[petalas]

Summary

Scoped written mathematical proof with exact finite corroboration. At a fixed dyadic helper cut, a scalar ADD paired with its inverse in the suffix changes unit-channel capacity by at most one. An ADD within either side preserves capacity exactly. The bound is attained for every number of crossing edits, with every arbitrary dirty input restored.

A' = T A,  B' = B T^-1,  T = I + a e_i e_j^t
fixed cut F, helper support G and complete endpoint BA

exactly one of i,j in F:  |capacity' - capacity| <= 1
both or neither in F:     capacity' = capacity

This answers a restricted changed-prefix opening in zxv's PR6064. Its fixed-prefix support-enlargement theorem is cited, not audited. Ali Romman's PR5615 supplies the saturated unit-pairing capacity criterion, with PR5602/5611/5599 and their retained helper/native/controller lineage. Module splitting, coordinate kernels and minor ideals are classical. This contribution was produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

No new kappa, supplier admission or selected-frontier change is claimed.

Evidence

• Before: the support-enlargement theorem leaves changed prefixes outside its scope. After: the proof gives the sharp one-channel bound for paired elementary updates adjacent to a fixed cut, plus same-side invariance and an attaining family.
• Exact local checks pass on 12,928 paired edits over 80 prefix instances: 4,032 same-side and 8,896 crossing edits. Capacity changes are -1 in 1,276 cases, zero in 11,625 and +1 in 27. The new checker computes full modules and every minor gcd without importing a predecessor executable.
• The sharp family passes full module/unit-block checks and literal execution on 7,380 signed dirty states. Six invalid ADDs reject, and six further algebraic controls pass.
• Warning-strict normal and optimized direct checks pass in 0.781385 and 0.787502 seconds locally on CPython 3.12.15/macOS arm64. These are timeout-budget observations, not a speedup measurement. Both registered checks have separate 60-second limits. Registered replay, metadata, committed bytes/modes and declared source history pass at 30314bb629a21f9ed5ed709021cdc808214265a1.
• Official export integrity and declared-input coverage pass. Optimized Git-free replay with empty PATH passes from read-only exported files, leaving all nine files unchanged. An intentional source-byte corruption rejects before mathematical execution. Receipts are generated from execution and bind source/package hashes; none is hand-written.

Exact reproduction from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/paired-prefix-cut-capacity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/paired-prefix-cut-capacity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/paired-prefix-cut-capacity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/paired-prefix-cut-capacity --check-source-history --check-modes

Another CPython 3.12 interpreter path may replace the local path. Direct commands use only the standard library and write nothing. Expected markers are PASS_PAIRED_PREFIX_CUT_CAPACITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The ADD must be elementary in the physical coordinates of the nominated cut. An earlier interior-prefix update may become a dense conjugate there and is outside this theorem. The cut/helper supports and complete endpoint stay fixed; inverse suffix pairing is required. The written proof covers arbitrary finite dimensions, while the executable domains are explicitly finite.

No actual selected-word installation, native common chart, timing, delivery, independent productive Gaussian target, precision/growth, storage, ownership, compiler or complete paid/target invoice is supplied. Scalar ADD counts are not recursive child counts. The original coverage-law-change card is explicitly narrowed to this algebraic side result. All five selected framework assumptions remain. Selected scientific replay, Lean and Docker were not run.

Open next questions

1. Determine the sharp change for a dense conjugate of an earlier prefix update. First pin its transport and both full eligible modules; the elementary-coordinate argument does not apply automatically.
2. Find a legal native cut where the extra capacity funds the complete signed invoice. First extract exact charts, timing, dirty endpoints, paired edits and delivery costs. Algebraic capacity alone supplies no productive target credit.

Related open work

• PR Prove sharp dyadic channel loss under frozen snapshots (no κ) #6465 studies the same dyadic unit-channel capacity under frozen snapshot constraints. It holds the pairing fixed and restricts insertion directions, while this result changes the prefix and suffix by paired elementary updates at fixed supports. No common physical instance or numerical value is computed or compared.

No inspected open PR computes capacity change for this fixed-support paired elementary-edit contract. The source PRs #5615 and #6064 are merged. The live open/merged/main and claim searches were refreshed before the commit; sparse historical notes and unpublished work remain coverage limits.

• PR Conditional κ = 0.003287014 (+1.65e-7 over #6417): a third group-1091 anchor on the clean mixed root #6453 changes an actual P104 helper trajectory and its complete mixed-root invoice. This proof supplies no edit or numerical saving on that word and does not audit its reported kappa.
• PR Mixed-query root: conditional κ = 0.00328621476572836015146308 (+4.28455% over #6375) #6398 changes the mixed-query root contract. This scalar cut lemma neither constructs that root nor computes its component saving.

The completion refresh inspected the other open account PRs created since the advance CLAIM, including PR6465. None was found to compute the same paired-edit instance or a conflicting common value. No extra computations checked another contribution.

CI notes

Both hosted checks passed at exact head 30314bb629a21f9ed5ed709021cdc808214265a1 in run 41550427580: plan and replay (research/paired-prefix-cut-capacity). The completed checks were read once; both report pass, and the PR still has the matching head.

Hosted replay certifies execution of the two registered normal and optimized finite checks at this commit. All local checks listed above also passed. This does not supply independent human or formal proof-kernel acceptance of the universal written proof or discharge the physical/supplier obligations. No CI failure remains.

Advance CLAIM.

Merge Danger

Door: two-way.

Blast radius: isolated. Six new files in one opt-in research package. Shared code, frozen sources and the selected result are unchanged.