[petalas]

Summary

A completed Gaussian readout can still affect retained dirty state. This scoped written proof gives the exact number of scalar channels that two invertible maps can isolate using the same input and output bases, with both cross blocks zero:

A = L^-1 R, N = A-I
completed readouts:     nullity(N)
independent isolation:  2 nullity(N) - nullity(N^2)

The proof constructs attaining bases over any field. For even-q bases over Q(i)[q^2] inside Q(i)[q]/(q^4-1), the exact rational capacity is the smaller of the two opposite-phase capacities.

The new synthetic example q^-1 (I+((1+q)/2)E12) has a completed central Gaussian readout and a literal inverse, yet zero Gaussian isolation capacity, even after adding any finite number of restored identity workspace coordinates. This is a scoped all-dimension obstruction with exact finite corroboration, not a search failure. No particular published caller is rejected, and no new kappa or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. Invariant complements, primary decomposition, rank-nullity and idempotent interpolation are classical; no global priority is claimed for the general direct-summand formula. The contribution is its explicit paired ordinary-chart criterion, constructive interface and sharp workspace-stable Gaussian example. Ali Romman's PR6398/6545/6551 and Cinna Davis's credited native/thin-query lineage supply context. Dillon / dillona's merged PR4623 has a different even-q exactification obstruction. Their constructions, identities, costs and credit are retained; none is replayed or audited here.

Only research/phase-paired-isolation/ is added. Base is canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected conditional kappa .003125804. Advance CLAIM.

Evidence

• Before, the example exposes one completed readout. After, the written capacity theorem proves no independent Gaussian block is possible under the stated common even-q charts, including arbitrary finite restored identity workspace.
• The exact checker enumerates 175 binary relative maps and 28,261 conjugations, 36 binary map pairs and 1,296 two-sided basis pairs, and 48 ternary relative maps with 2,304 conjugations. Constructive elimination agrees with separate exhaustive oracles.
• It checks 54 rational profiles, 64 four-phase branches including sixteen with positive capacity, twelve identity and twelve Gaussian workspace profiles, and eleven rejection controls. The universal conclusions rest on the written proof.
• Both warning-strict registered commands passed on CPython 3.12.15, macOS 27.0.1 arm64. Each was timed locally: normal 0.357 seconds, optimized 0.363 seconds, against separate 120-second limits. Their generated hash-bound receipts agree exactly.
• The package runner, syntax/function-size checks, committed-byte/source-history preflight and whitespace check pass. An optimized disposable nullity-only mutation is rejected. A read-only Git-free copy passes optimized replay with an empty PATH; all eight source files and modes remain unchanged. No external static analyzer was installed or claimed.

From the worktree root, reproduce with the available interpreter:

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

Other hosts may substitute an absolute CPython 3.12 path. Expected markers are PASS_PHASE_PAIRED_ISOLATION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Receipts are generated to stdout and bind every package file; no receipt is handwritten. There are no predecessor runtime inputs or scientific replays.

Limitations and unrun obligations

Both input and output bases must be common to the compared maps, and both cross blocks must vanish on arbitrary inputs. Later access to the retired input, odd-q chart coefficients or a changed consumer contract can evade the obstruction. The four-phase statement uses the complete formal central ring; a representation omitting some branches needs its actual spectrum analyzed.

The attaining alphabet grants arbitrary rational field bases. Dyadic Gaussian synthesis, an actual source-bound cut, intermediate chronology, native realization, ownership, source access, precision, stock, tapes and the complete paid caller bill remain unproved. All five selected framework assumptions remain. Selected cold replay, Lean and Docker are unrun. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Bind one actual cut to its complete phase maps, scalar target and allowed charts. The absent caller blocks a physical implication.
2. Determine when the rational attaining bases admit invertible dyadic Gaussian realizations. Begin with primitivity of the kernel/image complements; the field proof does not provide integral synthesis or its cost.

Related open work

• PR6398 constructs a mixed-query root. This result gives a synthetic common-chart criterion and makes no calculation about that root's capacity or invoice.
• PR6545 constructs an independent Gaussian retirement at an actual cut. Its identity is neither extended nor audited; the example here has a different map and boundary.
• PR6551 changes the child allocation under that retirement. This theorem changes no child allocation, paid profile or reported kappa.
• PR6554 independently audits that specific retirement. This contribution is an original abstract obstruction and does not duplicate its audit.
• PR6572 derives stored native phase tags. It concerns a metadata representation, while this result constrains simultaneous input/output bases; no common numerical quantity is compared.
• PR6573 changes spatial correlation in a Gaussian coarse-grid codec. Its prefix variance and residue capacity concern a different producer contract; no phase-chart isolation capacity is computed there or compared here.
• PR6574 constructs a reversible four-axis Gaussian grid map. Its rounding error and inverse concern a different operator and boundary; this result supplies no grid-error comparison.

The completion refresh inspected all fourteen other account open PRs created since the advance CLAIM at 2026-11-03T10:06:03Z, PR6572 through PR6586 excluding this PR. None computes the simultaneous phase-chart isolation capacity or supplies a competing value computed here. The adjacent phase-metadata and Gaussian-rounding work is distinguished above. No extra scientific computations audited or compared those contributions.

Current live claims, open/account PRs, main README and merged general-form/object searches and local reservations were checked before claiming and committing. The precise overlap refresh covered 528 open PRs and 6,540 Discussion records with reported complete pagination. Sparse notes, arbitrary diffs and unpublished work remain limits; these checks do not certify novelty.

CI notes

Both hosted checks passed at exact head 2907ad441b110029355a9c3c83cefc9ecc114eb6 in run 41618685171: plan in 4m59s and replay (research/phase-paired-isolation) in 1m22s. The completed check list was read once. The run is completed/success and its head matches the published contribution.

Hosted replay certifies the registered bounded normal and optimized finite checks. The local scientific, package, committed-byte/source-history, mutation and standalone checks listed above also passed. The all-field and all-workspace conclusions remain written mathematical proofs; independent human and formal proof-kernel acceptance are unclaimed.

Merge danger

Door: two-way.

Blast radius: isolated.

One opt-in proof and verifier package. Shared code, predecessor sources and credit, frozen evidence and the selected frontier retain their bytes.