Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460)

[DreamingOfClouds]

Conditional κ = 641739651875561/(5·10¹⁷) ≈ 0.001283479303751122, bit-bound: +0.711% over #460 (0.001274415506145115) and
+0.741% over #458. Whole-residual hosting on both sides: no cross-family residual split, no cross-copy lemma, no new complex lemma.

side	supplier	endpoint
bit (binds)	R715 E8 bit word, eleven stages, #442 whole-residual hosting, target mixing	c ≈ 1.285129·10⁻³ (+0.712% over #459's 1.276042·10⁻³, +0.742% over #454's)
complex	#458's paid-adapter kernel, unchanged	b ≈ 1.297204·10⁻³

#465 (Srikrishnabalaji) has independently certified this outer assembly (the step #463's draft left open) at 1.283479306671789·10⁻³
with a sharper bit charge, crediting #463; this PR supplies the full bit supplier and its verifier.

What is new: the bit word R = 715 (#436's R740 → 715).

1. A carrier lower bound, and a word that meets it. In this format a copy of source x_s that first acts above its line frame
   serves only targets in a hyperplane section of s^⊥ (frames contain everything they touched). The 63 lines of E7 = s^⊥ are a
   spherical 4-design, Σ_r ⟨r,u⟩²⟨r,v⟩² = 16(|u|²|v|² + 2⟨u,v⟩²) > 0, so no two hyperplanes contain them: every source needs ≥ 3
   registers, i.e. ≥ 240 carriers; likewise every target needs ≥ 2 helpers reaching its hyperplane (240, met since κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370).
   κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370-lineage words spend 272 carriers because each frame line 2e_m enters the D4 sums of seven blocks. Frame-pair D4 sums
   (D4(c) = Bcw[c] + FP + FP', FP = 2e_a + 2e_b over the 12 pairs of one AG(3,2) direction plane) meet the bound: R 740 → 724.
2. Eleven cancelling re-expressions (a rule rewritten as x ⊕ y of existing symbols; overlap cancels inside its frame): R 724 → 715.
3. Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's replica kernel rebuilt for the new word (deterministic search, coordinate entrances), and Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450's target mixing
   generalised to chained receivers (a donor that is itself a receiver); it reproduces Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450's mixed word byte for byte.

Verification (python -B research/e8-r715-paid-complex/verify.py /abs/out; CI workflow e8-r715-paid-complex): #458's full
wrapper and the R715 package's full wrapper from scratch, then the exact composition (two moment engines per side, six bootstrap
levels, all 47 strict outer inequalities, next-grid and parameter controls). The bit package regenerates the word from its frozen
hierarchy (legality of every rule and read checked), runs #370's independent all-column checker and mutants, a relaxed copy of that
checker on the kernel and mixed words, #443's C++ cohort legality and formal columns at five and eleven stages, mixing mutants, Gram,
eleven-stage template and hosting audits, and its own assembly. Two fresh-copy runs agree apart from timing fields (results, both
supplier RESULTs and the complex AUDIT byte-equal). The R715 word was also checked by an independent validator.

Review points

• Carrier bound (proof above): the 4-design identity for E7 and the frame-containment argument give ≥ 240 carriers; it is a
  scoped bound for this format, not used by the certificate.
• Rekid legality: each of the 11 re-expressed rules uses existing symbols whose frames lie inside the rule's frame, the
  hierarchy stays acyclic (the compiler's recursion requires it), every consumer frame contains the rule's frame, and content equals
  the F2 sum of its kids (checked by hier_word.py before compiling).
• Relaxed checkword (code/checkword_relaxed.py): drops only the two frame-0 assertions of κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's checker (y/helper start at
  frame 0, stock line 75; dirt compensation at frame 0, stock line 89) and skips its five-stage price for kernel words, because
  Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's kernel pivots start at their entrance frames. Labels, legality, exact F2 on all 955 columns and Gram are unchanged, and
  Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's C++ cohort legality and the five/eleven-stage formal columns still check the same words.
• Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450's Lemma M (bit target mixing, here with chained receivers); Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458's paid adapters (inherited).
• Contracts: the bit side keeps its own (halving depth 3, coefficient 525 bits charged 576, row coefficient 28,416 = 3(78²+1)+10,161,
  which sets the row allowance; the bootstrap and κ do not depend on either). Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445's sharper row bound gives 19,425 for R715, so the
  enlargement over the lineage's 20,161 is avoidable.
  Conditional like all parents; no unconditional theorem or Lean proof is claimed.

Credits: chafreaky (#458 paid adapters, #451/#454 whole-residual hosting and the cross-family obstruction, #443–445 kernel and
staging), rohanarun (#460 composition and verifier structure, #453, #459), Srikrishnabalaji (#465), DaysSky (#442 idle-axis hosting),
Dylan Crow (#370 bit word, producer and checkers), eumemic (#436 R740 word and reuse, #440/#441 retiming), higherbeing (#448),
icekylinx (#433), jacklightChen (#425), Jacob Sussman, Zhihao Chen and all source-pinned authors. Carrier bound, frame-pair
construction, cancelling re-expressions, kernel rebuild, generalised mixing and composition by DreamingOfClouds with
Claude Opus 5.5 (Anthropic), via Claude Code.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @DreamingOfClouds. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Carrier-bound-saturating R715 word and overlap cancellation.

Review and limits: Retains the carrier lower bound's compiler restrictions and full paid-complex composition. Earlier review of fourteen vendored files did not cover this entire contribution.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: cdf0c1c6540df987467988fd771340670f4136f3. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.