[JoyboyBrian]

Change and mathematical scope

Improve the selected #4470 compiler's mathematical nested-width upper coefficient from K/N < 3.639 to K/N < 3.070, answering #5026's truncated-source handoff. The selected κ, batching, completions, splitting and physical assumptions are unchanged.

exact root eligibility + nonroot missing-parent demand
  -> every source rank s > n/M
  -> S_n < (T1/(n/M)) sum_d P1^d T1^(G-d)
  -> fully charged P1 < T1 <= U
  -> ceil(log2 B_N)/(HN) < 3.070

All proof-relevant predecessor bytes are retained with original-head/path/hash attribution; live replay pins use this contribution's own history. Credit Sagar Thakrar (#5026), James Reed/jwr404, Mark/Matarener, zxv, Cinna Davis, Jacob Alan Sussman, Ryanson Jonathan and mihir-s-05. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. Investigation: #5061.

Validation

• Before: Bound #4470's complex row coefficient within 5/4 (no new κ) #5026 uses the untruncated full-layer upper envelope, with Xhi=233670787813430 and K/N<3.639.

• After: the uniform all-layer rank-weight inequality gives X=197115103572543 and K/N<3.070. Six actual selected v=1 boundary fixtures and twelve tiny literal chronological flows agree on complete source spectra. The concrete 3624 -> 3625 source-count decrease rejects monotone interpolation.

• Registered normal/-O replay passes; six semantic controls reject. Git-free replay passes; source and receipt mutations reject. Committed-byte/source-history preflight passes at 103f8130fe617e8c720af4cb6ce63cf638085d76. No product histogram or integer of size N is materialized.

• A separate Astra agent reviewed the exact-head proof, source bytes and checker without findings; its independent partial-Pascal fixture agrees on the complete source spectrum. This is AI-assisted review, not human or proof-kernel acceptance.

Merge Danger

Door: Two-way.

Blast Radius: Research.

Only a new scoped package. This is an author-checked uniform source-count argument with finite calibration, not physical stock/route admission, an optimal width, a new outer cutoff or a multiplication theorem. Native, ownership, whole-job, precision/tape and analytic/recovery hypotheses remain inherited and unrun.

Task and overlap

Refined selected-stock-envelope task b2_math_truncated_source_envelope, canonical claim. Scientific base 4c0a359d88f22f05189d51f48387750b801a45d4, refreshed unchanged before submission; priority revision 7. PR5026 exact head 6e1e7d47ed7fd7f3f7197cad7c19f8237c062b06 supplies the existing bracket and asks for retained eligibility. This proves the missing input source envelope; it does not redo the outer cutoff or the separate physical stock interface. Source heads and retained bytes are recorded in the package. Overlap refreshed 2026-10-23 07:37 UTC: 318 open PR bodies, 4159 Discussion records and all 6760 local notes; subsequent local claim check finds the canonical claim. Arbitrary diffs and unpublished work remain outside coverage.

Credit

See the retained notices and package attribution; predecessor credit and substantial AI assistance are stated above.

Open next questions

The package questions separate sharper fractional-moment source bounds from the unimplemented physical stock/ownership/routing interface. Start with the same nonroot absence indicator and fully charged completion mass; no new histogram or outer-root calculation is required.

Reproduce on macOS arm64 / CPython 3.12.7: python3.12 scripts/research_package.py --package research/selected-truncated-source-envelope. Targeted checks only; full selected/native, physical tape and proof-kernel validation were not run.