Audit fresh-row conditioning in the selected stopped cover

[adamgall]

Result and scope

This adds research/selected-cover-fresh-fibers/: a conditional all-depth address/counting theorem for the selected stopped bit cover, with exact adversarial controls. No new κ, supplier, numerical stock coefficient, or full all-size compiler admission is claimed.

A stock bound and a restored row endpoint do not alone justify the selected producer's per-edge bad-class average. The missing interface is that each invocation sees complete fresh row fibers, conditional on its ancestors. The package proves that implication from the source's field-dependency and complete-execution contract, and explicitly leaves the selected compiler's implementation of that contract conditional.

The theorem

For any integers W>=2, finite depth D>=1, and genuine row range Q>=S=W^D, pad once to P=ceil(Q/S)*S. Euclidean division identifies [P] with [ceil(Q/S)] × [W]^D.

Assume at every invocation boundary:

• invocation existence, width, fixed edge list and weights depend only on ancestor fields and fixed program indices;
• current-role routing is a bijection depending only on ancestors, fixing ancestors and unused descendant stock;
• a completed child restores its supplied row field and ancestor spectators before sequential reuse;
• execution uses the entire padded range, restricting to genuine rows only at the completed row-restoring endpoint.

Then every live invocation domain is a union of complete suffix cylinders. For each fixed ancestor prefix, every current role has exactly the same number of unused-stock completions. Prefix-dependent stopping and fallback are allowed: a parent's current residue becomes a child's ancestor. The proof is induction on invocation depth, not a finite-experiment generalization. There is exactly one volume factor P/Q<2, not one per recursive call. Endpoint restoration justifies unpadding; it does not justify pruning intermediate invocation domains.

Selected binding and unchanged paid bill

The frozen PR1087 assembler binds the selected PR589 record by Git blob e293ddc557742637f9f679f083a1a282d87b1b13 and SHA256. The actual ordinary producer has m=64, 28,416 live roles, 28,425 physical roles, 904,896 paid edge occurrences and maximum child rank 42. Dimension 73 belongs to the complex supplier and is not substituted into this bit argument.

For the selected role divisor

W(w) = 28425 * |GL_64(Z/q^w Z)|,

every invertible low matrix has exactly q^(4096*(w-1)) lifts. Right multiplication preserves these equal fibers. Fix an eligible odd prime q>2^80 before input widths, avoiding the fixed rational denominators and rank witnesses, as required by the inherited source contract. Its per-edge bad fraction is then bounded by 2*64^3/q < 10^-16 at every fresh invocation. Summing costs over fixed edge occurrences needs no independence assumption and no simultaneous union bound.

With selected histogram H, E=sum H_r=904896 and L=28416, the complete retained bill remains

(1/L) sum_r H_r*(r/64)^(1-a)
  + (1/10^16)*(32*64^2)*E/L*(1/64)^(1-a).

All good terms and the full fallback stay charged. There is no root refinement, displaced-good subtraction, or claimed change to selected κ.

Exact counterexamples

1. Adaptive current-role edge selection: in GL_2(F_q), index predicates by projective lines [v] and let B_v={g:first_row(g)·v=0}. Every fixed edge is bad on exactly 1/(q+1) of the group. Choosing the kernel line of the current first row makes the chosen edge bad everywhere. Identity routing and final row restoration remain valid. The only removed hypothesis is that the edge choice be fixed before reading that invocation's current role. This is a linear-minor witness, not an arbitrary rare-set oracle.
2. Pruning dummy rows: W=3,D=2,Q=10 requires padded range P=18. At prefix (slab,current)=(1,0), the genuine prefix contains only fresh value 0, whereas the padded execution contains all three values. Conditional frequency changes from 1/3 to 1 even with identity routes and exact endpoint restoration.

These expose invalid generalizations of the averaging argument, not a demonstrated defect in the selected compiler.

Sources, predecessors and novelty boundary

The README maps every theorem hypothesis to exact pinned sections of three-stage-cover-bit.tex, three-stage-cover-rows.tex, the retained PR185 proof, the selected PR589 record and frozen PR1087 assembler. The old dimension-67 note supplies a structural contract, not a selected numerical profile. Portability to the selected implementation stays conditional; finite scalar/controller records do not themselves emit an all-size read/write certificate.

Josie Elliston's #1956/#1965 and ToikY's #1990 isolate cutoff and stock-fit obligations. This contribution addresses the fresh-field/counting implication, not their headroom arithmetic. r480-sharp-charge535 and its PR461/464/465 predecessors already sharpen the numerical bad-class allowance; that refinement is not claimed here.

The finite source bytes and archived assembler member are authenticated before parsing; predecessor Python is never imported or executed. Selected frontier files, frozen evidence, shared indices and existing packages are unchanged.

Evidence and reproduction

Python 3.12, standard library only:

uv run --python 3.12 --no-project python -I -B research/selected-cover-fresh-fibers/verify.py
uv run --python 3.12 --no-project python scripts/research_package.py --package research/selected-cover-fresh-fibers

Observed smoke and registered optimized (-I -O) replay:

PASS_SELECTED_COVER_FRESH_FIBERS cases=60 cuts=150 GL_rings=4 controls=9
PASS_REGISTERED_RESEARCH_PACKAGE research/selected-cover-fresh-fibers outcome=completed

Coverage: 14,120 padded rows, 2,370 conditional fibers, four full GL_2 enumerations over moduli 3, 9, 5 and 25, adaptive-minor witnesses at primes 3, 5 and 7, and nine rejected semantic/source mutations. RESULT.json is required and compared canonically with fresh reconstruction. Assertions are not the acceptance gate. Separate code/proof review and registered replay corroborated the result; final review edits clarified prime eligibility and removed a redundant tautological check without changing the scientific receipt.

The first registration attempt rejected a non-enumerated metadata outcome; it was corrected to completed, and the registered replay above passed. No full selected supplier cold replay, formal-kernel check, hosted CI pass or independent human review is claimed.

Provenance and overlap

• Scientific base: 564148b82ae9df72163f22106f15e73efca8aff2.
• Submitted head: 1d3ee0926656d5260c6cb1c0b81964ffe9fe1377.
• Original selection: seed cc5bf62436a3476502c7769e29c2990c, adversarial-audit, priority revision 4; complete output in SELECTION.json, no redraw.
• Card scope changed from numerical selected-stock-envelope work to its explicit field-independence implication; the reason and stop condition are in TASK.json.
• Upstream checkpoint 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 changed guidance only; selected record, catalogue and scientific inputs are unchanged.
• Shared refresh: 353 open PRs including drafts at 19:00:33 UTC and 738 discussion/comment/reply records at 19:02:09 UTC, complete pagination. Targeted offline/local search at 19:09:29 UTC and direct live PR1990 disposition reading are recorded. Cached coverage can miss changes; no absence-based novelty guarantee is asserted.

Remaining question and credit

The concrete next step is an emitted all-size field-dependency certificate: verify that each call guard/edge-list/weight reads only ancestors and each route fixes unused descendant fields. Start with the retained child-weight rule and current chart routing, then annotate the selected all-size lowering. Stock magnitude or scalar endpoint restoration cannot substitute for this obligation. A deliberately current-residue-adaptive selector would instead need its own fully paid fiberwise charge.

Credit to the retained three-stage-cover framework and eumemic's source527 packaging; Rohan Arun's PR185/PR387 and selected PR1094 lineage; Jeremy Kothe's PR1087 assembler; PR589's credited punctured-E8, nine-stage and hosting lineage; and the named cutoff and numerical-charge predecessors. Prepared for Adam Gall with substantial OpenAI Codex assistance in investigation, proof, code and writing. Apache-2.0. No predecessor result is claimed as new.