Certify complete block-cascade conditioning and audit paid-word admission (no new κ)

research/complex-block-schedule-admission-audit/HANDOFF.md

@@ -0,0 +1,13 @@
+# Handoff: actual shared-register realization before κ accounting
+
+Status: exact finite endpoint conditioning correction and one conditional-rule misuse witness complete locally; publication coordinated by parent. No new κ.
+
+The most promising distinct constructive test is #4813's first two overlapping native ports, because its local two-/three-call operators come from the documented #4743 interface. Compose those two ports on the actual current five-bank state, track every independent original input, and compare with the alleged observations. Either emit the necessary preservation/restoration work and price it, or exhibit the first failed bank substitution. Do not repeat the full rank solve on ideal observations. For #4826 a genuine three-call port is an earlier missing gate; its valid residual identity alone uses a two-call prefix plus a three-call operator.
+
+Keep the source heads in ORIGINS.json frozen; refresh new PRs and discussions before taking this handoff. Reuse #4828's distinct two-bank conditioning theorem where applicable, and this package's full-cascade factor3 only on its stated model. A successful physical word must supply its raw inverse transpose, disjoint productive census and complete bill. A favourable caller moment and funded κ then remain separate milestones.
+
+Proposed public footer after replacing artifact with the publication URL:
+
+```json
+{"schema":"integer-mult-coordination/v1","kind":"HANDOFF","task":"complex-rank-shape-decoder","scope":"Emit first two overlapping native ports of frozen PR4813 on actual shared banks, with all preservation and restoration priced","base":"624de9dc66858af0945e5f8a0942488d62280d43","expiry":null,"artifact":null}
+```

research/complex-block-schedule-admission-audit/ORIGINS.json

@@ -0,0 +1,47 @@
+[
+  {
+    "pr": 4813,
+    "head": "f982ecc06826e7087e69b9468d3ff52589fd0822",
+    "source_path": "research/complex-block3-five-in-fourteen/verify_schedule.py",
+    "path": "inputs/pr4813-verify_schedule.py",
+    "sha256": "6216fd67f1a762303f0eed0e37b90ea0c72cf8e0cb7db70b1b27b2001ac20069",
+    "git_blob": "17a8439633effba74fff63a8fe4c332b8d709d80",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/blob/f982ecc06826e7087e69b9468d3ff52589fd0822/research/complex-block3-five-in-fourteen/verify_schedule.py"
+  },
+  {
+    "pr": 4818,
+    "head": "fd29d065bbe12f5be5af02fdf5a8b4d86dda3381",
+    "source_path": "research/complex-block4-three-in-ten/verify_schedule.py",
+    "path": "inputs/pr4818-verify_schedule.py",
+    "sha256": "7fab42730ebb5e1cd88dc369df526a002e91a70dc99cdbf4a7c0133fefa7e33d",
+    "git_blob": "8f9a2ef3cdba4c83be7a4d931c1070fef73ba896",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/blob/fd29d065bbe12f5be5af02fdf5a8b4d86dda3381/research/complex-block4-three-in-ten/verify_schedule.py"
+  },
+  {
+    "pr": 4819,
+    "head": "a7b42c2f8ffe5a18d305ca4dda5729d2f1d72ea8",
+    "source_path": "research/complex-block4-thirteen-in-fortyone/verify_schedule.py",
+    "path": "inputs/pr4819-verify_schedule.py",
+    "sha256": "4ddb2d11a4da6a05db3f06a400bd74059ce1448142df028e82a09d50eca06567",
+    "git_blob": "e22dd9afd88837705cba14195f6c9acb4a97c6e7",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/blob/a7b42c2f8ffe5a18d305ca4dda5729d2f1d72ea8/research/complex-block4-thirteen-in-fortyone/verify_schedule.py"
+  },
+  {
+    "pr": 4826,
+    "head": "d7b53d3992e658f0dd49d0207ba33bbc39272bac",
+    "source_path": "research/complex-block5-three-in-nine/verify_schedule.py",
+    "path": "inputs/pr4826-verify_schedule.py",
+    "sha256": "de0256a33d4007615b8dc0ab4c45ce3582bfb2cf963b499f84b0362935c96c4e",
+    "git_blob": "ea1511ae49f4e9b9d58033d7d69056227bf4b53f",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/blob/d7b53d3992e658f0dd49d0207ba33bbc39272bac/research/complex-block5-three-in-nine/verify_schedule.py"
+  },
+  {
+    "pr": 4826,
+    "head": "d7b53d3992e658f0dd49d0207ba33bbc39272bac",
+    "source_path": "research/complex-block5-three-in-nine/verify_ports.py",
+    "path": "inputs/pr4826-verify_ports.py",
+    "sha256": "bd874188e6d32cb692a345186c563e8f23d92918871ac2780a3b3502dda79a86",
+    "git_blob": "a455b4d5466614de2c528996b523fd0bb6b3bc90",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/blob/d7b53d3992e658f0dd49d0207ba33bbc39272bac/research/complex-block5-three-in-nine/verify_ports.py"
+  }
+]

research/complex-block-schedule-admission-audit/PR-DRAFT.md

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+## Change and mathematical scope
+
+Exact finite audit and elementary norm proof: the complete cyclic observation matrices proposed in #4818, #4819 and #4826 have sharp Euclidean condition number **3** (norm3/2 and inverse norm2). Their reported9/7 and8193/8191 bounds control a loop closure factor; they do not bound the complete cascade. Invertibility and the weaker bound16 survive.
+
+The package also gives a concrete counterexample to using #4813's conditional plus-swap-sector cancellation on a general input: e1 produces coordinate-zero error−(1+i)/4. This describes a conditional-rule misuse, not a claim that the source asserts the conditional identity without its premise. A positive exact check confirms #4826's monomial residual factorization at its literal32-coordinate width.
+
+Only `research/complex-block-schedule-admission-audit/` is added. No paid-call optimum, new κ or selected-frontier change follows. Actual chronological call realization, all-input ownership and restoration, raw inverse transpose, all-width lift, finite work/precision, complete caller and compiler/outer transfer remain open. Source matrix construction and injective recovery are not a priced physical word.
+
+## Task and overlap
+
+Base `624de9dc66858af0945e5f8a0942488d62280d43`; task `complex-rank-shape-decoder`, priority revision7. Scope changed by explicit assignment to auditing newly proposed small-unit schedules before using their headline savings.
+
+Audited heads: #4813 `f982ecc06826e7087e69b9468d3ff52589fd0822`; #4818 `fd29d065bbe12f5be5af02fdf5a8b4d86dda3381`; #4819 `a7b42c2f8ffe5a18d305ca4dda5729d2f1d72ea8`; #4826 `d7b53d3992e658f0dd49d0207ba33bbc39272bac`. Five unchanged source files are retained, with original Git blobs, exact hashes and immutable URLs in ORIGINS.json; no unmerged Git object is required at runtime.
+
+Bounded live PR-body queries and all four comment timelines were checked; no comments were present. #4828's exact sharp conditioning result concerns #4743's two-bank port and explicitly excludes these cascades. #4835 concerns a characteristic-zero finite-test kernel. This work checks different concrete matrices. The heads and canonical main were refreshed unchanged at2026-10-22 21:42 UTC. This is not exhaustive novelty or unpublished-work coverage; refresh discussions before taking the handoff.
+
+## Validation
+
+```sh
+python3 -I -B research/complex-block-schedule-admission-audit/verify.py
+python3 -I -O -B research/complex-block-schedule-admission-audit/verify.py
+python3 -I -B research/complex-block-schedule-admission-audit/mutation_check.py
+python3 -B scripts/research_package.py --package research/complex-block-schedule-admission-audit
+python3 -B scripts/check_research_commit.py --package research/complex-block-schedule-admission-audit --check-source-history
+```
+
+Local environment: Python3.12.3, Linux. Exact eigenvector witnesses attain both norm bounds. Every entry of the48- and96-square assembled matrices and all208 columns of the thirteen-bank forward method are compared to an independently evaluated permutation cycle. Source constructors are reused and disclosed; source bytes are authenticated before execution, and AST prefixes use those same authenticated bytes. The source's complete rank/test suites are not replayed.
+
+Actual separate-process mutations reject a changed retained-source byte and a false `new_kappa` receipt in a temporary Git-free export under optimized Python. The scientific checker uses explicit exceptions. No Lean, all-size native replay, complete compiler, selected-frontier scientific replay, hosted CI or independent human review is claimed.
+
+## Credit
+
+Prepared for Chris R / Clear1nsight with substantial OpenAI Codex assistance, Apache-2.0. Credit hexapode for the audited proposals; Dugongue #4743, Yiyang/Lyy-iiis #4710, lee-phys #4688 and Jacob Alan Sussman #4628 for the retained construction/bound lineage. Nick Petalas #4828 addresses the distinct complete-port conditioning question. The norm method is elementary; the result is its exact application to these literal source matrices.
+
+## Open next questions
+
+See `HANDOFF.md` and README. The smallest constructive next test is the chronological composition of #4813's first two overlapping native ports on actual shared banks, tracking all original inputs and charging preservation/restoration. For #4826 the earlier missing witness is a real three-call two-bank primitive containing the proposed rank-five target, with its dirty companion output and costs. A successful word then needs complete-caller pricing and funded compiler/outer transfer.

research/complex-block-schedule-admission-audit/PROOF.md

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+# Exact scope and mathematical checks
+
+## 1. A sharp bound for the literal full cascade
+
+Let V=(C^n)^W, and let X_s be the address permutation x -> x xor s. In each of the three source models,
+
+    M = I + U/2,       (Uz)_k = X_(s_k) z_((k+1) mod W).
+
+The source shift lists are (1,2,4) for W=3,n=16 and W=3,n=32, and (1,2,...,13) for W=13,n=16. U is a permutation, hence unitary. The triangle and reverse triangle inequalities give
+
+    ||Mz|| <= (3/2)||z||,       ||Mz|| >= (1/2)||z||.
+
+Thus M is invertible, ||M||<=3/2, ||M^-1||<=2 and cond_2(M)<=3. This is an endpoint norm ceiling, **not** a paid-call or κ ceiling.
+
+Both inequalities are attained. The all-ones vector has Uv=v. For the lower extreme, take the address character h(x)=(-1)^(x_0) and block signs a_0=1, a_(k+1)=-(-1)^(s_k bit0) a_k. Because W is odd and the xor of all shifts has bit zero set (7,1,7 in the three cases), this recurrence closes around the cycle. Then z_k=a_k h satisfies Uz=-z. Hence Mv=(3/2)v and Mz=(1/2)z. Exact norms are 3/2 and inverse norm2, so the sharp condition number is3.
+
+For both three-cycles the low signs are (1,1,-1). For the thirteen-cycle they are (1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1). These are literal integer witnesses on the actual finite address spaces; RESULT.json records them. The checker verifies the permutation bijection and both eigenvector equations independently, after comparing every source coupler to the specified permutation matrix. It also compares every entry of #4818's assembled 48-square and #4826's assembled 96-square matrices, and all 208 input columns of #4819's actual forward method.
+
+The small loop norm 2^-W controls a Schur complement I +/- (product G_k) after elimination. Elimination/back substitution also acts on the full vector; its norms cannot be discarded. Therefore (1+2^-W)/(1-2^-W) is not a valid bound for cond(M). This refutes9/7 and8193/8191 as **full-cascade** bounds while preserving invertibility and the weaker bound16. No condition claim is made here for #4813's different observation model or for the overdetermined O_full matrices.
+
+## 2. Explicit block-3 sector misuse witness
+
+Use the exact operators in #4813 `verify_schedule.py:198-235`. They satisfy
+
+    B = M0 C2 + 2 K a,       H K a = K/2.
+
+For a plus-swap-sector input, C2 x = T A x; thus H(B-M0 T A)x=Kx. This implication requires Tx=x. For arbitrary input e_1, exact evaluation instead gives
+
+    [H(B-M0 T A)-K]_(0,1) = -(1+i)/4.
+
+The checker also verifies Te_1 != e_1. This is a counterexample to the hypothetical misuse of that conditional plus-sector rule on all inputs. It does not assert that the source states this conditional identity without its sector premise; the source admission gap is its absent complete shared-bank word. It is **not** a disproof of sector splitting with its copies, recombination, calls and storage charged, nor of another all-input decoder. #4813's test suite supplies known sector signs and forms ideal Kx+Ex observations directly. It does not emit a full shared-bank word that maintains those independent original inputs across overlapping ports. Dense Gaussian elimination followed by K on the recovered x is an algebraic recovery check; its operation price needs a separate admissible realization.
+
+## 3. Positive block-5 check and remaining realization boundary
+
+#4826's `verify_ports.py:149-168` defines F=C0(P+2Q)C0 and J=C4 C3 C2 N. It computes R=JF-2K but supplies no assertion that R has the claimed form M0 C4 C3 C2. This audit computes
+
+    M0 = R (C4 C3 C2)^*,
+
+checks one nonzero in every row and column, and checks M0(C4 C3 C2)=R exactly. This finite identity succeeds and should be retained. The test does not establish an all-width tensor identity or make the three calls in J include the two calls already needed for F.
+
+The schedule constructor (`verify_schedule.py:155-216`) sets K=C4C3C2C1C0, picks G_k=X_(2^k)/2, sets E_k=KG_k, and inserts K,E_k into O_primary. The count s=9 is assigned independently. Full rank and exact recovery show that these **postulated observations** determine targets. They do not demonstrate three paid calls per port, ownership-safe chronological composition, a true raw inverse transpose, or priced physical implementation. #4818 and #4819 use the same distinction between assigned call counts and assembled target-containing observations. No universal impossibility of their call shapes is inferred.
+
+The source tests also do not discharge arbitrary tensor width, field/phase lifting, control/movement/storage, finite precision at internal cuts, complete recursive profile, compiler reserve or outer transfer. The printed unit exponent formula cannot be imported as a complete integer multiplication κ until those obligations are bound to a real word. Passing a lower-bound capacity inequality only removes one necessary obstruction; it does not synthesize the word.

research/complex-block-schedule-admission-audit/README.md

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+# Complete block cascade conditioning and admission audit
+
+The literal cyclic matrices in hexapode's #4818, #4819 and #4826 have exact Euclidean condition number **3**, with norm **3/2** and inverse norm **2**. Their stated bounds 9/7, 8193/8191 and 9/7 respectively apply to loop closure factors, not the whole cascade. The corrected bound is still below 16; this finding does **not** make the observation systems singular or prove their proposed call counts impossible.
+
+A second exact check exhibits the arbitrary-input failure of applying #4813's plus-swap-sector cancellation to a general bank. On input e_1 the output-coordinate-zero error is -(1+i)/4. A third check positively confirms #4826's residual identity at the literal 32-coordinate width, although its verifier computes the residual without asserting its monomial factorization. This preserves a valid algebraic starting point.
+
+No new exponent saving is established. A rank computation on an observation matrix containing K does not produce a chronological paid-call realization of that matrix. The proposed complete unit still needs that realization, an all-input owner inventory, raw inverse transpose, all-width justification, finite work/precision bills, and compiler/outer transfer. See [PROOF.md](PROOF.md) and [HANDOFF.md](HANDOFF.md).
+
+## Source and overlap scope
+
+Canonical base: `624de9dc66858af0945e5f8a0942488d62280d43`, priority revision 7. Exact audited heads:
+
+| PR | Head | Coverage |
+|---|---|---|
+| #4813 | `f982ecc06826e7087e69b9468d3ff52589fd0822` | Native block-3 operators, sector misuse witness, assembly/accounting boundary |
+| #4818 | `fd29d065bbe12f5be5af02fdf5a8b4d86dda3381` | Literal 48-square cyclic model and claimed bound |
+| #4819 | `a7b42c2f8ffe5a18d305ca4dda5729d2f1d72ea8` | Literal 208-square cyclic model and claimed bound |
+| #4826 | `d7b53d3992e658f0dd49d0207ba33bbc39272bac` | Literal 96-square cyclic model, residual identity, assembly/accounting boundary |
+
+Five unchanged Python source files are retained in `inputs/`; their original paths, immutable URLs, Git blob IDs and SHA256 hashes are in ORIGINS.json. They are self-contained retained inputs, not dependencies on unmerged Git objects. SOURCES.json binds the task/selected provenance at canonical base. The verifier authenticates retained inputs before loading any code. It uses source matrix constructors plus an independent permutation-cycle action and exact integer/rational witnesses; it does not claim a wholly independent reconstruction of every source Clifford primitive.
+
+Bounded live GitHub PR-body queries for `4826`, `4819 audit`, `4818 audit`, `4813 audit`, and `"condition" "cascade"` found these proposals and neighbouring #4828/#4835. All four proposal comment timelines were empty at inspection. #4828 (`299b6092a3a4f6c617dd509cda2d590932d8be1e`) proves sharp conditioning for #4743's **two-bank port** and explicitly excludes complete-cascade conditioning. #4835 concerns the address-algebra finite-test kernel. Neither is reproduced here. Unpublished work, exhaustive discussion coverage and novelty are not established; refresh these heads before publication. Root agents coordinate the ongoing Discussion/handoff check.
+
+## Reproduction
+
+```sh
+python3 -I -B research/complex-block-schedule-admission-audit/verify.py
+python3 -I -O -B research/complex-block-schedule-admission-audit/verify.py
+python3 -B scripts/research_package.py --package research/complex-block-schedule-admission-audit
+```
+
+Expected: `PASS_COMPLEX_BLOCK_SCHEDULE_ADMISSION_AUDIT`. RESULT.json records exact witnesses. `--write-result` explicitly regenerates that receipt. Ordinary checks are read-only. `mutation_check.py` copies the package to a temporary Git-free directory, runs optimized Python with no tools on PATH, corrupts a retained source byte and then the scientific receipt in separate subprocesses, and requires both mutations to be rejected. Original package bytes remain untouched. Checks stay active under optimized Python. The predecessor constructors contain assertions; this audit's independent comparison and rejection predicates use explicit exceptions. No original complete native suite, Lean, full phase proof, compiler, hosted CI or independent human review is claimed.
+
+## Open next questions
+
+1. Emit one actual shared-register chronological word for #4813's five banks, starting with the first two overlapping ports. Identify which input to port two is still the original bank and charge every preservation/restoration operation. This is the smallest discriminating test of the alleged 14-call schedule.
+2. For #4826, emit one actual three-call two-bank primitive realizing the stated rank-five K block, including dirty companion output, or replace the claimed interface. Dense insertion of K into an observation matrix is not that witness.
+3. If a whole word is obtained, derive the full norm/rounding bill using the corrected factor 3 and all internal cuts, then price the complete recursive caller. Endpoint conditioning alone cannot fund a κ improvement.
+
+Prepared for Chris R / Clear1nsight with substantial OpenAI Codex assistance. Apache-2.0. Credit hexapode for the audited packages; Dugongue #4743, Yiyang / Lyy-iiis #4710, lee-phys #4688 and Jacob Alan Sussman #4628 for their predecessor construction/bound work, retained by citation. Nick Petalas #4828 distinguishes the earlier complete-port conditioning problem. The permutation-cycle norm proof is elementary; no claim of a new general spectral method is made.