[petalas]

Summary

Scoped algebraic proof with finite controls: the exact triangular parity compiler can have exponential proper-prefix amplification for every fixed real eta>1, despite a completed forward norm bounded independently of dimension.

a = (eta+1)/2 > 1, b = a-1, f(z) = 1+bz+az^2
||T_N(f)||_infinity = 1+eta
||first right-shear prefix||_infinity = a^(N/2-1)

Together with Brian's cited #4737 theorem at eta<=1, this makes one the sharp threshold for uniform forward macro-prefix control over the complete tail-budget class. The new result is the exponential family above one. No new kappa, supplier admission or multiplication theorem is claimed. The selected frontier stays unchanged.

Evidence

Before: #4737 supplies a single eta=2 counterexample to extending its cap. After: a geometric-series proof supplies exponential growth for every eta>1, on a three-term nonnegative symbol. An explicit bounded input attains the norm. For eta=3 and N=16, the prefix norm is exactly 128 while the completed norm is 4.

Local warning-strict normal and optimized verification each pass all 25 exact rational fixtures and seven controls through N=64. Every fixture compares the root factorization's entire matrix to its literal Toeplitz endpoint, checks both shear inverse orders and an attaining input. Each registered command takes under one second locally; the registered replay takes about 1.4 seconds. These timings establish timeout headroom, not a performance improvement. Generated JSON receipts bind every package file by SHA256. No receipt is hand-authored.

From the repository root, CPython 3.12, standard library only:

/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
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/triangular-parity-tail-threshold --check-source-history

Expected marker: PASS_PARITY_TAIL_THRESHOLD fixtures=25 controls=7 max_n=64 no_new_kappa.

Limitations

The all-real-eta conclusion is a written scoped proof; finite rational fixtures corroborate it. The obstruction is specific to this parity factorization at completed-shear boundaries. The family is not claimed to have a uniformly bounded inverse endpoint. The retained eta<1 theorem and its actual caller are not contradicted. Rounded FFT/shear gates, alternate compilers, Gaussian integration, precision/tape costs, native admission, Lean, Docker and selected full replay remain outside scope and unrun. Independent human and formal proof-kernel review are unclaimed.

The package pins Brian's forward theorem and Vlado Vojdanovski/Cinna Davis's compiler as historical sources. Credit also Brian #4731, James Reed #4699 and the finite lineage, Ryanson Jonathan and unpaidmoon #4371. No predecessor executable or result is audited. This contribution was produced by Nick Petalas.

Open next questions

• Can a changed factorization avoid this amplification when the endpoint and inverse endpoint are separately bounded? Start with a specified zero-free symbol and its even-part zero; no such class is constructed here.
• Can one caller afford normalization into near-identity factors after coefficient generation, storage and extra shears are all charged? Start with the caller's literal symbol and complete invoice.

Related open work

• #4822: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4821: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4820: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4816: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4815: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4812: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4811: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4802: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4800: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4799: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4798: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4796: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4782: one half-tail eight-wire prefix example; this result uses a different three-term family at every eta>1, with no shared numerical instance claimed.

• #4823: one tail-5/8 eight-wire prefix example; this result concerns every tail budget above one and does not compute its instance.

• #4824: the finite half-tail signed cube; this proof concerns a different all-size, superunit-tail family and does not replay its certificate.

• #4829: sharp Euclidean gate-cut bounds inside one dirty Toeplitz shear; this result concerns infinity-norm growth at a completed root macro with a growing split-symbol budget. No common numerical instance is compared.

• #4831: exact binomial forward-suffix norms, including the unit-tail boundary; this result uses three-term symbols above tail one and a forward prefix. No common numerical instance is compared.

The related PR titles and bodies were read. No extra predecessor computation was performed for comparison, and no numerical agreement or disagreement is asserted on their different symbols.

CI notes

Both hosted checks passed on publication head 3f8fc3482ebed70149996a502a839b13ec91bfcf in run 39855955622: plan in 2m43s and the registered package replay in 1m25s. Run metadata confirms this exact head and a successful completed run. The replay executes both registered warning-strict normal and optimized commands.

Both registered commands, committed-byte/source-history preflight and official export integrity also pass locally. An isolated optimized replay from the read-only Git-free export produces exactly the same generated receipt as both registered modes. No check was left solely to hosted CI; no CI failure was observed or diagnosed. This records execution of scoped package checks, not independent human or proof-kernel acceptance of the all-size statement.

This account's open PRs created since the CLAIM were refreshed before DONE. Related #4824 is already listed; newly published #4829 and #4831 have been read and added above. Their symbols, cut operators or norms differ, so no matching numerical value is asserted or extra predecessor computation run.

Merge danger

Two-way door. Blast radius: package. Only research/triangular-parity-tail-threshold/ is added.