Lower the selected BIT rank-42 residual to one contiguous weighted child

[commdata2338]

Change and mathematical scope

The conditional selected BIT stock proofs (PR #2084, #2120) leave residual lowering and child-weight dependencies as implementation obligations. This package binds the first unhosted rank-42 idle residual of the selected PR589 pipeline and emits its generic local-ring lowering to one contiguous 42f-atom weighted child, checked on the complete two-bank operator.

• Object: idle child 6, copy 213, replica 0, pair 0, side 1, port 102, with source-bound projector P0 = diag(0_22, I_42) on 64 coarse coordinates. The frozen hosting patterns account for all 288 rank-42 children and select this one deterministically.
• Lowering: on the generic class (successive northeast pivots of P C are units over Z/q^w Z, q an odd prime, no pivot permutation or scaling granted), unit lower ADD normalizers, a lower cross-bank shear, one weighted reversed interchange on the contiguous 42f head prefix and tail suffix, and the inverse adapters execute G_A exactly on arbitrary dirty inputs. The child descriptor is A_child[kf+j] = A[kf+j] · p_k; it reads frozen parent-derived values and the atom index, never fresh child fields or unused stock. No gather, interleaving, extra child or pivot-normalization pass.
• Descriptor boundary: two GL64 parent charts that agree modulo 5 require first child pivots 1 and 11 modulo 25, so the parent's low matrix alone cannot determine the descriptor. Substituting low-only pivots keeps rank and call shape but changes the operation. This is a counterexample to a weakened interface, not a defect in a selected compiler.
• Adapter count: 2f·(nL + nR + nK) coordinate ADD updates plus one rank-42f child, with nL, nR ≤ 1785 and nK ≤ 4096 (a loose bound of 15332f); exact counts are in RESULT.json. Converting them to the ordinary-call and tape toll remains open.
• Status: a source-bound algebraic compiler component with a scoped proof for the generic unit-pivot branch. The generic normal-form theorem and contiguous index rule are the retained source527 three-stage-cover framework's; the new parts are the selected binding, the executable ordered lowering, an independent direct-operator oracle, the explicit ancestor descriptor and the high-field counterexample. No stock coefficient, supplier or κ change.

Remaining obligations.

• Physical prefix-controlled ADD execution and its ordinary-call and tape bill; nongeneric (nonunit-pivot) fallback integration; complete padding and fresh-field cylinders; internally borrowing ordinary leaves.
• Every selected prime, denominator and minor exclusion; native precision, time and space bounds; full selected cold replay.
• Formal-kernel or independent human review; hosted CI. Spectator preservation is shown algebraically, not for a physical stream.

Task and overlap

• Task ID / scope key: selected-stock-envelope (eee4a5467dd91c47ef45), refined to selected-PR589/rank42-idle-residual/local-ring-weighted-child/ancestor-descriptor. Chooser nonce db5d733383c70adb154598e6abf1bb5c, track bit-mechanisms, priority revision 4, preserved in SELECTION.json. Following revision 4's preference for a live central handoff, this was taken over the chooser's suggestions (bit-new-register-model, whose series this account already worked; bit-helper-contract, whose immediate scopes are covered). No reroll.
• Base and exact heads: base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, rechecked unchanged at 21:24 UTC; contribution 3ad4f746258a36c80db5869d7beb0f88c71a350e. The selected PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 (a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46) is bound through the selected assembly and manifest. Context heads: PR Prove a conditional GL64 parent-stock envelope for the selected BIT bootstrap #2084 8001a23b003bec5dff5e61edec907b652c475932, PR Audit fresh-row conditioning in the selected stopped cover #2097 1d3ee0926656d5260c6cb1c0b81964ffe9fe1377, PR Tighten the conditional selected BIT parent-stock bound #2120 378593574261e0b0915e7e8a83147aa640274fc8, PR Certify one selected BIT q-adic route and its carry failure mode #2140 7ec858702da04d02748a5c05d439f602741a869a. No pending predecessor executable is imported; eight live inputs on main are pinned by Git blob and SHA-256 in SOURCES.json.
• Scope status: changed, from numerical stock estimation to one selected compiler implication that PR Prove a conditional GL64 parent-stock envelope for the selected BIT bootstrap #2084 and Tighten the conditional selected BIT parent-stock bound #2120 leave as a realization premise.
• Nearest work: PR Prove a conditional GL64 parent-stock envelope for the selected BIT bootstrap #2084 and Tighten the conditional selected BIT parent-stock bound #2120 price conditional stock; PR Audit fresh-row conditioning in the selected stopped cover #2097 proves fresh-fiber counting under field dependencies; PR Certify one selected BIT q-adic route and its carry failure mode #2140 lowers one fixed data-port q-adic row route and leaves residual and child descriptors open. Ross Richardson's complementary claim covers the nonunit-pivot fallback; nothing from it is used here. PR Lower selected BIT fixed routes to charged stream primitives #2157, opened after this package's final snapshot, compiles the selected header routes into stream primitives and does not lower residual children.
• Claim and final check: the claim is in discussion #2114. A final overlap check of open PRs and Discussions (complete pagination), plus a live check of newer PRs, found no competing work on this residual.

Validation

python3 -B scripts/research_baseline.py
python3 -I -B scripts/research_package.py --package research/selected-bit-weighted-child --check-only
python3 -I -B scripts/research_package.py --package research/selected-bit-weighted-child
python3 -I -O -B research/selected-bit-weighted-child/test_child.py

Linux x86_64, CPython 3.12.3, standard library. The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about 10 seconds and left the worktree unchanged.

PASS_SELECTED_BIT_WEIGHTED_CHILD rank=42 cases=10 basis=768 controls=10 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/selected-bit-weighted-child outcome=completed

• Ten ring cases (q = 5, 7, 101 and the Lucas-Lehmer-certified 2^89 − 1; w = 1 to 3) check all 128 weighted-operator columns. At q = 5, w = 2 the scalar ADD interpreter matches a separate direct formula on all 768 basis inputs for f = 1, 2, 3, plus dirty inputs in every case.
• All 3,888 GL2(Z/9) charts and their 108 rank-one projectors are enumerated: the 63 generic projectors pass every basis check for f = 1, 2, 3, and the 45 nongeneric ones are rejected.
• Ten semantic controls (weight order, missing tail reversal, omitted restoration, nonunit weights, nongeneric projectors, rank, idempotence, normalizer order and low-only pivots), live-source mutation, missing and corrupted receipts, and an optimized read-only export without Git all reject or pass as expected.
• The finite fixtures support the implementation; the ordered written argument carries the all-width, all-f statement. Executable bounds: dimension at most 64, width at most 8, f at most 3.

Credit

• The retained three-stage-cover framework (eumemic's source527 packaging and its lineage): the weighted-child algebra, fresh-field contract and contiguous index rule.
• Zhihao Chen: the nine-stage geometry. DreamingOfClouds: PR Punctured star bit unit: bit c = 0.00165947204563779652541 (+4.54% over #510) #545 and Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's selected BIT pipeline on eumemic's base, with the retained hosting and compiler lineage.
• Adam Gall (@adamgall, PR Prove a conditional GL64 parent-stock envelope for the selected BIT bootstrap #2084, Audit fresh-row conditioning in the selected stopped cover #2097), Nick Petalas (@petalas, PR Tighten the conditional selected BIT parent-stock bound #2120, Certify one selected BIT q-adic route and its carry failure mode #2140), Josie Elliston (PR Audit: selected bootstrap cutoffs omit the carried constant (κ unchanged) #1956, Proof repair: bootstrap cutoffs need only a stock constant (repair (b) of #1956; κ unchanged) #1965) and ToikY (PR Audit selected bootstrap stock fit: conditional factor-16 coefficient extension #1990): the stock and field-dependency work this builds on.
• Rohan Arun (PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094) and Jeremy Kothe's preceding assembly bind the selected source. This account's PR Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307 and Independent verification: PR589 paid rank ledger by exact rational projectors (no new kappa) #1421 are earlier hosting and ledger audits; they are not reused as new evidence.

New material is Apache-2.0; retained notices are unchanged.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. Can the emitted prefix-controlled ADDs receive a complete ordinary-call and tape bill? First step: one nonconstant K_A coefficient, keeping the full parent high field and unused quotient. Blockers: the physical ordered-affine implementation and the internally borrowing ordinary leaf.
2. Can the retained unit-pivot fallback be emitted for a nongeneric selected residual with the same descriptor dependency? Coordinate with the active fallback claim in DONE: conditional selected BIT stock bound 8194 to 6748 #2114 and compare the two interfaces on the same selected projector, checking the full 128-dimensional operator and the charged child list. Treating residue-zero entries as zero modulo q^w is invalid.

Workflow feedback (optional)

• What helped: the DONE: conditional selected BIT stock bound 8194 to 6748 #2114 thread and PR Certify one selected BIT q-adic route and its carry failure mode #2140's explicit open question pointed straight at an unowned interface.
• Friction: the chooser's output was lost in a long batched tool output and recovered from the session log without rerolling.
• Suggestion: write the chooser JSON to a file before printing a compact suggestion list, so the nonce, scope keys and declined options survive truncation.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG