Certify all 128 signed n8 prefix maxima at 23/16 (no new κ)

research/triangular-parity-sign-cube/README.md

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+# Every full-support signed n=8 symbol has prefix maximum 23/16
+
+**Claim type: complete finite certificate**, with an elementary attaining-row
+argument. For every one of the 128 symbols
+
+```text
+f = (1, s1/8, s2/16, s3/16, s4/16, s5/16, s6/16, s7/16),
+sj in {-1,+1},
+```
+
+the largest proper forward-prefix infinity norm of the even-first triangular
+parity compiler is exactly **23/16**. Every symbol first attains this value
+after macro 11 and retains it through macro 21. All 128 symbols are extremizers.
+Their tail l1 is 1/2. The completed Toeplitz matrix has norm 3/2.
+
+This replaces individual sign examples by an exhaustive result for this fixed
+magnitude family. Brian's [general forward-prefix theorem](../triangular-parity-forward-bound/README.md)
+already gives 3/2 at tail budget 1/2. The new finite envelope is sharper by 1/16.
+This certificate does not establish a sharper bound for other magnitudes, interior
+points of a coefficient box, arbitrary dimensions, or child interleavings.
+No new kappa, supplier admission or complete multiplication theorem is claimed.
+
+## Schedule and exact certificate
+
+Use ordinary power-coordinate indices 0,...,7 throughout. At each node split
+`f(z)=e(z^2)+z o(z^2)` and compute truncated series
+
+```text
+k = o/e,       s = e-z o k,
+U = [I, T(zk); 0, I],       L = [I, 0; T(k), I],
+T(f) = L diag(T(e),T(s)) U.
+```
+
+The displayed blocks use even/odd order. The verifier embeds them directly
+on the corresponding ordinary coordinate indices. Execute `U`, the entire
+even child, the entire Schur child, then `L`, repeating this order at each
+node. Include one identity factor at each scalar leaf. Thus the schedule has
+22 factors and 21 nonempty proper prefixes. The identity initial cut, if
+included, has norm one and does not change any maximum.
+
+[verify.py](verify.py) independently implements this definition with exact
+`Fraction` arithmetic. It constructs the fixed 15-node tree without recursion,
+multiplies every chronological 8-by-8 prefix, and compares the final product
+to the literal lower-triangular Toeplitz matrix. It imports no predecessor
+verifier and does not run a predecessor's certificate.
+
+The maximum at each cut, taken over the complete 128-symbol domain, is:
+
+| Proper cut | Maximum infinity norm over all signs |
+|---|---|
+| 1 through 5 | 2601/2048 |
+| 6 through 9 | 21/16 |
+| 10 | 85/64 |
+| 11 through 21 | 23/16 |
+
+These are exact maxima from all 2,688 prefix matrices. They are not interval
+upper estimates or floating-point results. Each symbol's own maximum is
+23/16, even though its earlier cut norms can differ from this table.
+
+The attaining row has a short explanation. At cut 11 the even child is
+complete, so the even output rows are the upper block of
+`diag(E,I) U`, namely `[E, zO]`. Its last row, which is ordinary row 6,
+contains `f0,...,f6` in permuted order. Its absolute row sum is
+
+```text
+1 + 1/8 + 5/16 = 23/16.
+```
+
+The Schur-child macros change only odd rows. Row 6 therefore retains that
+sum through cut 21. The finite exact enumeration supplies the matching upper
+bound at every earlier and later proper cut. It also verifies that no symbol
+attains 23/16 before cut 11. The completed last row includes f7, giving 3/2.
+
+## Reproduce
+
+From the repository root, with standard-library CPython 3.12:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/triangular-parity-sign-cube/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/triangular-parity-sign-cube/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/triangular-parity-sign-cube
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/triangular-parity-sign-cube --check-source-history
+```
+
+Expected status is `PASS_TRIANGULAR_PARITY_SIGN_CUBE`, with `symbols=128`,
+`proper_prefixes=2688`, and `maximum_distribution={"23/16":128}`.
+Adding `--details` to either direct invocation emits every sign tuple and
+all of its 21 exact cut norms. The generated receipt includes the SHA256
+of this entire ordered certificate and of all five package source files.
+It writes nothing. No receipt is hand-written or retained as a separate claim.
+
+Eight controls reject the wrong root-left sign, a missing root-left shear,
+reversed macro order, an omitted symbol, a duplicated symbol, an omitted cut,
+a wrong reported maximum, and inclusion of the completed cut as a proper cut.
+Explicit exceptions keep every check active under `-O`.
+
+The verifier uses Python-managed allocation and exact rational objects. This
+is a Power of Ten rule-3 deviation: literal allocation-free steady execution
+conflicts with `Fraction` and the interpreter. All mathematical work is bounded
+by 128 inputs, a 15-node tree, 22 factors and 8-by-8 matrices. The five control
+copies are bounded by the same domain. The fixed finite input and no-write
+interface constrain live storage; no arbitrary input size is accepted.
+Functions stay below 60 lines, loops have explicit finite domains, and there
+is no recursion, dynamic evaluation or external command. Warning-strict normal
+and optimized replay are registered. No separate static analyzer was available on PATH.
+
+## Scope, sources and credit
+
+This contribution is produced by Nick Petalas. Independent human and formal
+proof-kernel review are unclaimed. Apache-2.0.
+
+Credit James Reed for the one-split bound, the nested-prefix question and the
+single-symbol series, including
+[#4811](https://github.com/CrocSwap/integer-mult-bounds/pull/4811),
+[#4812](https://github.com/CrocSwap/integer-mult-bounds/pull/4812),
+[#4815](https://github.com/CrocSwap/integer-mult-bounds/pull/4815),
+[#4816](https://github.com/CrocSwap/integer-mult-bounds/pull/4816),
+[#4820](https://github.com/CrocSwap/integer-mult-bounds/pull/4820),
+[#4821](https://github.com/CrocSwap/integer-mult-bounds/pull/4821) and
+[#4822](https://github.com/CrocSwap/integer-mult-bounds/pull/4822).
+Their individual values remain their results. The original addition here is
+the complete sign-family certificate and its sharp cut envelope.
+
+Credit JoyboyBrian for the
+[all-depth four-cut envelope](../triangular-parity-uniform-prefix/README.md)
+and its [sharp forward-prefix successor](../triangular-parity-forward-bound/README.md),
+which already settles the general 1+eta question; Vlado Vojdanovski for
+the shallow parity compiler retained by Cinna Davis in
+[SHALLOW-TOEPLITZ.md section 3](../unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md);
+and Ryanson Jonathan for the caller's error-ledger context in
+[one-factor-of-t](../one-factor-of-t/README.md). No predecessor credit line is changed.
+
+Canonical base is `624de9dc66858af0945e5f8a0942488d62280d43`, priority revision 7.
+[SOURCES.json](SOURCES.json) pins exact source bytes; [TASK.json](TASK.json)
+records the changed scope and bounded overlap coverage. The card's historical
+baseline command has been superseded by `scripts/research_selected_baseline.py`,
+which passed as a source-binding positive control. The selected conditional
+record remains kappa=60403/20000000. That orientation check is not a theorem replay.
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19132735)
+reserved this finite sign-family question.
+
+## Open next questions
+
+1. Is 23/16 still the exact maximum for this finite family under every legal
+   child interleaving? Enumerating pairs of legal child cuts is a feasible first
+   step. This certificate fixes even-first execution; the general theorem gives
+   only the weaker 3/2 bound for these schedules.
+2. Can one derive an analytic coefficient-sensitive envelope that explains
+   the 1/16 improvement? Start with the row support omitted by the unfinished
+   root left shear. The fixed-magnitude enumeration does not supply that formula.
+
+Full gate-level stability, generator normalization, the completed Gaussian
+stage, fixed-tape implementation and the dyadic p^100 precision ledger were
+not run or proved. There is no complete recursive invoice to admit here.

research/triangular-parity-sign-cube/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/triangular-parity-uniform-prefix/README.md",
+      "usage": "input",
+      "git_blob": "6e64fb48fbd33b9742f0ba81de77bfb661b80403",
+      "sha256": "dd0259414df08b5b462cec450df5a0618c30ba112ddf87b2acbb2e39a1c69414"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md",
+      "usage": "input",
+      "git_blob": "bc4349b2daea548ffa09e1f38a31f1df86b0b7be",
+      "sha256": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310"
+    },
+    {
+      "path": "research/triangular-parity-forward-bound/README.md",
+      "usage": "input",
+      "git_blob": "d008710585a897b9b568e74c7b867d8c784e4aa8",
+      "sha256": "745c2105fc614e6ba453327a37e483b808425a40ce3e2b44e2643b0ea1db3781"
+    }
+  ]
+}

research/triangular-parity-sign-cube/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact proper forward-prefix infinity norms for all 128 full-support n=8 symbols with f1=+/-1/8 and f2,...,f7=+/-1/16, using even-first macro execution.",
+  "scope_change_reason": "The all-size audit card is narrowed to an original complete finite sign-family result suggested by James Reed's single-sign examples. This does not discharge a quantified transfer premise.",
+  "bottleneck_link": "Scoped precision side result: identifies the exact finite envelope before the root left shear. All-size, gate-level and Gaussian caller precision costs remain open.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first original verified finite law; do not expand to other magnitudes or dimensions.",
+  "outcome": "completed",
+  "evidence": "All 128 distinct sign symbols, 2688 proper macro prefixes and 128 independent Toeplitz endpoint identities are checked exactly. Every symbol has maximum 23/16, first attained at cut 11. Eight enumerated mutation controls must fail.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Is 23/16 still the exact maximum for this finite family under every legal child interleaving?",
+      "why_it_matters": "The retained parallel compiler permits other child schedules.",
+      "first_step": "Enumerate pairs of legal child cuts for the fixed 128-symbol domain.",
+      "blocked_by": "The certificate fixes even-first execution; the general theorem gives only 3/2."
+    },
+    {
+      "question": "Is there an analytic coefficient-sensitive envelope explaining the 1/16 improvement?",
+      "why_it_matters": "It could replace finite family-specific enumeration.",
+      "first_step": "Analyze the row support omitted by the unfinished root left shear.",
+      "blocked_by": "The fixed-magnitude census does not supply a coefficient-variable formula."
+    }
+  ],
+  "coordination": {
+    "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19132735",
+    "upstream_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+    "priority_revision": 7,
+    "open_prs_at": "2026-10-22T21:08:49Z",
+    "open_pr_records": 348,
+    "discussion_records": 3780,
+    "discussion_fetched_at": "2026-10-22T21:09:37Z",
+    "coverage_limit": "Remote snapshots are paginated but not atomic; sparse local metadata is incomplete and arbitrary diffs are not searched. Distinctive README searches use all of origin/main.",
+    "precommit_upstream": "624de9dc66858af0945e5f8a0942488d62280d43",
+    "precommit_overlap": "Fresh open PRs and 30 latest handoff comments checked at 2026-10-22T21:13Z; no same-question claim. Distinct concurrent eta>1 and gate-cut tasks are excluded."
+  }
+}

research/triangular-parity-sign-cube/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/triangular-parity-forward-bound/README.md",
+    "research/triangular-parity-uniform-prefix/README.md",
+    "research/unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "All 128 sign symbols, all 2688 proper prefixes, all 128 Toeplitz endpoints, the sharp cut envelope, and eight rejecting controls; source-hash-bound generated receipt."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "All 128 sign symbols, all 2688 proper prefixes, all 128 Toeplitz endpoints, the sharp cut envelope, and eight rejecting controls; source-hash-bound generated receipt."
+    }
+  ]
+}