[thakrarsagar]

PR #4972 left a coarse bracket for the selected complex stock coefficient. This adds a written two-sided bound for the actual #4470 product/completion/splitting rule under #4436's nested divisible-width realization, without changing the supplier or compiler.

For H = 64215200000000, N = 2^H and K = ceil(log2 B_N)/H:

2.943 N < K < 3.639 N

The endpoint ratio is below 5/4. The proof counts distinct root sources, bounds height using the actual productive floor, and charges all original paid calls, split rounding and completion calls. This bounds one permitted realization; it does not identify a minimal or uniquely selected width.

In #4972's same conditional outer accounting, retaining c_row = 1, Cw = 232960, BIT/old-stock tariffs, nesting and sum allowances, the sufficient log2 C upper bound falls from 64215200000024 to 64215200000010; the displayed fit bound becomes log2 p = 64473100659036. The factor-5/4 bracket concerns K, not optimal C. The fit deliberately retains the predecessor's stricter beta = 69999991/70000000; selected #4470 uses 1399999997/1400000000, so this is a conservative formula comparison. No new κ: the accepted conditional exponent stays 60403/20000000.

Task and sources

• Card selected-stock-envelope, drawn track adversarial-audit, priority revision 7; scope refined to The complex row coefficient of the selected outer reserve is one (log2 C −64 bits, no new κ) #4972's open next question 1. Claim / canonical investigation.
• Seed: 12681598eca9c3a85296c48193125739.
• Exact base: bee95ce88f9f19802f0613f92838895cd0e14a1f.
• Publication head: 6e1e7d47ed7fd7f3f7197cad7c19f8237c062b06.
• Canonical main refreshed to f259dabc7d70e0db8d6d36bfda888f008bbfecc1 before publication; no guidance, selected-record or live-input drift.
• Only research/selected-complex-width-envelope/ is added: six files. SOURCES binds twelve live inputs and two historical provenance records; README supplies the proof and limitations.
• Bound compiler width by exp(O(N log N)) and route overlapping ranks #4436/Independently check #4436 §3 on raw p7: width factor, exact routing-slot identity, N log N growth (no new κ) #4476 are predecessors. Reduce direct-flow batch widths while preserving chronological calls #4507/Independently recompute #4507's chronological batch widths (p7 rank 1000: 17,504 -> 2,524 bits; 175 vs 700; no new κ) #4519 least chronological batching and Remove tensor padding (199.32× smaller uniform-rank exponent) #4483 changed-height work remain separate. Live handoff/discussion and targeted PR checks found no competing claim on this exact follow-up; sparse/partial overlap coverage is documented, not a novelty certificate.

Validation

Linux x86_64, CPython 3.12.15, standard library only:

python3.12 -B scripts/research_package.py --package research/selected-complex-width-envelope
python3.12 -B scripts/check_research_commit.py --package research/selected-complex-width-envelope
python3.12 -B scripts/export_research_package.py export --package research/selected-complex-width-envelope --output /tmp/complex-width-envelope.tar
python3.12 -B scripts/export_research_package.py check /tmp/complex-width-envelope.tar

Both registered normal/optimized commands pass: 366 small source flows, 12 literal comparisons, 332 nested lower checks, 16 height checks, 12 exact product/pad/split cases, six reduced-power (v=1) calibrations of the actual #4470 invoice, and five mathematical rejection controls.

Committed-byte and official export checks pass for all 18 declared files. Git-free direct normal/optimized replay passes with read-only file modes and Git absent from PATH. Six private mutations each reject in both modes: omitted completion, unqualified floor credit, wrong height rounding, changed source bytes, changed receipt and missing source. Code mutations use --emit so failure is mathematical rather than a stale-receipt mismatch. All 19 exported files including the export receipt remain byte-identical.

Export SHA-256: 2f85c372b5cea6193b0b9a48159474f223b336d5df13e7bb4d0f1251050762b0.

Evidence is author-checked. No independent human review or proof-kernel acceptance is claimed. Hosted plan and the sole registered-package replay both passed at the exact publication head. Hosted execution corroborates this bounded author verifier, not independent scientific acceptance.

Limits and credit

Not checked: exact B_N, alternative/minimal batching, a materialized native/all-column program, owned-bank/route bodies, precision/fixed tapes, BIT/outer admission, analytic/prime/recovery premises, or the selected cold replay. The optional full-history ancestry preflight was not run in the shallow clone; direct export needs no Git or foreign source objects.

Credit James Reed / jwr404 (#4972/#4970), Mark / Matarener (#4612 and lockstep-live-role-audit), zxv (#4436 and strictification/native/stock work), Cinna Davis / 1337isnot1337 and zxv (#4440/#4470), Jacob Alan Sussman (Theorem 1G), Ryanson Jonathan (#4476), and the distinct #4507/#4519 batching lineage. Source notices and Apache-2.0 are retained.

Prepared by Sagar Thakrar with substantial OpenAI Codex assistance, solo. Open next questions identify a quantified truncated-source census and a separate physical fixed-program stock/route implementation. No selected result or frozen predecessor is edited.