `c` replaced by `1/c` and
+each shear coefficient negated. PR #4957 does not fix this word as the physical
+return. It is the inverse of the published elimination, and the count below is
+the intermediate quotient of that word.
+
+Every scale or shear coefficient in the inverse word lies in `{1, -1, 1/2, -1/2}`.
+`det S = -16`, so every denominator of `S^{-1}` divides 4. The residue used
+below is unique for every odd integer `q`, because 2 is invertible modulo `q`.
+
+## One carry digit per scale and per shear
+
+A low row is an 8-tuple with coordinates in `{0, ..., q-1}`. An elementary
+right factor acts on that row as follows.
+
+- A swap exchanges two coordinates. Both digits are already reduced, so the
+  PR #4957 quotient of the swap is 0.
+- A scale multiplies one coordinate by a fixed rational and leaves the other
+  seven unchanged. The quotient of each unchanged coordinate is 0, so the
+  factor emits one carry digit.
+- A shear adds a fixed rational multiple of one coordinate into one other
+  coordinate. Again only one coordinate changes, so the factor emits one
+  carry digit.
+
+The carry digit is the quotient `(z - h) / q` in `Z[1/2]`, where `z` is the
+unreduced value of the updated coordinate and `h` is the unique element of
+`{0, ..., q-1}` for which the difference is `q` times an element of `Z[1/2]`.
+The digit is nonzero when that quotient is nonzero as a rational.
+
+The inverse word therefore has 19 arithmetic sites on one row. At most 19 of
+them are nonzero. In particular the count is strictly less than 48, for every
+low row and every odd integer `q`. This is the site bound. It is not a claim
+about a longer factorization, and it is not the 48-correction closed form of
+`S^{-1}` proved in #7104.
+
+## Named row
+
+PR #7104 attains its return envelope on the low type
+
+```text
+u(q) = (q-2, q-2, q-5, q-5, 0, q-1, 0, 0),
+```
+
+which is a legal row for every integer `q ≥ 5`. The closed-form correction of
+`u(q)` under `S^{-1}` has six nonzero entries. The calculation here follows
+the inverse word instead of that closed form.
+
+For every odd integer `q ≥ 7`, each updated coordinate stays inside an affine
+formula `A + s q` with `s ∈ {0, 1/2, 1}` and `A ∈ Z[1/2]`. The slope is the
+unique value in `{0, 1/4, 1/2, 3/4, 1}` for which `A + s q` is an integer in
+`{0, ..., q-1}` for every odd integer `q ≥ 7`. Integrality is checked on a
+complete set of odd residues modulo the power-of-two denominator. Along each
+residue class the value and the slack `q - value` are nondecreasing, because
+the class step adds a nonnegative integer to both. The resulting digits are:
+
+| site | operation | unreduced value | residue | carry |
+| ---: | --- | --- | --- | --- |
+| 0 | `x7 += x3` | `q-5` | `q-5` | 0 |
+| 1 | `x7 += -x2` | `0` | `0` | 0 |
+| 2 | `x7 += -(1/2) x1` | `1 - q/2` | `1` | `-1/2` |
+| 3 | `x7 += -(1/2) x0` | `2 - q/2` | `2` | `-1/2` |
+| 4 | `x7 *= 1/2` | `1` | `1` | 0 |
+| 5 | `x5 += -x3` | `4` | `4` | 0 |
+| 6 | `x5 += (1/2) x1` | `3 + q/2` | `3` | `1/2` |
+| 7 | `x5 += (1/2) x0` | `2 + q/2` | `2` | `1/2` |
+| 8 | `x5 *= -1/2` | `-1` | `q-1` | `-1` |
+| 9 | `x3 += -x7` | `q-6` | `q-6` | 0 |
+| 10 | `x3 += -x2` | `-1` | `q-1` | `-1` |
+| 11 | `x3 += -(1/2) x1` | `q/2` | `0` | `1/2` |
+| 12 | `x3 += -(1/2) x0` | `1 - q/2` | `1` | `-1/2` |
+| 13 | `x3 *= -1` | `-1` | `q-1` | `-1` |
+| 14 | `x2 += x7` | `q-4` | `q-4` | 0 |
+| 15 | `x2 += -x6` | `-3` | `q-3` | `-1` |
+| 16 | `x2 += -x4` | `-2` | `q-2` | `-1` |
+| 17 | `x1 *= 1/2` | `(q-2)/2` | `q-1` | `-1/2` |
+| 18 | `x0 *= 1/2` | `(q-2)/2` | `q-1` | `-1/2` |
+
+Swaps sit between these sites and emit no digit: `x5` with `x6`, then `x4`
+with `x5`, after site 8; `x3` with `x4` after site 13; `x1` with `x6` after
+site 17; `x0` with `x5` after site 18. Thirteen of the nineteen digits are
+nonzero. The final low row is
+
+```text
+(0, q-1, q-2, 0, q-1, q-1, q-1, 1).
+```
+
+Starting from high part 0, the same quotients update the high coordinates.
+The final high row is the constant
+
+```text
+(0, -3/2, -1/2, 0, -1/2, -1/2, -1/2, -1/2),
+```
+
+which is the closed-form return correction of `u(q)` from #7104. The
+intermediate word therefore emits 13 nonzero quotients on this row, and those
+quotients compose to the 6-nonzero closed form. Dropping the quotients while
+retaining the linear action misses that high row.
+
+The odd integer `q = 5` is the one odd integer `≥ 5` at which `q - 6` is
+negative, so site 9 is outside the tail formula. Direct evaluation of this
+single row gives the same digits, except site 9 is `-1` and site 10 is `0`.
+The nonzero count remains 13, and the final low and high rows are the same
+`q = 5` specializations. That evaluation visits each of the 19 sites once.
+
+The zero row emits the zero quotient at every site, so 13 is the count on
+`u(q)`, and it is a maximum only in the sense of the separate censuses below.
+
+## Censuses
+
+For an odd prime `q`, the domain `{0, ..., q-1}^8` is finite. The base-`q`
+code of an integer in `{0, ..., q^8 - 1}` visits each row once. On every
+coefficient that occurs, the integer nonzero test agrees with the rational
+quotient for every digit the operation reads; that agreement is checked for
+`q ∈ {3, 5, 7}` before the scan.
+
+| q | domain | maximum nonzero sites | rows at the maximum |
+| ---: | ---: | ---: | ---: |
+| 3 | 6561 | 18 | 9 |
+| 5 | 390625 | 18 | 575 |
+| 7 | 5764801 | 19 | 245 |
+
+The maxima are attained by `(1,1,1,2,0,0,0,1)`, `(1,1,1,4,0,0,0,1)` and
+`(5,5,6,3,0,0,0,4)` respectively. Each of those three counts was recomputed
+with the rational quotient, independently of the scan. The full histograms
+are the constants in `verify.py`.
+
+These are exhaustive censuses of those three domains. They do not by
+themselves determine the maximum at any other prime. The site bound of 19
+already covers every odd integer `q`. The prime 7 shows that 19 is attained,
+so no smaller uniform site bound holds for every odd prime. Every counted row
+is still strictly below 48.
+
+## What this does not change
+
+The 5/5/14 word, the owners, ROW restoration, P1–P3, the native compiler and
+the exponent gaps stay as PR #4957 left them. The closed-form envelopes 48
+per route and 96 for the pair, and the `q = 3` pair maximum 88, stay as
+#7104 proved them. Repeating `u(q)` on all eight high rows emits
+`8 * 13 = 104` nonzero intermediate quotients. That route total is a
+different number from the closed-form 48, and this note does not put either
+number into a charged bill.
+
+The selected frontier stays `363005817/100000000000`.
+
+## Controls
+
+The range test rejects residue `q - 6` at `q = 5` and accepts it for every
+odd integer `q ≥ 7`. The zero row pays nothing. Omitting the quotients in the
+high update misses the closed-form correction on `u(q)`. A scale or shear
+coefficient outside `{±1, ±1/2}` is rejected before a census. The published
+factor list, the helper-462 endpoints and the selected-frontier string are
+read from the pinned inputs.

research/frame506-inverse-row-carries/README.md

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+# Intermediate carries on one row of the frame-506 inverse elimination
+
+**Proof, plus three exhaustive finite censuses. No new κ.** The selected
+frontier stays `363005817/100000000000`.
+
+## Result
+
+The published frame-506 elimination has 5 swaps, 5 scales and 14 shears. Its
+inverse is an elimination word for `S^{-1}`. PR #4957 does not fix that word
+as the physical return. Each scale and each shear emits one carry digit on a
+low row, under the quotient `(z - h) / q` from PR #4957, section 3. A swap
+emits none.
+
+- On every low row, and for every odd integer `q`, at most **19** of those
+  digits are nonzero. Nineteen is strictly less than 48.
+- On the #7104 return row `u(q) = (q-2, q-2, q-5, q-5, 0, q-1, 0, 0)`, exactly
+  **13** digits are nonzero for every odd integer `q ≥ 5`. The digits are
+  constant for every odd integer `q ≥ 7`, and the single row `q = 5` differs
+  in two sites while keeping the same count. The digits are attached to the
+  19 factors in [PROOF.md](PROOF.md).
+- Those 13 quotients compose to the 6-nonzero closed-form correction of #7104.
+  The closed-form route envelope remains 48. This note counts one row of one
+  word.
+- Exhaustive base-`q` censuses give maximum 18 at `q = 3` (domain 6561),
+  maximum 18 at `q = 5` (domain 390625) and maximum 19 at `q = 7` (domain
+  5764801). The bound 19 is attained, and every row in those domains is still
+  below 48.
+
+## Task
+
+Catalogue card `supplier-boundary-audit`, scope **refined**. The open question
+answered here is question 1 of PR #7104: the intermediate digits of an
+elimination word for `S^{-1}` on one high row. The 48/96 closed form and the
+published 64/128 envelope are left as proved. Question 2 of #7104, a charged
+bill that moves an exponent, is not taken.
+
+Base `e54523517d0f03e58352e647c403a5af3e525a57`.
+
+## Reproduce
+
+From the repository root, Python 3.12, standard library only:
+
+```sh
+python3 -B research/frame506-inverse-row-carries/verify.py
+python3 -B scripts/research_package.py --package research/frame506-inverse-row-carries
+```
+
+The direct checker prints `PASS_FRAME506_INVERSE_ROW_CARRIES`.
+
+## Open next questions
+
+1. For which odd primes is the one-row maximum exactly 19? The censuses give
+   18 at `q = 3` and `q = 5`, and 19 at `q = 7`. The next finite domain is
+   `q = 9`, with `9^8 = 43046721` rows, under the same base-`q` scan. No
+   algebraic blocker is known; this package stops after the three recorded
+   domains.
+2. Does some other elimination word for `S^{-1}` emit 48 or more nonzero
+   intermediate digits on one row? The 19-site bound uses this inverse of the
+   published 5/5/14 word. A longer factorization has more sites. PR #4957
+   does not designate another return word, and this package does not search
+   one.
+
+## Credit
+
+The matcher, the 5/5/14 elimination and the carry formula are from PR #4957
+(Chris R / Clear1nsight, with the retained #4760/#4814 lineage). The named
+row and the closed-form correction it composes to are from PR #7104.
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

research/frame506-inverse-row-carries/SOURCES.json

@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "75d77b328ea02ded535e7f23aee5b1e55d0bb1a8",
+      "sha256": "154c3d30455d778fd3343b6b4851f0fa4665f0f7625b0adb3675e2bafbbf6bf1"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/bit-completed-backend-api/independent/inputs/WORD.json.gz",
+      "usage": "input",
+      "git_blob": "776ddd13d67b2259e90d39de5b42e4df4b1447f1",
+      "sha256": "69fc50430f0062c040e0686f59ad32ee5af2387ae9a97d7d11b4af1aa994ed5b"
+    },
+    {
+      "path": "research/bit-completed-backend-api/independent/inputs/frames.json",
+      "usage": "input",
+      "git_blob": "a9a932b9268da28af9ba4bccdab205c5bcd50ad3",
+      "sha256": "d006b3bfe0c04afecfea7aecfa59dfe4298b99501c70da3938e40289bc2f5010"
+    },
+    {
+      "path": "research/bit-completed-backend-api/science/RESULT.json",
+      "usage": "input",
+      "git_blob": "55dca30281e3375d63a128c3de041e46c8a3c5ba",
+      "sha256": "965b650c0e49572e028f89a60943c64ad8dff9032fc93272b86cd1f9715a77b7"
+    }
+  ]
+}

research/frame506-inverse-row-carries/TASK.json

@@ -0,0 +1,97 @@
+{
+  "schema_version": 1,
+  "task_id": "supplier-boundary-audit",
+  "priority_revision": 8,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "original_card": {
+    "id": "supplier-boundary-audit",
+    "question": "Does one changed supplier have complete source-bound native, caller, ordinary and outer admission coverage?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "Pin the current sources and one obligation; the historical owner audit is an example, not evidence of a current bug.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Choose one latest uncovered candidate and build a delta coverage map against the selected review. For P104, #5717 changes owned-lifetime assumptions and #5719 covers exact #5713 causal premises; neither automatically transfers to #5718. For BIT, #5679/#5708 need component admission and compatible outer use. Check controls and exact source identity before reuse; stop with one resolved gap or explicit remaining obligation. A heavy full replay requires a separate serial resource budget.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "supplier admission",
+      "5717",
+      "5719",
+      "5708",
+      "owned lifetime"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4088: author-checked local words, native receipts, owner/phase fill and exact moment.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Rebind and validate ordinary/outer admission with all four providers; full mixed release not established.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #3801: successor scatter-boundary decision package.",
+        "evidence": "research/scatter-boundary-decision/README.md",
+        "remaining": "Consult its exact acceptance scope before any new exporter audit; not proof of a selected failure.",
+        "status": "author-checked",
+        "source_commit": "911f8800ba7a4b03603ef62c5ffc6240e53cb5fd",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "For the inverse of the published 5/5/14 frame-506 elimination, attach one carry digit to each of the 5 scales and 14 shears and count the nonzero digits on one low row.",
+  "scope_change_reason": "PR #7104 leaves open the intermediate-carry count of an elimination word for S^{-1} on one high row. This package answers that count for the inverse of the published word. It does not claim supplier admission.",
+  "bottleneck_link": "BIT is the next cap after the selected complex supplier. One row of intermediate carry digits is a finer reading of the same frame-506 route. The predecessor already treats a finite carry constant as lower order, so the count does not by itself move kappa.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the site bound, the closed form on u(q), and the exhaustive censuses at q=3, 5 and 7. Do not retune an exponent or replace the 5/5/14 word.",
+  "outcome": "partial",
+  "evidence": "PROOF.md range argument for every odd integer q>=7 on u(q), direct evaluation of the one row q=5, and exhaustive base-q censuses of all 3^8, 5^8 and 7^8 low rows. The selected frontier string is read and left unchanged.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Substantial Cursor Grok assistance. The proof and checker were written with that assistance; no separate human review is claimed.",
+  "open_questions": [
+    {
+      "question": "For which odd primes is the one-row maximum of this inverse word exactly 19?",
+      "why_it_matters": "The site bound 19 is attained at q=7 and missed at q=3 and q=5, so the uniform bound is sharp while the small primes are not.",
+      "first_step": "Run the same base-q census on the domain of q=9, which has 9^8 = 43046721 rows.",
+      "blocked_by": "Nothing algebraic. The q=9 domain did not fit beside the q=7 census in this replay."
+    },
+    {
+      "question": "Does another elimination word for S^{-1} emit 48 or more nonzero intermediate digits on one row?",
+      "why_it_matters": "The 19-site bound is special to the inverse of the published 5/5/14 word. A longer word has more sites.",
+      "first_step": "Fix one other explicit factorization of S^{-1} into swaps, scales and shears, and count its nonzero quotients on u(q).",
+      "blocked_by": "PR #4957 does not designate a second return word."
+    }
+  ]
+}