Prove exponential parity-prefix growth above tail one (no κ)

research/triangular-parity-tail-threshold/README.md

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+# The sharp tail threshold for forward parity prefixes
+
+For every fixed real `eta > 1`, the exact triangular Toeplitz parity compiler
+has a family whose completed map has infinity norm `1+eta`, but whose first
+proper macro-prefix has norm
+
+```text
+((eta+1)/2)^(N/2-1),       N = 4,8,16,... .
+```
+
+This is a **scoped algebraic proof with finite controls**. It rules out even a
+polynomial-in-`N` forward-prefix bound based only on a fixed coefficient-tail
+budget greater than one. Together with Brian's
+[#4737](https://github.com/CrocSwap/integer-mult-bounds/pull/4737) theorem for
+`eta <= 1`, it identifies one as the exact threshold for a dimension-independent
+forward macro-prefix bound over the complete tail-budget class. The new part is
+the exponential family for every `eta > 1`; the complementary theorem is cited,
+not re-proved or replayed.
+
+## Proposition and proof
+
+Use the exact parity recursion in
+[`SHALLOW-TOEPLITZ.md`, section 3](../unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md).
+A macro is a completed Toeplitz shear. At every node the right shear runs first,
+the two children run next, and the left shear runs last. The proof only needs
+the root right shear, so it applies to every legal child interleaving.
+
+Fix `eta > 1` and put
+
+```text
+a = (eta+1)/2 > 1,       b = (eta-1)/2 = a-1 > 0,
+f(z) = 1 + b z + a z^2,  N = 2m >= 4.
+```
+
+Let `T_N(f)` denote multiplication by `f` modulo `z^N` in coefficient order.
+Its tail coefficient l1 norm is `a+b=eta`. Every row has absolute sum at most
+`1+a+b`, and every row with index at least two attains it. Thus
+
+```text
+||T_N(f)||_infinity = 1+eta = 2a,
+```
+
+independently of `N`. In even/odd order the root split has
+
+```text
+e(z) = 1+a z,       o(z) = b,
+k(z) = o(z)/e(z) = b sum_(j=0)^(m-1) (-a)^j z^j mod z^m,
+U = [ I T_m(z k) ; 0 I ].
+```
+
+All inverses here are finite triangular inverses, since `e(0)=1`. No analytic
+convergence is assumed. The first macro-prefix is exactly `U`. Its upper block
+row with within-block index `i` has absolute row sum
+
+```text
+1 + b sum_(j=0)^(i-1) a^j = a^i,       0 <= i < m.
+```
+
+For `i=0` the sum is empty. The largest row is `i=m-1`, so
+
+```text
+||U||_infinity = a^(m-1) = ((eta+1)/2)^(N/2-1).
+```
+
+The shear is a proper prefix because it precedes the children and the nonzero
+root left shear. Coordinate permutation preserves this norm. Also
+`U^(-1)=[I -T_m(zk);0 I]`, so the same exact norm holds for its inverse.
+The norm is attained on a unit infinity-norm input: choose each input coordinate
+to have the sign of its coefficient in the largest row, and zero when that
+coefficient vanishes. Thus the growth is an actual amplification of bounded
+data, not merely a loose product estimate.
+
+For fixed `a>1`, the logarithm of the displayed norm is `(N/2-1) log(a)`.
+It grows linearly in `N`, whereas the logarithm of any fixed polynomial grows
+as `O(log N)`. Therefore no dimension-independent bound, or polynomial bound
+with constants depending only on `eta`, can cover every symbol in this class.
+This conclusion already holds with three nonzero coefficients and nonnegative
+input-symbol coefficients. The proof is valid for every real `eta>1`; the
+finite checker uses rational instances only.
+
+For example, `eta=3` gives `f=1+z+2z^2`. At `N=16` the first prefix has norm
+`128`, while the completed map has norm `4`. These are exact values checked
+by the verifier.
+
+## What this changes and what it does not establish
+
+The completed map's bounded forward norm alone cannot justify increasing the
+parity compiler's permitted tail budget beyond one. Any such extension needs
+an additional restriction or a changed factorization. Brian's #4737 gives a
+single `eta=2` violation of the `1+eta` cap; this result strengthens that domain
+observation to exponential growth for every fixed superunit budget.
+
+The retained triangular caller in `SHALLOW-TOEPLITZ.md` assumes `eta<1`.
+Nothing here contradicts that theorem or changes the selected frontier.
+This family is not asserted to have uniformly bounded inverse endpoint norm.
+The obstruction is specific to the exact parity factorization and its macro
+boundaries. It does not exclude other compilers, settle internal FFT or shear
+gate accounting, provide a physical supplier, or prove a complete integer
+multiplication theorem. No new kappa is claimed. Formal proof-kernel and
+independent human review are unclaimed.
+
+The original `uniform-transfer-audit` card changes to this algebraic domain
+side result. The original card, scope reason and remaining obligations are in
+[TASK.json](TASK.json). The current selected source-binding orientation passed
+with `scripts/research_selected_baseline.py`; the card's older balanced-schema
+control is historical. No predecessor executable was replayed.
+
+## Reproduction and finite evidence
+
+Run from the repository root with CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/triangular-parity-tail-threshold
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/triangular-parity-tail-threshold/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/triangular-parity-tail-threshold/verify.py
+```
+
+Expected marker:
+
+```text
+PASS_PARITY_TAIL_THRESHOLD fixtures=25 controls=7 max_n=64 no_new_kappa
+```
+
+The 25 fixtures are the full Cartesian product of tail budgets
+`17/16, 5/4, 3/2, 2, 3` and dimensions `4,8,16,32,64`. For every fixture the
+checker independently divides the finite series, applies the root right,
+diagonal and left factors to every input column, and compares the complete
+product with the literal Toeplitz endpoint. It checks both inverse orders for
+the first shear, its exact norm, an attaining signed input, the endpoint norm,
+and the dyadic growth identity `p_(2N)=a p_N^2` on every adjacent size pair.
+Seven controls cover the tail boundary, the finite upper tail limit,
+non-dyadic size, size exhaustion, nonunit division, a missing shift and the
+wrong Schur sign. These are controls of the new family, not rechecks of a
+predecessor's claimed counts.
+
+Each run prints a generated JSON receipt containing these results and SHA256
+hashes of all five package files. No receipt is hand-authored or committed.
+The all-size claim rests on the preceding proof, not these finite samples.
+Lean, Docker, internal-gate compilation and the selected full operator replay
+are outside scope and unrun.
+
+The checker has no imports from predecessor code, network calls, subprocesses
+or writes. Source documents are pinned as historical provenance in
+[SOURCES.json](SOURCES.json). Its loops have fixed finite bounds, with matrices
+of at most `64 x 64` and 25 fixed rational fixtures. Functions are under 60
+lines, and explicit contracts remain active under optimization. Python's
+managed allocation and arbitrary-precision arithmetic are a Power of Ten
+deviation: live dimensions and coefficient inputs are bounded by this fixed
+grid. The rejection helper's indirect calls have five explicit local targets.
+Warning-strict normal and optimized execution are the available diagnostics.
+No standalone linter or type checker is installed in this environment.
+
+## Open next questions
+
+- Can a changed factorization retain polynomial forward-prefix norms on a
+  class with tail budget above one and a separately bounded inverse endpoint?
+  This would distinguish a compiler limitation from unavoidable conditioning.
+  Start with a concrete zero-free symbol whose even part has an interior zero,
+  then price the replacement factorization. Such a class and its inverse bound
+  have not been constructed here.
+- Can a caller normalize a superunit-tail triangular symbol into near-identity
+  factors without repaying the gain through coefficient generation, storage
+  or additional shears? Start with one caller's literal symbol and complete
+  invoice. Caller-specific spectra and implementation costs remain unknown.
+
+## Credit
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Brian's #4737 supplies the complementary forward theorem and the prior
+`eta=2` example; #4731 supplies the common four-cut envelope. James Reed's
+#4699 and subsequent finite parity-prefix packages supply the one-level
+budget and individual examples. The parity compiler is retained from Vlado
+Vojdanovski's #1673 and Cinna Davis's #4100 caller. Credit also Ryanson Jonathan
+and unpaidmoon #4371 for the preceding conditioning question. The new
+contribution is the sparse exponential family and the universal superunit-tail
+obstruction. All predecessor files and credit lines are unchanged.

research/triangular-parity-tail-threshold/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-forward-bound/README.md",
+      "usage": "provenance",
+      "git_blob": "d008710585a897b9b568e74c7b867d8c784e4aa8",
+      "sha256": "745c2105fc614e6ba453327a37e483b808425a40ce3e2b44e2643b0ea1db3781"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md",
+      "usage": "provenance",
+      "git_blob": "bc4349b2daea548ffa09e1f38a31f1df86b0b7be",
+      "sha256": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310"
+    }
+  ]
+}

research/triangular-parity-tail-threshold/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": "For every real eta>1, prove an exponential first forward macro-prefix in the exact triangular parity compiler, despite a completed forward norm of 1+eta.",
+  "scope_change_reason": "This is an original algebraic domain side result following Brian #4737, not an audit of the retained eta<1 caller. It isolates the universal forward-prefix threshold, separate from finite half-tail examples and internal gate bounds.",
+  "bottleneck_link": "Precision side result: bounded completed forward norm cannot alone extend this parity factorization to a superunit coefficient-tail class; a different restriction or factorization is needed.",
+  "budget_minutes": 60,
+  "stop_rule": "Ship the first proved original obstruction, with bounded rational controls; do not expand to a replacement compiler.",
+  "outcome": "negative-result",
+  "evidence": "README.md contains the all-real-eta proof; verify.py checks 25 exact root factorizations, attaining inputs and seven controls through N=64.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can a changed factorization retain polynomial forward-prefix norms above tail one when the inverse endpoint is separately bounded?",
+      "why_it_matters": "This separates avoidable compiler amplification from unavoidable inverse conditioning.",
+      "first_step": "Construct one explicit zero-free symbol whose even part has an interior zero, then specify the replacement factorization.",
+      "blocked_by": "No such symbol class with a uniform inverse bound or replacement invoice is supplied here."
+    },
+    {
+      "question": "Can a concrete caller normalize a superunit-tail triangular symbol into near-identity factors within its paid budget?",
+      "why_it_matters": "A normalization might avoid the obstruction, but additional shears and coefficient preparation can repay its gain.",
+      "first_step": "Use one literal caller symbol and price coefficient generation, storage and every additional factor.",
+      "blocked_by": "Caller-specific spectra and complete implementation costs remain unknown."
+    }
+  ]
+}

research/triangular-parity-tail-threshold/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact 25-fixture root factorization, norm, attaining-input and growth-recurrence controls, plus seven domain/algebra controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Repeat the same complete finite domain under optimization; every explicit contract must remain active."
+    }
+  ]
+}