Audit every scalar event in the actual P104 tensor code

research/p104-tensor-word-audit/CERTIFICATE.json

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+{
+  "schema_version": 1,
+  "ports": 104,
+  "pivot_ports": 8,
+  "invoices": {
+    "boundary": {"add": 319488, "half": 43264, "double": 43264},
+    "coded_pair": {"add": 26240, "half": 3328, "double": 3328},
+    "reference_pair": {"add": 241280, "half": 41600, "double": 41600}
+  }
+}

research/p104-tensor-word-audit/PROOF.md

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+# Source-bound audit of the actual P104 tensor words
+
+## Scope, sources and operator order
+
+Chris R / Clear1nsight's PR3979, scientific head
+`a8a1d744811cd7d1f9c43ec7ccbae85ef9695b61`, supplies the construction, words,
+counts and comparison audited here. Canonical main is
+`20ecf2891d80b5acf18d980b2164fb44ea7faa90`. The new result is a compositional
+exact check at the actual 104 ports and all 21,632 data coordinates. It is not
+a new code, a criticism of the predecessor's written derivation, or a full
+multiplication/native audit. The unchanged exported sources retain their
+authorship and earlier P104/native/caller credit.
+
+All arithmetic below is coordinatewise over the Gaussian dyadic ring
+`R=Z[i,1/2]`, under the predecessor's common-FULL-chart scalar contract.
+Temporarily modified donors must return. No physical chart transport or owner
+reassignment is inferred. Number data by
+`index(bank,i,j)=(bank*104+i)*104+j`. The first eight axis coordinates are the
+original pivot ports `(0,1,3,6,22,31,93,103)`; the remaining ports retain
+canonical lexicographic order. Operators act on column vectors, the rightmost
+operator executes first, and `K_r=K tensor I`, `K_c=I tensor K`. Axis zero
+varies i at fixed j; axis one varies j at fixed i.
+
+The canonical P104 fixture is authenticated by SHA256
+`516c06f6ec3cb84be588c546108de897650288576dd5fbf5fe44b9c5546ad8a4`.
+The primary checker obtains the frozen source's actual words and B2=2B; the
+independent checker reconstructs roots from a different full ternary/sign
+enumeration and never imports a primary or predecessor executable. Both bind
+all 768 B2 entries to SHA256
+`88e9a726e8c8aec055bfd7cd024883b5c8ecbda0cd6471e591a08c43a108a535`,
+using ASCII `json.dumps(b2,separators=(',',':'))` without a newline. This is the
+explicit bridge between literal-word checking and the independently rebuilt
+geometry, rather than agreement only on dimensions or support counts.
+
+## Every literal event has an exact local meaning
+
+The authenticated source builders generate four complete words: encoder,
+decoder, coded Gram pair and specified uncoded reference Gram pair. No
+predecessor evaluator, result routine or receipt supplies their semantics.
+For every event the primary checker verifies opcode, arity, integer indices
+and signed-unit ADD coefficient. Its localization checks that both donors and
+recipients lie in the declared bank/fiber. Every word is consumed to its end;
+truncation, extra events and a move to another fiber cannot be ignored.
+
+On the 104 independent local coordinates, an exact sparse Fraction interpreter
+certifies the four sweep maps `U,U^-1,U^t,U^-t`. On 208 independent two-bank
+local coordinates it certifies the lower/upper M shears and lower/upper A
+shears, including complete source return. These are eight distinct local maps.
+Every other fiber's full localized event sequence is checked equal to its
+certified pattern. Within a fixed sweep, different fibers have disjoint
+supports, so these local maps compose to their stated global tensor factors.
+
+The coverage is 416 encoder fibers, 416 decoder fibers, 16 coded-pair fibers
+and 208 reference-pair fibers: **1,056 actual fiber instances**. The four
+words together contain **763,392 literal events**, all checked. This is the
+sum of one copy of each of the four audited words, not the length of a single
+two-pair experiment. Pattern reuse avoids redoing identical arithmetic; it
+does not replace inspection of any event or restrict input values.
+
+Each event index is below `2*104^2`. Therefore the words extend by logical
+identity on every additional independently arbitrary helper coordinate,
+including the named 72-per-fiber auxiliary banks. This is stronger than
+checking only zero dirt or two sentinels, but is solely a logical index fact.
+It does not execute their physical completion, native return or ownership.
+
+## Global algebra on all coordinates
+
+Let P select the first eight coordinates. The independent reconstruction
+checks the actual Gram matrix A, all B entries, and
+
+```
+U = [I B;0 I],   U^-1 = [I -B;0 I],
+A = U^t P U,    M = U U^t,    N = M^-1 = U^-t U^-1.
+```
+
+The encoder sweeps execute `U_c^-t` before `U_r` on X and `U_c` before
+`U_r^-t` on Y. Thus their global map and the certified decoder are
+
+```
+L = U_r U_c^-t,
+C = diag(L,L^-t),        C^-1 = diag(L^-1,L^t).
+```
+
+There is no cross-bank slot permutation. The uncoded lower stage and subsequent
+upper stage are
+
+```
+S = [I 0; A_r I],        R = [I -A_c;0 I].
+```
+
+Their chronological pair is RS. The two-bank coded fibers implement
+
+```
+S_hat = C S C^-1 = [I 0; P_r M_c I],
+R_hat = C R C^-1 = [I -M_r P_c;0 I].
+```
+
+For example, the lower coefficient factors as
+`(U^-t A U^-1) tensor (U U^t)=P tensor M`. The upper coefficient is
+`M tensor P`. These factors retain the displayed order; replacing M by
+`U^t U` is generally incorrect. The independent checker proves both C
+inverse identities, both Gram conjugations, and the full endpoints
+
+```
+C^-1 (R_hat S_hat) C = RS,
+C^-1 (R_hat S_hat)^2 C = (RS)^2.
+```
+
+The same C is used on both sides of the internal boundary. These equalities
+hold on every independent real/imaginary component of both banks, including
+the Gram kernel. They are not tests of a sampled dense input.
+
+### Exact contraction without a huge dense matrix
+
+Each bank-to-bank block is stored as a finite sum `T=sum_k c_k A_k tensor B_k`
+of real dyadic 104-by-104 factors. Matrix products preserve this form. Its
+exact Frobenius square is
+
+```
+||T||_F^2 = sum_(k,l) c_k c_l
+           <A_k,A_l>_F <B_k,B_l>_F.
+```
+
+This follows by expanding each entry and separating the two tensor indices.
+All factor entries are integers divided by powers of two; the implementation
+performs integer multiplication and common-denominator addition. There is no
+floating point, tolerance or probabilistic contraction. Since T is real,
+this sum is zero exactly when every entry of T is zero. Checking all four
+two-bank residual blocks therefore proves equality on all **21,632 data
+coordinates**, while avoiding their 467,943,424-entry dense operator.
+The independent checker verifies eight global block identities in this way.
+Its zero contractions combine with the literal-fiber audit above; neither
+checker alone is claimed to certify the entire evidence chain.
+
+## Counts, comparison and the omitted identity requirement
+
+Counting every generated event gives the predecessor's invoices:
+
+| Word | Unit ADD/SUB | Halves | Doubles |
+| --- | ---: | ---: | ---: |
+| Entry plus exit | 319488 | 43264 | 43264 |
+| Coded uncorrected Gram pair | 26240 | 3328 | 3328 |
+| Specified uncoded Gram pair | 241280 | 41600 | 41600 |
+
+For k repeated identical **uncorrected Gram pairs**, the charged addition
+comparison is `319488+26240k` against `241280k`; each scaling category is
+`43264+3328k` against `41600k`. At k=1 the code loses 104448 additions and
+4992 operations in each scaling category. At k=2 it saves 110592 additions
+and 33280 in each scaling category. Since the respective inequalities are
+`215040k>319488` and `38272k>43264`, two is the first positive integer winning
+all three columns. This compares precisely these existing words, not optimal
+uncoded algorithms, runtime, or an identified physical repeated consumer.
+
+The required identity shears are different. Their coded coefficients are
+`N_r M_c` and `M_r N_c`; the Gram words contain only `P_r M_c` and `M_r P_c`.
+The independent checker verifies the identity conjugations and
+
+```
+N-P = U^-t (I-A) U^-1,        A^2=13A.
+```
+
+Over Q, `(I-A)^-1=I-A/12`. Hence N-P has rank 104 and, since M is invertible,
+each omitted tensor side has rank **10816**. This is an algebraic-rank fact,
+not a dyadic primitive, native-child tariff or cost lower bound. A nonzero
+kernel payload `(-Bz,z)` is retained by U even though its pivot query vanishes;
+the omitted identity side acts nontrivially on it. The original PR already
+owns these completion warnings and rank claims.
+
+Positive exact residual norms additionally separate reversed U/transpose
+order, reversed Gram-stage order and omitted lower/upper identity completion.
+The original modulo-13 spectral-normalization result is outside this audit.
+
+## Remaining obligations and stopping rule
+
+This closes a bounded all-coordinate software/algebra audit of the pinned
+scalar words. Physical chart conjugation, actual adjacent/repeated consumers,
+identity-side implementation, copied-center endpoint correction/wrappers,
+dirty-helper completion, owners/routing/holding/control/tapes, signed-floor
+precision and all-size/native admission remain unresolved or inherited. Scalar
+ADD/scale counts are not native rank units. No new physical bill, passing
+supplier inequality, kappa change, human acceptance or proof-kernel certificate
+is claimed. Stop dependent construction until a named consumer and favorable
+complete allowance are supplied.
+
+Prepared by ToikY with substantial OpenAI Codex assistance, including separate
+AI implementation/review contexts. Apache-2.0; all predecessor credit is retained.

research/p104-tensor-word-audit/README.md

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+# P104 tensor-code literal-word audit
+
+This independently checks the actual scalar programs in [Chris R / Clear1nsight's
+PR3979](https://github.com/CrocSwap/integer-mult-bounds/pull/3979) at frozen head
+`a8a1d744811cd7d1f9c43ec7ccbae85ef9695b61`. Every one of **763,392 literal
+operations** is covered by **1,056 fiber instances, disjoint within each sweep**, reduced to eight
+exact local linear maps. A separately implemented exact tensor calculation checks
+eight global identities, comprising 32 bank blocks, on all **21,632 data
+coordinates**. No coordinate sampling or floating tolerance is used.
+
+The contribution is source-bound implementation coverage. The tensor code,
+algebraic conjugation and protocol savings belong to PR3979. Its written proof
+already addresses arbitrary inputs; this audit adds exhaustive actual-instance
+word coverage beyond its small-grid basis tests and actual dense fixtures. It
+does not supply a new code, physical repeated caller, admitted supplier or κ.
+
+## Evidence and reproduction
+
+From the repository root, with Python 3.12 and no third-party requirements:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-tensor-word-audit
+```
+
+The three registered commands are `verify.py`, `independent.py` and
+`test_verify.py`. Direct replay also works from a Git-free committed package
+export containing the declared canonical P104 fixture. A package-directory copy
+alone omits that external input. `SOURCES.json` pins its exact bytes and the
+three package-local predecessor exports; neither an unmerged predecessor commit
+nor network access is needed for replay.
+
+`verify.py` authenticates the frozen constructor before calling its word builders
+and geometry constructor. Those builders are the object under review. It never
+calls the predecessor evaluator, invoice or result routine. The checker consumes
+every event, validates its global data index, bank, axis, sign and chronology,
+and evaluates one full basis map for each distinct 104- or 208-coordinate fiber.
+Every other fiber is checked for exact program equality after its own physical
+index localization; a cached pattern cannot hide an altered later fiber.
+
+`independent.py` imports neither primary nor predecessor code. It reconstructs
+the roots and all 768 signed-half coefficients by a different enumeration. Both
+checkers bind this complete ordered coefficient array to the same canonical
+digest. It proves the global block identities by exact factored Kronecker
+products and Frobenius-square contractions; zero means every matrix entry is
+zero. Four deliberately wrong global expressions have strictly positive norms.
+See [PROOF.md](PROOF.md) for the composition argument and conventions.
+
+`test_verify.py` runs 12 semantic word mutations and seven actual CLI damaged
+source/invoice cases under both normal and optimized Python: 38 rejections,
+plus four separate independent-checker fixture rejections. It also replays the
+actual 104-port primary checker normally and optimized from an inode-disjoint,
+read-only Git-free closure and verifies unchanged source bytes. These are
+software checks and a scoped exact algebra certificate; no human or proof-kernel
+review is implied.
+
+## Recomputed original invoice
+
+| Named word | Unit ADD | Half | Restoring double |
+| --- | ---: | ---: | ---: |
+| Encoder plus decoder | 319,488 | 43,264 | 43,264 |
+| One coded Gram pair | 26,240 | 3,328 | 3,328 |
+| One specified reference Gram pair | 241,280 | 41,600 | 41,600 |
+
+The claimed two-pair saving is confirmed: 110,592 ADDs and 33,280 halves and
+doubles each, with entry and exit paid. One pair loses 104,448 ADDs. These
+are PR3979's results against its specified reference, with no optimality claim.
+The correct identity-shear coefficients remain `N tensor M` and `M tensor N`,
+where `M=UU^t`, `N=M^-1`; the omitted side maps each have rational rank 10,816.
+Algebraic rank does not price native calls. Helpers are untouched by the literal
+data indices, which proves logical identity extension only.
+
+## Open next questions
+
+1. **Actual repeated consumer and shared charts.** Pin one chronological physical
+   producer/consumer pair and write every data/helper owner and ordered chart
+   conjugate. This determines whether a shared code can be used at all. No such
+   physical pair is constructed here; scalar conjugacy does not discharge it.
+2. **Complete paid identity caller.** After that interface is fixed, compile the
+   required `N-P` side maps and code boundaries and price every native transport,
+   helper, routing and finite-width precision charge. First test the complete
+   favorable family gate; the isolated repeated-Gram saving cannot justify
+   construction expansion. All-size and multiplication-theorem admission remain
+   separate obligations.
+
+## Attribution and scope
+
+Chris R / Clear1nsight owns PR3979's tensor construction, scalar word builders,
+original proof, invoice and side-map analysis. Its source files are retained
+byte-for-byte in `source/`. PR3961 supplies the preceding representation mismatch;
+PR3882/3940/3943/3954 retain the caller lineage. ToikY's PR3956 adapter comparison
+and PR3966 phase gauge are context, not dependencies of this certificate.
+The retained source credits Rohan Arun/PR1972, Chafik Boukhalfa/PR1991, Adam
+Gall/PR1940, Josie Elliston and the E8 predecessors, Sagar Thakrar/PR1012,
+Douglas Colkitt, Dominik Scholz and the native/compiler/coverage/terminal-floor
+contributors named in the preserved source. Their claims outside this scope
+were not replayed here.
+
+New audit, packaging and tests: ToikY with substantial OpenAI Codex assistance,
+including separate AI math and test contexts. Apache-2.0. Only this scoped package
+is added; canonical main, frozen predecessors and the selected frontier are
+unchanged. `TASK.json` records handoff-first selection, overlap coverage and the
+change from the catalogue card's original chronological-correction question.

research/p104-tensor-word-audit/SOURCES.json

@@ -0,0 +1,54 @@
+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "research/p104-native-channel/inputs/P104-FIXTURE.json",
+      "usage": "input",
+      "git_blob": "2e16f99b295bb386fd13bd8fbd57b42ef54143a9",
+      "sha256": "516c06f6ec3cb84be588c546108de897650288576dd5fbf5fe44b9c5546ad8a4"
+    }
+  ],
+  "source_exports": [
+    {
+      "path": "research/p104-tensor-word-audit/source/README.md",
+      "origin_repository": "Clear1nsight/integer-mult-bounds",
+      "origin_commit": "a8a1d744811cd7d1f9c43ec7ccbae85ef9695b61",
+      "origin_path": "research/p104-tensor-code/README.md",
+      "git_blob": "be374d54df5d481bdf8d83671c0a3843149dec11",
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+      "usage": "package-local exact source export; primary hardcoded hash authentication; word/geometry builders only are objects under review"
+    },
+    {
+      "path": "research/p104-tensor-word-audit/source/PROOF.md",
+      "origin_repository": "Clear1nsight/integer-mult-bounds",
+      "origin_commit": "a8a1d744811cd7d1f9c43ec7ccbae85ef9695b61",
+      "origin_path": "research/p104-tensor-code/PROOF.md",
+      "git_blob": "b570c8bdf5acec263c63562ddb5e384ff3616c3f",
+      "sha256": "259ba7b0926dd6db4e7c45ff7ccb10659dee06aef0a573547236c994ad304756",
+      "usage": "package-local exact source export; primary hardcoded hash authentication; word/geometry builders only are objects under review"
+    },
+    {
+      "path": "research/p104-tensor-word-audit/source/verify.py",
+      "origin_repository": "Clear1nsight/integer-mult-bounds",
+      "origin_commit": "a8a1d744811cd7d1f9c43ec7ccbae85ef9695b61",
+      "origin_path": "research/p104-tensor-code/verify.py",
+      "git_blob": "7b8194642450100fafefde2e3bbb6d4a3daa7194",
+      "sha256": "27bae82fe247fe49b8fc3b5efbd429e2c8c0ee8f56cc4ace748cbb681a5abacc",
+      "usage": "package-local exact source export; primary hardcoded hash authentication; word/geometry builders only are objects under review"
+    }
+  ],
+  "source_export_contract": "The pending foreign commit is not a Git-history dependency. All three exact exports are retained in this package and authenticated by the primary checker before import. Its evaluator, invoice and result routines are never called. Independent checker imports no primary/predecessor code and separately reconstructs the canonical live fixture; both bind the complete ordered B2 array. Generic history checks cover canonical files, not foreign origin availability or scientific acceptance."
+}