[ywz65]

PR2569's unchanged source-directory producer reconstructs the radix exponent for each of its m² high-field coordinates. That makes the CH cardinality setup cubic in w, rather than the stated O_q(w log(C+1)+w²). This source-pinned audit gives the exact division write bill and corrects the setup bound. It also proves that complete directory production remains Theta_q(D) in its actual serialized output extent, including this setup and local cleanup.

For L-bit words, every retained unsigned division writes exactly 3L²+8L+4+(3L+2) popcount(quotient). The high_power call performs exactly m²w divisions and m²wL full-width shift iterations, even when the dividend has leading zeros. Its sharp setup order is Theta_q(w³+(w−1) bitlength(C)); making the fresh C copy gives Theta_q(w³+w bitlength(C)). At w=1 the unchanged high_power performs no multiplication and need not read C.

The complete Context/produce/Context.close interval performs ((2m²+1)w+1) divisions. Its output has D=CH(h+1)(B+2+m²(L+1)) cells, with H=q^(m²(w−1)). The proof absorbs the corrected cubic setup into D directly. The separate payload statement D=O_q(V) still needs PR2569's h/w/R premises; this audit does not refute its final volume-based asymptotics.

Validation: native CPython 3.12.7 registered optimized primary and all four export/integrity groups pass; native CPython 3.14.5 optimized primary passes. Sixteen setup cases cover w through 32, a 129-bit C and m=64/w=1. Eleven complete directory cases check exact serialized order, repeated/empty physical prefixes, ownership, cleanup and meters. Four semantic controls and typed receipt/source corruption tests accompany the written all-size argument.

Independent AI review found no material issue in the stated scope. The reviewer separately derived the division formula and cubic absorption, checked 166 direct-source divisions at L=2..11, authenticated all 11 retained source/proof files, and passed the registered replay plus all four export groups.

python3.12 -I -O -B research/source-directory-setup-cost/verify.py
python3.12 scripts/research_package.py --package research/source-directory-setup-cost

Scope: fixed q>=3, fixed 1<=m<=64 and finite compiled request catalogue, canonical cardinality tapes, and the retained fixed-alphabet Tape/Word meters. Input fixture/vocabulary compilation and returned directory/Count disposal lie outside the measured interval. Python controller/allocation costs, a literal fixed-tape compiler, numerical reserve constants, full selected packet/route execution, recursive hosting and supplier/exponent admission remain open. The producer and selected record are unchanged.

Source pins: PR2569 0b116cb34f27a4b064f0d8dd65549ea5705cf9c0; credited primitive/division audit PR2566 bfe8317a38f03a5693470e873b29101f3f7a55e4. Every retained source has SHA256 and Git-blob authentication before execution. Direct base is current canonical main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. The bit-helper-contract picker refinement and bounded overlap coverage are recorded in TASK.json. Canonical advance claim: #2569 (comment).

Credit Jose Manuel and the retained source lineage, and ywz65's prior cost audits. Prepared for ywz65 with substantial OpenAI Codex assistance. No human or formal-kernel verification is claimed. Open next questions concern sharing the encoded exponent traversal with a complete bill, and pricing the full caller's directory reserve.