[unpaidmoon]

Change and mathematical scope

Conditional coefficient identity for the complex coarse-depth envelope. No new κ. Selected κ stays 0.001672660197076918, and every actual_stock_coefficient in the bootstrap receipt stays null.

Under the coarse-depth expansion, #4141's strict digit bound log_q C < 1321/40 (primes q > 2^80) and #3667's conditional leaf hypothesis S_leaf = 1,

log_q S_parent(e,w) < T(s) * w * (1 + log2(e))

for integer atom width w >= 1 and width e >= 2, where T(s) = (43/26)*(1-s)*(1321/40). The shape step is exact: w*(1+L) - L = w + (w-1)*L with L = log2(e), and both the constant w and the slope w-1 are nonnegative for w >= 1.

#3667's published reading-(a) coefficient multiplies the base-2 length 2642 instead. On both recorded savings that value is exactly 80*T(s). Native stage 1 displays it as ≈ 4362.14. rho = 0 here is log_q(1) = 0. A one-digit external leaf still has log_q(q) = 1 and is not bounded by the parent term alone.

This does not re-prove the layer closure, does not identify |Sp(72,2)|*14826 as C(w) at two widths, and does not adopt a selected-stage coefficient. #3667 already records that the selected bootstrap does not call this layer.

Task and overlap

Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7. Card selected-stock-envelope, scope refined: the card asks for one selected-stage coefficient; this is the complex-coarse coefficient obtained by joining two merged notes, not that stage field.

Ordered list. The guidance order used here is the eight current leads in docs/research/research-family-status.md. Each lead still has a stated next piece that is not closed by a merged result. Structured #1166 handoffs were checked first and did not supply a free item: the typed actual-width H handoff is this account's #4123/#4127, and the n=7, L=22 census handoff is open #4221.

1. Audit one #4100 proposition (unaudited obligations remain)
2. Finish #4088 four-provider admission (#4180 is open and does not close the full engine)
3. QQ16 matched protocol or a cheaper demand (#4177, #4172, #4167 are open)
4. R437 compiler, caller, or stock integration (#4176 is open; the complete bill remains stated there)
5. P104 changed producer/consumer premise (the five scoped audits are already delivered)
6. Fixed-inventory ceiling / paid-transition check (#4170 and #4181 are open; residual search stays paused)
7. New geometry and recursive boundaries
8. Selected stock and formal interfaces

Item 8 is this package. Inside that lead, the compact-switch audit is #4321, the log_q parent correction is #4141, the loose rho = 0 reading is #3667, and the stage-one recurrence and stock boundary are open #3898 and #3920. The distinct remainder is the join: #4141 leaves rho unbound, and #3667 sets rho = 0 while still multiplying 2642.

Overlap at 2026-10-21 13:28 UTC (458 open PRs, 3225 discussion records, complete pagination) hits those two packages and the #4141 discussion note, which still leaves rho separate. No open PR states T(s) under S_leaf = 1.

Validation

Python 3.12, standard library only. Registered replay passed:

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

Marker:

PASS_SELECTED_STOCK_LOGQ_RHO_JOINT digits=1321/40 factor=80 bootstrap_T=174485626749443905524883973467613250556094475116027/3200000000000000000000000000000000000000000000000 conservative_T=4362140668748600743658666134074946636014640956718493/80000000000000000000000000000000000000000000000000 rho=0 S_leaf=1 coefficient=null layer_not_bootstrapped headroom_not_adoption no_new_kappa

test_verify.py rejects width 0, a zero stock gap, and treats both a one-digit leaf and an 8194 leaf coefficient as outside rho = 0 (controls=5). The same certificate passes under python3 -I -O -B.

Not run: the rho-ledger's full verifier, the PR2287/PR3385 layer replay, cutoff-margin arithmetic, and any Lean or native build. S_leaf = 1 is used as #3667's pinned conclusion.

Credit

Parent term: unpaidmoon, research/selected-stock-logq-parent (#4141), from Wentao Li's A = 0, B_bits = 2642 and Sricharan Vagvala's coarse-depth composition. Leaf hypothesis and the loose coefficient: Mainzu, research/selected-stock-rho-ledger (#3667). Stage receipt: Josie Elliston, via the bootstrap stock audit. Prime floor: the selected bootstrap scope map.

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

Open next questions

1. Does a later selected assembler bootstrap this complex layer? T(s) is a stage obligation only for that call. The pinned rho-ledger says the current assembler does not. Blocked until code/assemble.py changes.
2. Is |Sp(72,2)|*14826 the uniform divisor C(w) at two atom widths? Read the open stage-one recurrence audit before another divisor search. The cover note still has no W(w) formula.

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