[JoyboyBrian]

Complete the actual admitted Count boundary

The original R437 root q=2^127-1, e=n=64, w=2, h=0, with authenticated precision L255/I8/Q520213, now executes through its completed saved 32,526-bit fibre Count, including the original suffix product, before payload peel. The exact output is 9475*(q^2)^128. This answers Open next question 1 of #4987, whose full original exponent diagnostic was interrupted after 90.505 seconds.

A new opt-in sequential unary radix cache validates the immutable modulus once and retains the original encoded exponent Counter and every original Count multiplier. The source-bound proof applies uniformly to valid finite contexts. Read-only actual-call observations confirm one invocation, two Word divisions and all 256 Count.multiply_small calls, versus the original 256 radix divisions for this exponent segment. No runtime Word.inspect or host numerical decoding drives execution. The completed local run is evidence of completion, not an asymptotic runtime claim or a controlled cross-revision speed comparison.

The literal original NODE AST is retained through its 23 statements before peeled: actual ingress, quotient, stock guard, saved leading, initial fibre, full exponent, suffix product and saved fibre. Only the previously credited #4794 quotient adapter and the new radix adapter replace globals. Full Context/codebook construction remains unrun; the structural context view is authenticated to the admitted root and is not claimed as execution of that constructor.

Validation and a real rejection-path correction

The malformed-context controls found that the frozen original radix guard raises before freeing its returned quotient and remainder. The opt-in validator keeps its exact arithmetic/guards and frees those two Words on rejection. The original leak is confirmed by a positive control; frozen sources are unchanged. A failing Count primitive retains its partial output and original exception; the cache and Counter are erased. This is not general output rollback or an allocation-exhaustion guarantee.

• The actual root checks complete output, source handle/key, borrowed Program/Count/directory bytes and heads, Ring modulus/one plus runtime size/depth Words, unrelated dirty Count173/Word37, numerical live sets and erased owned backing before fixture disposal.
• 81 original-versus-cached source-primitive pairs preserve the exact Count modification word, including zero exponent. Invalid moduli reject before Count writes; an injected failure after one completed write preserves the partial output and exception while restoring temporary ownership.
• Two separate read-only peer reviews found no valid-context or source-cut blocker. A separately written bounded check authenticated all 408 inputs, tested 36 further pairs including zero Counts/composite radix, nine invalid contexts, the original two-Word leak and a partial-write exception. Peer evidence is AI review, not human/proof-kernel acceptance or a second q127 replay.
• Registered Python 3.12 normal/optimized full-root replays pass. Exact committed-byte/source-history preflight and official export integrity pass at 764b06d86b1ec48b74e5ae2d29c7e521cf69301a (419 files). Git-free cold normal/optimized full replays pass in 58.106/56.436 seconds. Source corruption rejects; a bool-for-integer receipt mutation also rejects after a complete 56.632-second execution. All input/receipt bytes are restored.

Reproduce: python3.12 -B scripts/research_package.py --package research/r437-admitted-count-power. Both full-root commands have a 120-second bound.

Scope, sources and remaining obligations

Canonical investigation: Discussion #4682 claim. Base canonical main bee95ce88f9f19802f0613f92838895cd0e14a1f, priority revision 7, selected conditional kappa 0.00302015 unchanged. Initial overlap covered 310 open PRs, 4096 Discussion records and all 6690 tracked notes. Before submission, canonical main refreshed to f259dabc7d70e0db8d6d36bfda888f008bbfecc1 at 06:37 UTC; governing guidance, selected record and declared inputs were unchanged. Fresh complete remote snapshots cover 315 open PRs (06:38:22 UTC) and 4130 Discussion records (06:39:11 UTC); scoped radix/cache/exponent searches found the predecessor diagnostic, this claim and the round hub, with no other same-boundary delivery. The local scoped scan retains all 6690 notes at the frozen bee95 base; root also checked the newer 6710-note inventory. Arbitrary diffs and unpublished work remain limits. Family decision: narrow to the incomplete measured Count boundary, then stop before peel.

Complete Context/GL64 directory production, gather/payload, original peel, full fixed-tape/allocator pricing, the inherited large quotient-divisor preparation, descendant precision/stock, compiler and supplier/outer admission remain unknown. All 256 Count multiplications and every cache/Counter/data movement remain paid. No full historical suite, selected cold replay, Lean, Docker, admitted supplier, multiplication theorem or new kappa is claimed.

Credit David Arias, Chris R/Clear1nsight, Jose Manuel, Nick Petalas #4854, ToikY, codyw912 and the retained Count/compiler/caller/stock lineage. The root fixture adapts JoyboyBrian #4987; #4981's prefix/consumer fusion is distinct. New cache, composition and checks: JoyboyBrian, substantial OpenAI Codex assistance with separately assigned reviews. Apache-2.0.

Supplemental AI peer review is local-only evidence; committed source, written proof, actual observer and registered verifiers are the public reproduction artifacts. No independent human acceptance is claimed.