[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. No new kappa or selected-frontier change.

For forward seven-axis B/C whole calls, grant a free central odd Gaussian input gauge on at most q main fields while retaining the cited paid profile and separate helper contract. The exact optimal integrated sign-cut capacities are:

d = 8^t - 7^t, 0 < t < 1
a = min(q,56)
b = min(56, floor(max(q-56,0)/2))
K = 28 + 140*7^t + 84*8^t

B: K + d*(a+b)
C: K + ((1+d)/2)*a + d*b

At saving 17/5000, one B call's optimistic signed bill remains negative through q=30 and first has a positive upper bound at q=31. C remains negative through q=22 and first passes at q=23. These are exact thresholds of a necessary capacity screen. Passing does not construct independent productive outputs or a profitable caller.

The new step is the complete 12-choice local gauge classification, its optimal budget allocation and the exact threshold certificates. It takes the odd-input-gauge direction in PR6639. Ali Romman's capacity theorem and P/H profiles are cited inputs, not audit deliverables. Credit unpaidmoon, Ali Romman, Dugongue, Cao Yifan/intqwq, Chris R/Clear1nsight and the retained B/C lineage. Dugongue's fine-gap-gaussian-boundaries consumer constructs related chronological incidences and an alternating schedule; those are not reverified here.

Advance claim.

Evidence

Before, the retained whole-caller theorem excluded free odd input gauges. After this change, the forward central-input case has an exact capacity envelope for every budget 0..224 and every exponent 0<t<1.

Local CPython 3.12.15 normal and optimized direct checks passed, timed at 0.768 s and 0.771 s. Both registered commands passed, as did metadata validation, committed-byte/source-history preflight and a Git-free empty-PATH export. A corrupted cited source and two actual implementation mutations, an undercharged A gauge and a substituted B endpoint, were rejected.

The checker enumerates 384 literal matrix rank cases, 3,072 sign-mask cases and 29,700 budget comparisons against a separate dynamic program. Integer-power inequalities certify all threshold endpoints. A freshly generated stdout receipt binds the package files by SHA256.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -X dev -W error -B research/bc-central-input-gauge-budget/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -X dev -W error -O -B research/bc-central-input-gauge-budget/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/bc-central-input-gauge-budget
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bc-central-input-gauge-budget --check-source-history

Another CPython 3.12 path may replace the displayed interpreter. Expected output includes PASS_BC_CENTRAL_INPUT_GAUGE_BUDGET B=31 C=23 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The free gauge count is per call occurrence, including reused prepared fields. Mixed callers use the sum of per-call envelope gains minus the cited deficits, not a universal integer threshold per call.

Only forward central input-gauge placements are covered. The separate helper contract and cited paid profiles remain hypotheses. A physical gauged producer, output gauges, inverse/transposed placements, different axes or shared helper/main resources are not supplied. Native replay, compiler/hosting/precision/tape and all-size admission, the selected cold replay, independent human review and proof-kernel review are unrun. The five framework assumptions remain.

Open next questions

Can an arbitrary-input native producer supply these odd-gauged main inputs for less than their separate H1 fees while retaining all independent helpers? First give that producer and its complete preparation/restoration bill, then apply its actual occurrence budget to the envelope. Separate H1 fees already exceed the capacity gain. A favorable upper bound alone does not justify another caller search.

Related open work

• PR6639 bounds post-call ordinary helper feedback and explicitly leaves odd input gauges outside its algebra. This changes that input hypothesis in the forward central-axis case; it does not audit its helper counts or recompute a number it reports.

A refresh on 2026-11-03 found no additional B/C gauge-capacity study among this account's open PRs created since the advance claim.

The final overlap refresh covered 619 open PR bodies, 6,724 Discussion records and all tracked local note paths. Main and merged-work searches found no earlier central-input budget envelope. Those searches do not prove global novelty.

CI notes

Both hosted checks passed at head e0b9e4f: plan and replay (research/bc-central-input-gauge-budget), in run 41671115344. Local normal/optimized, metadata, source-history, export and corruption checks also passed.

Merge Danger

Door: two-way. Blast radius: isolated. Only the new research package is added; no shared code, frozen predecessor package or selected record changes.