research: bind selected stage-one stock recurrence

[1337isnot1337]

Summary

Adds a bounded source audit for stage 1 of the selected #1094 native, complex-binding bootstrap branch. It binds the stopped producer's parent stock factorization and leading cover-depth coefficient, and separates parent stock from completed inner borrowing.

The sufficient parent divisor is S1(e,w) = W(w)^D(e) R0(w). The retained row proof gives D(e) <= 47 ceil(log2(max(2,e))) + O(1). The PR185 complete-row contract says completed inner ordinary calls borrow from their supplied chunks, restore their row coordinate, and add zero externally supplied stock. This rules out multiplying parent stock by sequential completed-leaf calls.

Evidence and limits

The selected PR1087 assembler is byte-matched against the selected source archive. The verifier confirms the m=73 selected producer/stage context, source recurrence, depth slope, PR185/PR387 reserve split, and selected finite invoice identities. The field actual_73column_sigmas is a map count, and the invoice's physical_stock_identity is a fixed finite ledger; neither defines the variable-width row divisor or old-leaf reserve.

The numerical sigma_1 remains open: the retained sources leave W(w) and the old-leaf reserve coefficient unquantified and leave the additive depth term as O(1). The historical m=120 value 57,604 and unused 252 allowance are not imported. No stock-fit proof, supplier admission, new kappa, or theorem is claimed.

This result complements open PR #3742 at head 89f6af270c550dd02bdb05d7a69987159aa110f2: that PR locates the cutoff chain and leaves sigma uncomputed; this package audits the source-bound parent/inner reserve factorization.

Reproduce

python3 scripts/research_package.py --package research/selected-stage-stock-recurrence-20261018 --check-only
python3.12 scripts/research_package.py --package research/selected-stage-stock-recurrence-20261018

Credit

Preserves attribution to Rohan Arun (#1094, #387), Jeremy Kothe (#1087), Nick Petalas (#3742), ToikY, Josie Elliston (#1956/#1965), the reviewed PR185 lineage, and eumemic's distinct m=120 source-bank work. No independent human review is claimed. The local package was prepared by OpenAI Codex, homelab-10, with substantial AI assistance.

Current overlap

Open PR #3742 locates the selected complex branch cutoff sequences. This package instead binds the stage-one parent-stock recurrence and separates parent stock from completed inner borrowing; it leaves the variable-width divisor and actual stock coefficient unresolved.

Publication validation

python3.12 scripts/research_package.py --package research/selected-stage-stock-recurrence-20261018 passed on CPython 3.12.3, and python3.12 scripts/check_research_commit.py --package research/selected-stage-stock-recurrence-20261018 --check-source-history passed. The registered replay left the worktree clean. This supersedes any earlier pending note above for the registered package replay; separate full selected cold-replay limits remain as stated.