[petalas]

Summary

Scoped all-depth proof with exact finite corroboration. For the retained triangular Toeplitz parity compiler on f(z)=1+c z, the exact maximum forward-suffix infinity norm, over all completed-shear cuts and legal child interleavings, is

N = 2^h
M_h(c) = |c| + M_(h-1)(-c^2),   M_0(c)=1
M_h(c) = 1 + sum_{j=0}^{h-1}|c|^(2^j)

At |c|=1, suffixes therefore grow exactly as 1+log_2 N while the completed operator has norm 2. Brian's cited forward-prefix theorem also bounds prefixes by 2. A proper nontrivial cut attains the formula for c!=0, h>=2. This adds the exact binomial law and critical-tail suffix obstruction beyond #4731's four-wire witness and common four-cut envelope. No new kappa or supplier admission is claimed.

The result was produced by Nick Petalas. Credit Vlado Vojdanovski's retained parity compiler, Cinna Davis's later Gaussian caller, Brian's #4731/#4737 macro-cut results, James Reed's one-level accounting and finite-prefix examples, and their retained predecessor lineage. No predecessor executable is imported or audited.

Evidence

Before: a general 1/(1-eta) envelope for eta<1 and a four-wire binomial suffix example. After: the exact all-depth formula, an explicit attaining row and the unit-tail logarithmic separation.

The self-contained standard-library matrix checker passes 48 rational fixtures through N=32, enumerating 223 cut pairs and eight semantic/domain controls. Every constructed pair satisfies QP=T; the maximum and attaining row match the formula. For c=1/2, N=8/16/32 give 29/16, 465/256, 119041/65536. For c=1, they give 4, 5, 6.

Both registered warning-strict normal and optimized commands pass. Each direct run took about 0.39 seconds locally; registered replay took about 1.12 seconds, well below the command timeouts. These are verification durations. Generated stdout receipts bind all five package files by SHA256; normal and optimized receipts agree. Package metadata/source validation, committed-byte/source-history preflight and whitespace checks pass on CPython 3.12.15.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/triangular-parity-binomial-suffix
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/triangular-parity-binomial-suffix/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/triangular-parity-binomial-suffix/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/triangular-parity-binomial-suffix --check-source-history

Expected direct marker: PASS_BINOMIAL_SUFFIX fixtures=48 controls=8. Any CPython 3.12 executable can replace the local interpreter path.

Limitations

The all-complex, all-depth statement rests on the written induction; finite rational fixtures corroborate it. This concerns only binomial symbols and cuts between completed shears. Internal FFT gates, rounded arithmetic, Gaussian gauge conversion, fixed-tape costs, native admission, formal proof-kernel checking and the full selected replay are unrun obligations. No optimum over general symbols with a given tail budget is asserted. The retained eta<1 stability assumptions and selected frontier are unchanged. Independent human and formal proof-kernel review are unclaimed.

Open next questions

What is the sharp forward-suffix envelope over all unit symbols with tail l1 at most eta<1? Start with exact four-wire even/odd coefficient optimization and a correlation-preserving row invariant. The identity even child is essential to this proof, so the general case remains open.

Related open work

• #4782: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4796: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4798: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4799: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4800: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4802: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4811: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4812: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4815: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4816: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4820: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4821: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4822: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4823: a finite triangular macro-prefix example at a different symbol; this result gives all-depth binomial forward-suffix norms and does not recompute that example.
• #4824: the full signed eight-wire prefix family; different support and cut operator from this binomial suffix law.
• #4827: superunit-tail forward-prefix growth on a three-term symbol; this independently claimed binomial suffix result includes the critical unit-tail boundary.
• #4829: sharp Euclidean cuts inside one dirty Toeplitz shear; this result concerns composed macro suffixes in infinity norm.

These related titles and bodies were read. Their different symbols/operators were not computed for comparison, so no numerical agreement or disagreement is claimed. The merged #4731/#4737 predecessors are pinned in SOURCES.json.

This account's open PRs created since the CLAIM at 2026-10-22 21:13:47 UTC were refreshed before DONE. Related #4824, #4827 and #4829 are already listed above; their symbols, cut operators or norms differ. The other new contributions study address-algebra kernels, connector or projector tariffs, complex-port conditioning, partial-channel rank or integer job splitting. No common numerical instance is compared and no extra predecessor computation was run.

CI notes

Both hosted checks passed on contribution head ffd229598f3a1a87437e02c74a459498f257f6d5 in run 39856254958: plan in 3m47s and registered package replay in 1m14s. Run metadata confirms that exact head and a successful completed run. The replay executes both registered warning-strict normal and optimized commands.

All registered and direct local checks, committed-byte/source-history preflight and whitespace checks also pass. No check was left solely to CI and no CI failure was observed. These are scoped automated package checks; independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way. Blast radius: package. Only five files under research/triangular-parity-binomial-suffix/ are added.