[petalas]

Summary

Scoped constructive proof with exact finite corroboration, produced by Nick Petalas. For any dyadic cut matrix K and 0 < epsilon < 1, approximate unit-channel capacity in induced infinity norm is exactly rank_Q(K). Odd Smith factors still obstruct exact normalization. The changed approximate contract has an explicit dyadic precision bound and exact restoration of arbitrary dirty helpers:

h += Y x
y += A h
h -= Y x
y -= A h
endpoint: x unchanged, h restored, y += A Y x

Four nonzero block shears are necessary and sufficient in this two-kind alphabet. A fixed scalar reader A=[3] has sharp minimum relative error 2^-b with b fractional insertion bits. Every scalar coefficient is lowered explicitly to signed unit ADDs and SCALE 2/half, with a complete logical event count and intermediate guards.

Credit Ali Romman's PR5615 exact-capacity criterion and PR1135 same-grid borrowing distinction, PR5602/5611/5599, Gareth Davidson's PR1057, PR985/1054, Nick Petalas's PR5115 and the retained helper/native/controller/compiler lineage. Dyadic density, rational inversion, shear commutators and binary synthesis are classical; the contribution is this changed interface, quantitative normalizer and matching scoped bounds. No predecessor result is audited or replayed. No new kappa or selected-frontier change.

Evidence

Before, exact dyadic unit capacity can vanish at positive rational rank. After, the proof constructs arbitrarily accurate responses while restoring arbitrary dirty helpers exactly and charging both observations and deliveries.

Local CPython 3.12.15 checks pass:

• All 625 matrices in the declared 2-by-2 integer domain: 496 constructions and 129 singular rejections; six larger/rectangular/dyadic cuts through four channels.
• Complete literal elementary-gate endpoints on 4,022 basis/dirty vectors and 2,510 additional large-dirt states, compared with a separate matrix-product endpoint.
• 123,060 integer candidates for 195 sharp odd-scalar cases; factor-three depths 0 through 64; 5,425 two-role executions for 217 coefficient presentations.
• Nine exact-reason input rejections and four semantic controls. Warning-strict normal and optimized checks pass, individually timed at 0.38 seconds, within separate 60-second limits. These are replay-budget observations, not a performance comparison.
• Registered metadata/replay, exact committed bytes, executable modes and source history pass at 042b4ee85f64f4b3279426e2ed10191296448e6c. Official export integrity and declared-input coverage pass. Both modes pass from a read-only Git-free export with empty PATH; all eight exported files retain their hashes. Receipts are generated from executed checks and bind every package file.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/approximate-dyadic-dirty-channels/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/approximate-dyadic-dirty-channels/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/approximate-dyadic-dirty-channels
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/approximate-dyadic-dirty-channels --check-source-history --check-modes

Any CPython 3.12 executable may replace the absolute path. Expected markers: PASS_APPROXIMATE_DYADIC_DIRTY_CHANNELS, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

This uses exact dyadic gates on a sufficiently refined grid. Independently rounded gates are outside the proof. The algebraic response capacity is not independent productive-target credit. No actual physical P104 cut, native common-chart implementation, caller installation or favorable complete signed invoice is supplied. Native pricing, chronological access, bases, bounded dirty storage, stock, tape, complete rounding/recovery and uniform compiler obligations remain. The executable constructor has the explicit finite domain in README; the arbitrary-size theorem has a written proof.

Selected cold replay, Lean, Docker and physical hosting were not run. Independent human and formal proof-kernel review are unclaimed. All five selected framework assumptions remain. The original coverage-law-change card is explicitly changed to this approximate local interface result.

Open next questions

1. Nominate one actual cut and caller error allowance, then price both observations and both deliveries, compatible charts and every extra charge before crediting productive targets.
2. Give a fixed-grid implementation with incoming source/helper widths, enough fractional bits for the lowered gates and complete consumer error propagation. The grid and recovery budget are missing here.

Related open work

• Prove sharp one-channel capacity change for paired cut edits (no κ) #6462 bounds exact dyadic unit-capacity changes under paired cut edits; this changes the normalization tolerance and supplies no edited physical instance.
• Prove sharp dyadic channel loss under frozen snapshots (no κ) #6465 studies exact capacity with frozen snapshots; this imposes no snapshot restrictions and compares no common instance.
• Prove exact heterogeneous tensor-cut multiplicities (no κ) #6649 optimizes exact heterogeneous tensor-cut capacity; this does not pool or optimize tensor types and computes no common density.
• Exact unbiased native coarsening with a sharp residue codec #6540 gives unbiased native coarsening with retained residues; this uses deterministic exact-dyadic shears and no stochastic/native codec.
• Prove modular normalization metrics and a paid query completion (no new κ) #6491 studies exact modular normalization and its metrics, including a pairing-three cut; this uses real absolute approximation on D, not modular inverse normalization, and compares no common metric constant.
• Prove sharp dyadic two-wire approximation floors (no κ) #6614 gives approximation floors under exact Gaussian-dyadic unitarity; this uses nonunitary shears and does not replace an orthogonal two-wire gate.
• Prove exact isolation capacity for shared phase charts (no κ) #6585 studies exact common-chart isolation capacity; this does not establish a common physical chart or independent Gaussian isolation.

No extra computations checked another contribution's numbers.

After CI, this account's open PRs were refreshed and all 12 other submissions created since the advance CLAIM were inspected. None studies this approximate dyadic capacity contract or computes a conflicting common instance. The related entries above remain applicable.

CI notes

Both hosted checks passed at exact head 042b4ee85f64f4b3279426e2ed10191296448e6c in run 41670471284: plan and replay (research/approximate-dyadic-dirty-channels). The completed check list was read once; both report pass, and the PR retains that head.

Hosted replay certifies execution of the registered normal and optimized finite checks and package metadata validation. The local publication/source-history and portable export checks above also passed. The delivered scope matches the advance CLAIM. No check was waived. Independent human and formal proof-kernel review, physical supplier admission and the remaining mathematical implementation obligations are unclaimed.

Advance CLAIM.

Merge Danger

Door: two-way.

Blast radius: isolated. Seven files in one opt-in research package. Shared implementations, pinned evidence and selected-result records retain their bytes.