[unpaidmoon]

Change and mathematical scope

Partial coefficient correction. No new κ. Selected κ stays 0.001672660197076918.

research/selected-stock-shape-a-zero proves log2(C) < 2642 for the finite stock C = |Sp(72, 2)| * 14826 with shape slope A = 0, then displays

sigma <= (43/26) * (1 - s) * 2642 + rho

inside the composition from research/selected-coarse-depth-stock/PROOF.md. That composition takes A and B from the base-q hypothesis log_q C(w) <= A*w + B.

For every prime q > 2^80, log_q C = log2(C) / log2(q) < 2642/80 = 1321/40. The displayed 2642 substitution remains a valid sufficient digit bound (log_q C < 2642 for every q >= 2) and is exactly 80 times looser than 1321/40 on the stated prime floor.

Let T(s) = (43/26)*(1-s)*(1321/40). Under the coarse-depth depth bound and the finite-stock reading C(w) = C, any leaf hypothesis log_q S_leaf <= rho * w * (1 + log2 e) gives the sufficient envelope coefficient sigma = T(s) + rho. The inequality is strict before adding rho.

Exact parent terms, rho separate:

Branch	Sufficient parent term T(s)
bootstrap	174485626749443905524883973467613250556094475116027 / 3200000000000000000000000000000000000000000000000
conservative_bootstrap	4362140668748600743658666134074946636014640956718493 / 80000000000000000000000000000000000000000000000000

Both are < 64. Stage 1 of each branch still has headroom 6562783. That comparison does not adopt an envelope. Every actual_stock_coefficient stays null. rho = 0 is the homogeneous leaf bound only when S_leaf = 1; a one-digit leaf has log_q(q) = 1.

Remaining obligations: bind rho or prove S_leaf = 1; decide whether the assembler charges C as the uniform per-level divisor C(w) at stage 1.

Task and overlap

Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (upstream CrocSwap/integer-mult-bounds main). Priority revision 6. Selected κ unchanged.

Task card: selected-stock-envelope in docs/research/agent-task-catalog.json. Scope status: refined. The card asks for one actual selected-stage stock coefficient. This package translates the already proved base-2 bound into the log_q parent term the coarse-depth envelope uses, uniformly for primes q > 2^80. It does not fill actual_stock_coefficient.

Why the premise moved: research/selected-stock-shape-a-zero already proves A = 0 in base 2 with B_bits = 2642, then displays that integer inside the log_q composition. The missing piece left by that note is the base, not another statement that the coefficient is unresolved and not another cutoff rerun. The m=73 producer still binds κ; the factor-80 correction changes the stated parent term and leaves rho and finite-stock admission open, so κ does not change.

Handoff: Discussion #1166 comment 19047276, taken as item 19 of this session's chronological eligible-next-task list. Fork branch cursor/selected-stock-logq-parent-b48b at d9ba42bab118f34411541366f078ebbc5d82da04.

Nearest predecessors, and what this adds:

• research/selected-stock-shape-a-zero (Wentao Li): base-2 A = 0, B_bits = 2642, and the displayed 2642 substitution. This package replaces that substitution with 1321/40 on q > 2^80.
• research/selected-stock-envelope-m73 (Aaron Wang): records B = 2642 and the missing A, B, rho. A = 0 was later proved; this note does not restate that gap.
• research/selected-coarse-depth-stock: the log_q composition and the 43/26 depth factor.
• research/selected-bootstrap-stock-audit (Josie Elliston): both receipt branches, null actual_stock_coefficient, stage-1 headroom 6562783.
• docs/research/selected-bootstrap-scope-map.md: prime floor q > 2^80.
• research/stock-margin-chain (Hiroki KUMAZAKI) and Nick Petalas's tie-out: cutoff comparisons; coefficient left unresolved. This package does not replay that arithmetic.
• Open PRs through Add a typed one-round actual-width H and rounded return. #4123 were checked before selection. Later sibling PRs seen in flight (Prove the exceptional single-hub literal charge is 48. #4124–Price one typed actual-width physical H and its toward-zero return #4127) address other handoffs and do not contain this log_q correction.

Validation

Python 3.12, standard library only. Registered replay:

python3 -B scripts/research_package.py --package research/selected-stock-logq-parent

That runs verify.py (including python -I -O -B) and test_verify.py. Marker:

PASS_SELECTED_STOCK_LOGQ_PARENT bits=2642 q_floor=80 factor=80 ... rho=unbound coefficient=null headroom_not_adoption no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/selected-stock-logq-parent outcome=partial

The verifier pins the scope map, the bootstrap receipt, the coarse-depth proof, and the shape-a-zero proof by SHA256; recomputes |Sp(72,2)| * 14826 and its bit length; checks both receipt savings against gaps.stock; checks that every actual_stock_coefficient is null; checks stage-1 headroom 6562783; checks that the prime-floor term is exactly 1/80 of the displayed 2642 term and that a one-bit change of the bit length changes the term. test_verify.py checks the two exact fractions, that they differ, that a one-digit leaf is not <= 0, and that a zero log floor raises ValueError.

Finite arithmetic check of cited source identities. Not a Lean proof, not a cutoff replay, and not an admission that C is C(w) at stage 1. scripts/research_baseline.py was not rerun; the package pins the four live inputs and the catalogue/selected-record provenance instead.

Credit

Base-2 shape A = 0, B_bits = 2642: Wentao Li, research/selected-stock-shape-a-zero, from Aaron Wang, research/selected-stock-envelope-m73. Log_q composition and 43/26: research/selected-coarse-depth-stock (Sricharan Vagvala's conditional rate, as retained there). Savings and null coefficient fields: Josie Elliston, research/selected-bootstrap-stock-audit. Prime floor: selected bootstrap scope map. Cutoff chain left unresolved: Hiroki KUMAZAKI, research/stock-margin-chain, and Nick Petalas's independent tie-out.

Prepared by unpaidmoon with substantial AI assistance (Cursor Grok). No independent human review is claimed. Apache-2.0.

Open next questions

From research/selected-stock-logq-parent/TASK.json:

1. Bind rho, or prove S_leaf = 1. The corrected envelope is T(s) + rho. A one-digit leaf already rules out reading rho = 0 off the parent term. First step: one complex residual whose external leaf packet is counted. Blocked by: no per-leaf recurrence in the retained complex sources. The inherited slack 252 is not that coefficient.
2. Adopt C as the stage-1 divisor. Both the base-2 shape theorem and this translation assume the finite stock is C(w). First step: match one stage-1 residual to that cardinality at two atom widths. Blocked by: the complex cover note has no W(w) formula.

Workflow feedback (optional)

The shape note's displayed 2642 sits one sentence after a correct base-2 theorem, inside a composition whose hypothesis is base q. A package check that the cited proof sentence and the scope-map prime floor are still the pinned bytes would catch this class of unit slip without another cutoff suite. Feedback discussion: #1200