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Selected stock envelope at m=73: bound quantities and the exact missing shape lemma

Aaron Wang, with help from opencode (substantial AI assistance; no model named; no independent human review claimed). Card selected-stock-envelope (drawn via scripts/choose_research_task.py, scope unchanged: scope_status: unchanged, actual_scope equals the card question byte-for-byte). Base main 18aacb215fc3717b06be8ceb9fca62aeca0d3e99; branch research/agent-89-f5c32b23bcd01d2e356d-9de3; head 725d9ba226b388737fd02e636f10cd69701e8d58. No kappa, supplier, frontier or selected-record change.

Question and reading

The drawn card asks: what explicit uniform stock envelope does the actual selected stopped producer supply at one bootstrap stage? Its acceptance gate demands a binding to the selected producer and warns that a historical m=120 coefficient is not a selected m=73 proof. This package takes a construction reading at the selected m=73 complex producer (the producer that binds the shared saving and the selected kappa), at the native stage-1 domain (e >= 2^4785575287428, w = ceil(e^s)), and terminates at an exact, line-level missing lemma. It does not rerun cutoff-pair arithmetic and does not import the historical 57,604 or 252 allowances.

Established (exact, machine-checked)

python3 -B research/selected-stock-envelope-m73/verify.py prints:

PASS_SELECTED_STOCK_ENVELOPE_M73 sp72_bits=2628 families=14826 B=2642 ledger=16469->15555 halving=5329>4608 guard=18,17<=6562783 missing_lemma=A_B_rho_unbound controls=5 checks=45

1. Per-level stock. The charged per-level stock allowance is exactly B = 2628 + 14 = 2642 bits: the closed-form order |Sp(72,2)| = 2^1296 * prod_{j=1..36} (2^(2j)-1) has bit length 2628, and the physical-family split 14336 + 480 + 8 + 2 = 14826 has bit length 14. Both are recomputed from scratch; the retained stock bill reports the 2,642-bit allowance without decomposing it.
2. Depth. The halving guard is exact: 73^2 = 5329 > 2 * 48^2 = 4608, so two levels more than halve the child count at maximum child rank 48. The sharper coarse rate 43/26 per log2 unit stays the named conditional result of PR 2105.
3. Both row ledgers recomputed: 2*2642 + 9909 + 252 + 1024 = 16469 < 20161 and the conditional refinement 4370 + 9909 + 252 + 1024 = 15555 with 4370 = ceil(43*2642/26); the 914-bit difference reproduces exactly.

The exact obstruction

The retained fit chain needs H3 in the w-shaped normalization log_q S(e,w) <= sigma * w * (1 + log2 e). The retained coarse-depth proof reduces the m=73 case to shape premises log_q C(w) <= A*w + B and log_q S_leaf(e,w) <= rho*w*(1+log2 e), giving sigma <= (43/26)*(1-s)*(A+B) + rho, and states verbatim that A, B and rho are not established there. This package searched the retained cover sources for A and records the obstruction precisely:

• The only w-shaped per-level divisor in the retained notes is the ordinary cover formula W(w) = W_0 q^{m^2(w-1)} (notes/three-stage-cover-bit.tex), whose m^2(w-1) digits are charged in the row reserve; it is written for the ordinary cover and used at m=64.
• The complex cover note notes/three-stage-cover-complex.tex states no w-shaped divisor at all; the bounded no-match is machine-checked against the pinned bytes.
• The finite invoice charges 2*2642 bits per log2 unit with no w term.

So A, the w-shaped constant B, and the leaf reserve rho have no source binding at the m=73 complex stopped producer. The producer identification is itself recorded as an ambiguity: docs/research/selected-bootstrap-scope-map.md binds the ordinary stopped producer of the four conditional reserve proposals to PR589 BIT m=64, while the card gate demands an m=73 binding. The conditional m=64 coefficients 8194 (PR 2084, Adam Gall) and 6748 (PR 2120, Nick Petalas) are not m=73 proofs, and this package claims neither.

Sufficient-guard comparison (consistency only)

Under either conditional ledger reading, ceil(log2(8*sigma)) is 18 (at 16469) or 17 (at 15555), far below the recorded stage-1 sufficient guard H_1 = 6,562,783. A large numerical headroom is not a proved envelope; the contribution is the bound quantities plus the missing lemma.

Reproduce and evidence limits

python3 -B research/selected-stock-envelope-m73/verify.py
python3 scripts/research_package.py --package research/selected-stock-envelope-m73 --check-only
python3 scripts/check_research_commit.py --package research/selected-stock-envelope-m73

Local outputs: PASS_SELECTED_STOCK_ENVELOPE_M73 (45 checks, five rejected controls including |Sp(70,2)|, a wrong B, a rank-52 child and a wrong cover exponent), PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check), PASS_REGISTERED_RESEARCH_PACKAGE research/selected-stock-envelope-m73 outcome=partial, PASS_RESEARCH_COMMIT_BYTES. Standard library only, exact integers, Python 3.12; no retained script is imported or executed. This is a finite source-bound arithmetic audit and an exact recorded obstruction, not a supplier replay, cold replay, stock implementation or multiplication theorem.

Open next questions

1. Bind A, or prove the complex per-level divisor is w-independent. First step: read the retained invoice sources behind the 14,826-family identity for any per-row digit charge. Blocked by: no such sentence in the pinned cover notes.
2. Bind rho, the complex old-leaf reserve. Blocked by: the retained complex sources state no per-leaf reserve recurrence.
3. Maintainer wording decision on whether "the actual selected stopped producer" is the ordinary BIT m=64 chain or the complex m=73 producer.

Credit

Sricharan Vagvala (theon199), PR 2105, supplied the complex coarse-depth thresholds, the 43/26 rate and the conditional stock bill with the 16469 to 15555 comparison. Adam Gall (PR 2084) and Nick Petalas (PR 2120) supplied the conditional m=64 coefficients. Josie Elliston (PR 1965) and ToikY (PR 1990) supplied the fourteen headrooms and the factor-16 fit extension whose source boundary names the uncomputed coefficient. Hiroki KUMAZAKI (PR 2972) closed the cutoff-pair arithmetic and left the coefficient unresolved; Riquelme Soares (PR 2931) supplied the stage-3 pair. The selected BIT supplier is DreamingOfClouds's PR 589. Philippe Van Velzen's stage-1/stage-2 conditional envelope chain in discussion 1166 (2026-10-14/15) informed the ordinary-producer reading; no result of that fork chain is claimed here. PR authorship above was verified against current ground truth at writing time.