[zxv]

Change and mathematical scope

For the explicitly declared precision/stock/fallback policy, the actual PR4359 nine-row outer assembly is feasible exactly when

$$\rho(1+s)<2,\qquad \mathrm{OLD}<\kappa<\min(M/\rho,1/\rho-s),\quad M=\min(U,B,1/32),\quad s=\zeta/(1-\nu).$$

Here the prescribed guard is d^ρ polylog(d), total logarithmic stock is h^ν p^ζ polylog(p), and the individual-axis fallback below p^t, t>s, is paid. The proof gives rational witnesses for every interior point and exact positive-weight obstruction identities. It strengthens the source group_scans saving from 1−ε+β to 1−ε+β−t; that row also pays the full-width residual. Quadratic prescribed guards cannot fit the retained membership geometry. A loose quadratic upper bound is not a lower bound on necessary precision.

A concrete false-positive screen uses hypothetical U=B=.02, ρ=1, ν=1/2 and ζ=.4975. At κ=.011, all 58 original predicates, both new guard tests, largest-call stock fit and cutoff tests pass. The paid fallback alone rejects: saving 16993/7000000, with policy supremum .005. This falsifies treating the new fallback as free; it alleges no error in the existing ζ=0 ledger.

This is a scoped written proof with exact arithmetic evidence. No new supplier, κ, selected-frontier change or framework-assumption discharge is claimed. U and B are completed above-cutoff budgets for the hypothetical changed interface; they are not raw moments or newly admitted all-size suppliers. Native, compiler, restoration, tape, completed numerical error and analytic implementations remain obligations.

Significance for researchers

A κ>.01 supplier proposal needs a fitting representation as well as stronger complex and BIT budgets. The exact target is M>.01ρ, s<1/ρ−.01 and ρ(1+s)<2 under this policy. Linear precision and ζ=0 recover Nick Petalas's PR5984 region, which is credited rather than repeated. The new result exposes the paid residual that a largest-call stock test misses and identifies the guard, stock, fallback or membership contract that must change.

Task and overlap

Explicit backward-interface assignment; card uniform-transfer-audit, scope changed from closing an inherited assumption to a parameterized outer representation contract. Scientific base: canonical main 721c038, priority revision 8, selected PR6030. SOURCES.json pins every consumed input. Discussion6056 is the canonical investigation. This work does not extend local helper words or duplicate PR6047, PR6049 or PR6054.

The existing PR6050 navigation repair is included so shared planning reaches scientific replay after the selection change. Its changed navigation files can trigger stale source-pin failures in archived consumers; frozen packages are unchanged.

Validation

CPython 3.12.8 normal and optimized registered replays pass 4,056 rational witnesses, all 63 predicates per witness, independently restated nine-row formulas, 300 arbitrary-parameter dual-identity checks, strict-boundary exclusions and the paid-fallback control. Points 10^-24 below the supremum are included. Exact committed bytes pass; all six existing navigation tests pass. No selected cold replay, Lean, Docker, numerical Gaussian replay or independent human review ran. Optional full-history provenance checking is unavailable in this shallow checkout. The global sparse replay planner was stopped after more than two minutes without a matrix; targeted registered metadata and scientific replays pass.

Reproduce: python3.12 -B scripts/research_package.py --package research/outer-representation-budget. PROOF.md supplies the universal argument; finite tests do not establish it by extrapolation.

Credit

Prepared by zxv with substantial OpenAI Codex assistance in mathematics, implementation, source review, verification and writing. Retain Nick Petalas's PR5984, the PR4359 ledger and complete-stock/semantic-precision arguments, Vlado Vojdanovski's PR1673, Jeremy Kothe's PR1758, Chafik Boukhalfa's PR1786, DreamingOfClouds's PR6030, and the retained radix-union/Strassen/fine-prime, Harvey–van der Hoeven and OpenAI manuscript lineage. Apache-2.0 and source-specific notices are retained.

Open next questions

Bind one actual supplier's completed error, total simultaneous stock and paid residual to this contract. Alternatively, supply a faster below-cutoff routine or a complete membership implementation weakening β+2(1−ε)<1. The package states first steps and blockers; none of these implementations is assumed here.