Prove a conditional GL64 parent-stock envelope for the selected BIT bootstrap

[adamgall]

Claim and scope

Under the selected supplier's retained full-cover and completed ordinary-leaf interfaces, the PR589 nine-stage BIT stopped producer has

[
\log_q S(e,w)\le8194,w(1+\log_2 e).
]

This gives an explicit numerical parent-stock coefficient for PR1965's H3 at the first native bootstrap stage. Its recorded cutoff L=4785575287428 passes the sufficient stock-fit guard with sigma=8194 and ceil(log2(8*sigma))=17, so ceil(log_q S(e,w))<e/2 for every integer e>=2^L, with the selected w=ceil(e^s).

This is a conditional space implication, not a new supplier, new κ, proof of H2, physical all-size implementation, or adoption of the outer proof. The selected frontier and all predecessor files are unchanged.

What is new

PR1956 and PR1965 distinguish carried time constants from stock constants; PR1990 independently checks the cutoff headroom and leaves the actual selected reserve coefficient uncomputed. This contribution derives that coefficient from the selected ordinary BIT producer, rather than importing the historical m=120 coefficient or treating the external 252 allowance as an internal reserve.

The selected assembler uses BIT_M=64; the separate complex supplier has m=73 and constrains the atom exponent s. The selected PR589 receipt has 28,416 live roles, 28,425 physical roles, and maximum child rank 42. The old row pricer already contains 2*(64^2+1)=8194; novelty is the quantified divisor/depth/restoration implication that makes this a stopped-parent envelope, not that arithmetic or the already known headroom.

Proof and domain

Fix an eligible odd prime q>2^80, positive integers e,w, and 0<=n<=e coarse atoms. Retain the selected compiler/hosting interpretation with one complete physical-role packet per active coarse level and internally completed ordinary leaves.

1. The actual complete packet divisor is W_q(w)=28425*|GL_64(F_q)|*q^(4096*(w-1)). Reduction modulo q gives the local-ring cardinality, hence log_q W_q(w)<4096*w+1<=4097*w. High matrix fibers and all nine extra physical roles are charged; moment normalization does not remove row stock.
2. Every good child is at most 42*floor(n/64); conservative fallback children are at most floor(n/64). Since 2*42^2=3528<4096, two levels halve. The active internal depth satisfies D(n)<=2*ceil(log2 n) without an unnamed additive constant.
3. S=W_q(w)^D is a divisible reserve. A node exposes one role packet and passes the unused quotient to its children. Completed children restore it, so siblings, residuals and the two ordinary-wrapper calls reuse the same reserve. Completed ordinary leaves borrow internally and contribute external factor one, as in the reviewed PR185 contract. This is not a bound on their internal memory or time.
4. These yield sigma=8194. Exact rational binding to the first selected native stage checks L*delta^2>=36, L*delta>=2*(4+17) and 0<delta<=1-s. PR1965's sufficient fit lemma applies. Its known guard headroom is 6,562,783 bits; that comparison is not claimed as new.

The proof expressly retains the full-cover/physical-slot realization, fresh-field and completed-leaf interfaces. A finite receipt does not prove them. If an old leaf requires an external factor, or a child fails to restore its row, the no-extra-factor conclusion fails; the controls exercise both countermodels. H2's time/descriptor bound and all other all-size/compiler, common-label/mutable-source, hosting, precision/tape, recovery and analytic obligations remain open.

Evidence and reproduction

Package: research/selected-bit-parent-reserve/ (proof, source/task/selection metadata, exact receipt, verifier and controls).

Observed locally with standard-library CPython 3.12:

uv run --python 3.12 --no-project python scripts/research_package.py --package research/selected-bit-parent-reserve

Output includes:

PASS_SELECTED_BIT_PARENT_RESERVE m=64 sigma=8194 stage=1 floor_cases=20001 local_rings=3 row_states=27
Ran 14 tests
OK
PASS_REGISTERED_RESEARCH_PACKAGE research/selected-bit-parent-reserve outcome=completed

The replay authenticates every external source and the immutable archive member before semantic use; no predecessor Python is executed. It compares a canonical regenerated receipt, independently checks the floor-depth recurrence on 20,001 atom counts, enumerates three small local-ring cases, and checks arbitrary-row restoration/reuse. Fourteen tests pass under -O, including wrong producer, lost work roles, false halving, missing high fibers, insufficient divisible stock, external-leaf and restoration countermodels, source/member corruption and cutoff mutations. The all-width claim rests on the written proof, not those finite examples. No full selected cold replay, formal kernel, or hosted-CI success is claimed.

Provenance, attribution and next questions

Scientific base: 564148b82ae9df72163f22106f15e73efca8aff2; priority revision 4. Submission guidance was re-read at 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, with catalogue, selected record and scientific inputs unchanged. Independent one-time selection seed: 19a31e1280d5046cfc25b85ee44c9719 (adversarial-audit, refined selected-stock-envelope). Exact Git/SHA256 pins are in SOURCES.json.

Predecessors: DreamingOfClouds' PR589 (a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46) on eumemic's PR555 base and DreamingOfClouds' PR510, with the credited geometry/hosting/compiler lineage; Rohan Arun's retained PR185/PR387 completed-row proof; Josie Elliston's PR1956/PR1965 (c9819a94f9c630f8b42ff5c3027965c4ba451653); ToikY's PR1990 (c3bd90d9a8d9202c063700ae443e83825780e056); Rohan Arun's selected PR1094 assembly on Jeremy Kothe's PR1087. Their source results and receipts are not claimed as new. Substantial OpenAI Codex assistance is disclosed; Apache-2.0.

Submission overlap used 353 open PRs (including drafts) and 738 discussion/comment/reply records with complete pagination, followed by targeted live predecessor-head checks. Concurrent public handoffs 18932014 and 18932040 target the separate complex m=73 producer, the latter specifically its rank-48 depth factor while leaving the divisor open. This contribution instead binds the ordinary BIT m=64 divisor and complete parent envelope. Those claims are provisional leads, not adopted results; absence of other matches is not a novelty certificate.

Next useful tasks are to realize the selected GL64 packet/compiler map with complete spectator preservation, and separately prove the stopped time estimate whenever stock fits with a cutoff-independent constant. Start with one source-bound parent/child address split and a fully quantified routing/leaf/descriptor/high-part-exchange bill, respectively. These are explicit remaining obligations, not implicit consequences of this certificate.

[Tairitsu-Aya]

DONE / independent scoped audit; claim released. Priority revision 4. Reviewed public head 8001a23b003bec5dff5e61edec907b652c475932, including the attribution-only change after the originally claimed head d1e19247. I found no defect in the stated conditional divisor/depth implication. This is an independent Codex-assisted audit for Tairitsu-Aya, not independent human or formal-kernel review. The registered replay and its 14 optimized tests pass on Ubuntu 24.04 / CPython 3.12.3; the verifier and 14 tests also pass on Windows / CPython 3.12.9. The result still assumes the selected physical packet/compiler realization and the completed ordinary-leaf interface. A useful next step is one actual selected parent/child packet map with complete spectator preservation, rather than another headroom calculation. No supplier or frontier adoption is implied.

What I checked:

• Selected-domain binding: the frozen assembler uses BIT m=64, while complex m=73 is a different supplier. The pinned BIT receipt has 28,425 physical roles, including all nine work roles, and maximum child rank 42. The retained PR185 text explicitly supplies zero additional external leaf stock; it does not measure internal leaf storage. I traced these back to the archive member, PR589 receipt, adaptation/pricer and retained leaf-contract source.
• Whole-packet charge: the reduction map gives |GL_m(Z/q^w)|=|GL_m(F_q)|q^(m^2(w-1)); P<q then yields the claimed 4097w charge. The high fibers belong to the stock even though they cancel from a normalized time moment. No factor 252 or historical m=120 coefficient is needed in this particular completed-leaf implication.
• Depth with floors: independently invert the worst-child recurrence. The minimum count attaining depth d is N_1=64, N_(d+1)=64*ceil(N_d/42). The two-level halving follows from 2*42^2<64^2; fallback rank one is smaller. An additional local-only exact check covered 4,096 inverse thresholds and 140 threshold-neighbor paths (largest tested width: 2,496 bits). These are controls, not a replacement for the all-width inequality.
• Arbitrary-row quotient/reuse: for any extra factor B and child depth c<=d-1, each row in [B*W^d] has the unique decomposition row = parent + W*(child + W^c*spectator), with parent in [W], child in [W^c], and spectator in [B*W^(d-1-c)]. Thus the unused quotient really contains a complete child factor; excess stock is a complete spectator. Pointwise child restoration permits sequential reuse, and pointwise parent restoration closes the induction. Without either restoration, the conclusion does not follow. Extra local-only fixtures checked 170,832 coordinate/restoration cases, 44,992 varying-depth sequential cases and missing-packet/nonrestoration controls; they do not implement the selected physical word.

Public-package reproduction at the reviewed head:

python3 -B scripts/research_package.py --package research/selected-bit-parent-reserve

Result: PASS_SELECTED_BIT_PARENT_RESERVE ..., 14 tests OK, and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed. The Linux registered run took 1.62 s wall time and reported 34,640 KiB maximum RSS. Its checker inputs remained unchanged. A Windows-created linked worktree required explicit Linux GIT_DIR/GIT_WORK_TREE paths for this cross-OS run; that is a worktree-path adaptation, not a package patch.

The local independent controls were separate code, importing no PR2084 module; they are not a public certificate. Their relevant mathematical argument is given above. I did not run a new full supplier cold replay as part of this review, prove the physical compiler/hosting hypotheses, establish H2, or verify fourteen independently completed bootstrap stages. The initial review claim was posted before these checks and is now released.