[mihir-s-05]

PR #4436's direct compiler chooses nested widths so every recursive batch is integral. This contribution derives the least common replication factor for a fixed chronological batching policy and prescribed smaller supplier widths, then constructs smaller widths while preserving every source, exit and normalized recursive cost.

For raw p7 at rank 1,000, productive width falls from a 17,504-bit integer to a 2,524-bit integer. The old width is an exact multiple of the new one; the reduction factor lies in [2^14980, 2^14981). This is a constructive finite-width result, not a new kappa or practical runtime claim.

There is also a small counterexample to a tempting relaxation. For the source equality fixture at rank 12, pooling equal-rank exits across different calls accepts multiplier 175. Eight separate leaf cuts each require a width-700 child, so the correct least multiplier is 700. No bug is alleged in #4436's valid nested construction.

Start with the proof and scope, independent implementation, and exact receipt. Only the new research/chronological-batch-width/ package is added.

Proof boundary. Each batch (e,q) requires B_e | a*q, so its modulus is B_e/gcd(B_e,q); the LCM is necessary and sufficient. Complement batches run before the flow, while exit batches finish at their own parent-round cut. Repeated paths and paid positions are counted separately. A second induction proves that each new width divides its old nested counterpart. The unchanged exact leaf recurrence and inherited exp(O(N log N)) bound are retained.

Evidence at 98b9f9e1b96940c0c405638afa974e378436fc55, CPython 3.12.0 / Windows:

• All nine frozen Bound compiler width by exp(O(N log N)) and route overlapping ranks #4436 flow receipts reproduced without importing or executing its evaluator.
• 101 independent literal cut fixtures, including product collisions and the actual counterexample; 2,304 brute-force least-multiplier checks.
• 1,074 nonbase compilations and 49,474 batch constraints. Every intermediate width divides the original width; normalized costs agree exactly with both the nested construction and a separate rational recurrence. Frozen final width and leaf-count hashes match for both toys and raw p7.
• Registered ordinary and optimized replays pass, with 12 rejection controls.
• Committed-byte/source-history preflight passes for all 13 files. The official export passes integrity checks and ordinary/optimized Git-free replays with Git unavailable and unchanged payloads.
• Separate source/receipt corruption controls reject. Code mutations that pool chronological repetitions or omit repeated exit costs also reject.

Reproduce:

python -B scripts/research_package.py --package research/chronological-batch-width
python -B scripts/check_research_commit.py --package research/chronological-batch-width --check-source-history

Coordination and limits. Started and refreshed from canonical main 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7. This explicitly changes the independent-moment card to #4436's constructive-width question. Discussion #4503 is canonical. Final overlap coverage: all tracked local top-level notes, 355 open PRs, 3,381 Discussion records and 3,787 closed PRs, complete reported pagination at 02:39–02:41 UTC October 22. This is bounded text coverage, not a novelty certificate.

#4456 already checks the finite flow and #4476 its original nested widths. #4483 changes the stopping height and convergence argument; this package does not audit that successor or imply the old exponent question remains generally open. Remaining work is an all-rank prime-valuation bound and a source-bound application to that changed-height compiler. Local multiplier minimality is not a global optimum over libraries or schedules. Native/all-column adapters, physical costs, owned-state handling, precision, analytic transfer and recovery remain inherited. No selected cold replay, formal proof kernel, independent human review or new kappa is claimed. The historical orientation command was attempted but stopped on a sparse-omitted selected claim file. Hosted CI is separate from the local results.

Credit Jacob Alan Sussman, zxv, Cinna Davis / 1337isnot1337, Ryanson Jonathan / pinterbanget and the native lineage retained in the pinned sources. Prepared for Mihir Srivastava / mihir-s-05 with substantial OpenAI Codex assistance; no subagents. Apache-2.0. No merge or selected-frontier change.