Method + bounds (no frontier claim): kernel width rule, EXACT pivot maximum 112 and a seeded 112-pivot selection on the R715 word

[DreamingOfClouds]

Method and knowledge package on the R715 bit word (PR #463): what sets the width of a residual, how many narrow residuals the shared-donor kernel can give on the retimed R715 word with coordinate entrances (the verified residual-isometry format: exactly 112), and a 112-pivot kernel selection by seeded annealing. Conditional κ = 1294233365881801/1000000000000000000 ≈ 1.294233365881801·10⁻³, bit-bound: +0.838% over #463
and +0.165% over #472 (κ = 1.292099559·10⁻³). It is below the current lemma-less frontier: −0.108% vs #473 (κ = 1.295630·10⁻³, as updated on Oct 11, now the frontier) and −0.098% vs #478 (1.295509·10⁻³). This PR's point is the method and the bounds, not the frontier. The bit supplier is c = 1295910576589222380953/10²⁴ ≈ 1.295911·10⁻³ (+0.839% over R715). Same word, same
hosting rule, same mixing, #458's complex supplier unchanged: only the kernel selection changes, 84 → 112 pivots.

side	supplier	endpoint
bit (binds)	R715 word, 112 rank-1 kernel pivots, eleven stages, #442 whole-residual hosting, target mixing	c ≈ 1.295911·10⁻³
complex	#458 paid-adapter kernel, unchanged	b ≈ 1.297204·10⁻³

What is new

1. The width rule, checked exactly. A residual's width is 8 − entrance rank, and only kernel pivots have a nonzero entrance: the helper hierarchy and the
   target mixing do not set it. Checked three ways on all 715 helpers (kernel-word frames, exported records of the mixed word, the independent hosted audit).
2. The pivot ceiling and the exact maximum, for coordinate entrances (the verified residual-isometry format). A pivot needs a frame line inside its first frame (263 of 706 helpers), and its delivery vector must be an XOR of donors of one
   entrance class. Proved: pivots ≤ |C| − rank = 263 − 86 = 177, and rank-1 coordinate entrances lose nothing up to parity. Exact maximum on the retimed R715 word: 112, by a computer-assisted exhaustive search (independently re-audited), certified by
   code/exact_pivots.py (rerun by verify.py --light, asserted equal to 112, about 10 s): the pivot problem is a min-sum-rank list partition (max pivots = |C| − min Σ_i rank of the class
   vectors; NP-hard in general, not a matroid union/intersection/parity problem), solved exactly here by splitting the delivery matroid into components (208 / 28 / 14 elements plus 13 coloops),
   exhaustive span-tuple DP on the small ones (16 and 9 pivots) and, on the 208, an exact block DP with a coupling-assignment lower bound and an exhaustive search (minimum sum-rank 121, so 87 pivots;
   120 shown infeasible): 87 + 16 + 9 = 112. The annealing plateau (14 of 16 restarts at 112, never 113) is therefore optimal. Priced by the model, a rank-1 pivot is worth ≈ +0.033% of c,
   so the pivot count on R715 is exhausted within this format (177 would have been ≈ +2.9%; it is not reachable; the donor choice among 112-pivot selections still moves c, README §3).
   Scope: the 177 and 586 are valid but loose scoped bounds; the 177 does not cover non-coordinate entrances. Upstream kernel() itself accepts any flat; there all 706 helpers qualify and the same argument
   gives only 706 − 120 = 586, in a format this package does not verify (it needs coordinate residual isometries). The exact 112 is a statement about the same coordinate, one-class-per-donor model;
   entrances at arbitrary root lines were only searched.
   Follow-up: the same exact method gives 101 pivots on the R687 read-then-climb word (100 after the parity rule); to be reported in a follow-up PR.
3. The method: select_kernel.py anneals the class assignment on the exact objective Σ rank (self-contained xorshift64*, reproducible from a seed),
   kernel_from_selection.py builds the word with Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's kernel unchanged. Pinned: --seed 3 --iters 150000.
4. Honest size: by kernel choice alone, without retiming or reordering, the first 112-pivot selection (c = 1.295564·10⁻³) landed, in κ, within 0.003% of Conditional kappa 0.001467289440740487: rank-sector factory on 112 ports #473 as first posted (1.293922·10⁻³);
   the shipped best-of-eight seed was 0.027% above that. Conditional kappa 0.001467289440740487: rank-sector factory on 112 ports #473 has since been updated (1.295630·10⁻³) and is 0.108% above this PR. This is selection tuning, not a ≥ 1% result.

Verification (python -B research/e8-r715-kernel-cap/verify.py /abs/out [--light]; CI workflow e8-r715-kernel-cap): LIGHT (about two minutes) checks integrity and
pins, recomputes the ceiling analysis from the pinned R715 word, recomputes the exact pivot maximum and asserts 112, checks the selection against #443's exact kernel conditions plus the F₂ response relation and the coordinate-flat condition,
rebuilds the kernel word (replay, interface) and the mixed word and compares them, checks the width rule, re-solves the hosting (CP-SAT) and prices it with the
exact engine, and runs the screen composition with #458's pinned endpoint. FULL adds the seeded search, #443's regressions, the relaxed all-column checker,
C++ cohort legality and formal columns at five and eleven stages, mixing mutants, Gram, eleven-stage template/audits, hosting selection rediscovered with HiGHS and
required equal, hosted audit, assembly, the pivot-count scan, #458's complex supplier from scratch, and the exact composition (two moment engines per side, six
bootstrap levels, all 47 strict outer inequalities, next-grid / zero-η / zero-β / unit-β rejection).

Review points

• The ceiling's proof (items 1–4 of README §2) is elementary. The new claim is the exact maximum 112: its certificate is the code in code/exact_pivots.py (component split, block DP,
  coupling lower bound, exhaustive search), rerun in LIGHT; please review that code and the min-sum-rank formulation of README §2 items 2–3 it relies on.
• The seed choice among equal pivot counts is an exact-priced selection (c between 1.295564 and 1.295911·10⁻³ across eight seeds).
• Inherited and unchanged: Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450's Lemma M (bit target mixing, chained receivers), Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458's paid adapters, the R715 hierarchy and its carrier bound (Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463).
• The bit side is now 0.1% under Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458's complex endpoint; further bit gains here are complex-capped.
  Conditional like all parents; no unconditional theorem or Lean proof is claimed.

Credits: chafreaky (#417 shared-donor kernel and exact replay, #443 replica kernel and residual isometries, #451/#454 whole-residual rule, #458 paid adapters, #472),
rohanarun (#434 retiming with kernels, #435 banks, #460 composition and verifier structure, #467, #470 retimed kernel and reordered R715 deliveries, #473), eumemic (#436 R740 word,
#440/#441), DaysSky (#442 idle-axis hosting), Dylan Crow (#370 word, producer, checkers), Srikrishnabalaji (#465), Dugongue (#471, #476, #478), jacklightChen and Zhihao Chen
(#386/#425), icekylinx (#433), higherbeing (#448), Jacob Sussman and all source-pinned authors. Width-rule check, ceiling analysis, exact pivot maximum (min-sum-rank formulation, component/block decomposition, exhaustive search; related literature: submodular multiway partition, Svitkina-Fleischer 2008, Chekuri-Ene 2011),
annealing selection, kernel build from a selection, hosting re-solve, pivot scan, composition and verifier by DreamingOfClouds with
Claude Opus 5.5 (Anthropic), via Claude Code.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @DreamingOfClouds. Your work and attribution are retained in the published 100-PR review, credits and complete original source.

Covered: Exact coordinate-entrance pivot maximum and a 112-pivot witness. The ceiling is for the specified R715 word and coordinate-entrance format. Preserve method, negative results and finite selection without claiming a general optimum or selected frontier.

Keeping this PR open: The exact width rule/pivot ceiling has successful light checks. The actual full Linux job 114567592208 reaches the final discover check but fails: frozen hosting selection differs from recomputed optimum. Correct or explain that frozen selection with a source-bound completed full replay before closing the construction claim.

The source archive preserves the entire submission; it does not substitute for the remaining execution or review. The batch's machine-readable disposition records this as partial coverage. Reviewed head: af51ea7ad6570f04c8424be7cc6f4070bfd1e6ff.