Method + knowledge (no frontier claim): read-then-climb bit word R687 (c ≈ 1.315433·10⁻³), a copy lower bound, exact pivot certificates

[DreamingOfClouds]

Method + knowledge PR (no frontier claim). For the complex side it cites our #484: a pinned full-run result at #484 head
d30a05f. The branch itself contains #463 and #471.

It contains three things:

• an anatomy and lower bound for the copy term of κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370-lattice bit words;
• read-then-climb, a compile change that removes 28 copies;
• exact kernel-pivot certificates.

The resulting bit supplier is R687: c = 65771650815860982499/(5·10²²) ≈ 1.315433·10⁻³. It is priced at eleven stages with
#442 whole-residual hosting and target mixing, with no cross-family split. That is +2.36% over the R715 bit supplier (#463) and
+1.32% over #483's bit.

complex supplier	κ	binds
#484 (our draft, head d30a05f; Lemmas A, G), cited from its pinned full run	≈ 1.313705·10⁻³	bit
#471 (Dugongue), rerun unchanged, #471's own assembler	1303530097714685/10¹⁸	complex

Relation to #493. #493 (eumemic) builds a 489-helper word with sources as carriers and c ≈ 1.565·10⁻³. Its bit side no longer
binds, so #493 + #484 = 1.325031·10⁻³ is above our #484 figure, and #493 + #471 equals our #471 figure (#471's complex cap).

Our levers do not transfer to #493. #493 is a hand-built D4-quad skeleton, not a compiled pair ⊂ L ⊂ M hierarchy, so the copies
that read-then-climb removes do not exist there. And no bit-side gain can move κ while #493's c sits 18% above every whole-residual b.

What is new

1. Copy anatomy. Raw R = 9 + carriers + copies + fresh. On R715 that is 839 = 9 + 240 + 528 + 62.
   • Most copied symbols (427 of 467 on R687) have exactly two rule parents whose frames meet only in the symbol's own frame: a
     pair in two L's, an L in two M's, an M in two tops.
   • The exceptions: 28 L's that also build D4(c) (three parents), 10 pair and 8/4 sums with 3–4 parents, one 9/4 sum and one
     16/6 sum.
   • Given the DAG, the copies depend on the phase rule: 556 under κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's phases against 500 with read-then-climb.
2. Lower bound (transversal-cover lemma, any legal word). A register whose first use is at frame G feeds only targets in G^⊥.
   So χ(u) ≥ the least number of proper sections covering u's downstream targets (the carrier bound is the case of a source).
   • κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 lattice family (hand proof). Assumptions: rule gates at the hierarchy's frames; one nested read chain per target;
     distinct M frames. Results:
     • M copies ≥ 112, or ≥ 140 when every rule runs before the scatter.
     • L copies ≥ 168.
     • Pair copies ≥ 168: each lattice pair has two L readers at incomparable frames, and the carrier bound (3 chains per cw source)
       caps the distinct pairs at 168. Without that bound, pair copies + cw carriers ≥ 392.
   • Totals: lattice copies ≥ 448 = 112 + 168 + 168, or ≥ 476 with the D4 assembly. With all-A phases the bound is ≥ 504,
     against R715's 528.
   • Per DAG, machine-checked by verify.py (copy_bound.py, exact Dilworth plus the lemma only): R687 has 500 copies
     against a bound of 464; R715 has 528 against 492.
   • Scope: no bound on copies alone holds outside a fixed DAG, because copies trade one-for-one against recomputation and cancellation.
3. Read-then-climb + coset-split π-pairings. A rule that reads a low-read symbol moves after the scatter, so the symbol's
   register is read at its own D5 frame first and then climbs. Spreading the D5 slots over both π-cosets of each block brings M
   copies from 140 to 112, which is the new bound.
   • A per-block search over two slices of the flags (pairings × l5, and pairings × folds; raw R is block-separable) found no
     lower raw R. It ran outside the package and is not re-run by verify.py.
   • Stacked with Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463's re-expressions, R goes from 715 to 687.
   • mix_target.py generalises Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450 mixing to words whose phase B holds helper rules.
4. Exact kernel pivots. Pivot selection is a min-sum-rank list partition over a linear matroid. exactgen.py solves it by a
   computer-assisted exhaustive search per component, using block DP and coupling enumeration.
   • Scope: the maximum over Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443-format selections on the retimed pre-kernel word (Conditional κ = 1.128983115440183e-3: an E8 bit word with 740 helpers under #433's complex supplier #436 retime, 6 sweeps), with coordinate
     entrances and one class per donor.
   • R687: maximum 101; parity leaves 100. The greedy search had 71. This adds +0.887% to c.
   • R715: maximum 112. This equals the exact maximum in our draft Method + bounds (no frontier claim): kernel width rule, EXACT pivot maximum 112 and a seeded 112-pivot selection on the R715 word #485, recomputed here by a second solver.

Verification

python -B research/e8-r687-tensor-complex/verify.py /abs/out [--light]; CI workflow e8-r687-tensor-complex.

The bit package (research/e8-r687-read-then-climb/verify.py) checks:

• Regeneration: the word is rebuilt from the frozen hierarchy byte for byte; every rule and read is checked for legality;
  only the phase rule of κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's compile2 is substituted.
• Bounds and certificates:
  • the copy anatomy and bound on both pinned hierarchies;
  • the R687 pivot lower bounds and witness, checked against the pinned refutation log (--certificate reruns it, about 30 min);
  • the R715 log is pinned only.
• Kernel and mixing: the kernel is rebuilt from the frozen exact selection (Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443 replay and interface); mixing is regenerated
  byte for byte.
• Word checkers: κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's independent checker and mutants, the relaxed all-column checker on the kernel and mixed words, Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's
  C++ cohort legality, five- and eleven-stage formal columns, record mutants, mixing mutants and Gram.
• Eleven-stage pipeline: geometry, template and hosting audits, the frozen HiGHS selection (--discover-check recomputes it),
  and assembly.

An independent validator found the R687 word valid, and phase-order mutants are rejected. An independent audit then checked the
exact-pivot word:

• its tape check re-derived c (|Δc| = 7.3·10⁻²⁵; profile and program digest equal);
• an independent solver recomputed the R687 maximum of 101.

Review points

• Read-then-climb (code/hier_word_rtc.py). Phase-B rules read helpers after the scatter. κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's compile already supports
  phase B. The emitted word passes κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's checker unchanged: 927 columns, B gates hold 100 helper rules.

• B-rule-aware mixing. Prefixes are compared per (register, version). On words without B rules it makes Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450's choices; it
  passes mix_check with 4/4 mutants rejected.

• Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing #471 manifest. Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing #471's MANIFEST lists a .gitattributes absent from its PR tree. Our wrapper accepts it absent (recorded in
  the receipt), hash-checks it when present, and checks every other file.

• Inherited. Complex supplier b ≈ 1.327·10⁻³ (+4.7%) on whole-residual hosting: shared-donor kernel and entrance gauges with paid residual wrappers (conditional on two lemmas) #484's figure carries its Lemmas A and G.

  • From Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450 only Lemma M (bit labels) is used; Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450's cross-copy hosting is not.
  • Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450's B-constancy step does not apply, because B holds 100 helper rules. The mixer's scalar identity is checked exactly on
    all 927 columns instead.
  • The usual compiler, dirty-helper and analytic interfaces are inherited.

  No unconditional theorem or Lean proof is claimed.

Credits:

• Dugongue: Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing #471 complex supplier and assembler, Conditional kappa = 0.001291542455423938: two-layer target factoring on R715 (+0.1461% over #470) #476/Conditional kappa = 0.001295508560071773: fourteen target factors on PR473 (+0.1226%) #478/Conditional kappa = 0.001296606691405309: response-kernel cancellation and nested-donor flags on R715 #483.
• rohanarun: Conditional kappa 0.001467289440740487: rank-sector factory on 112 ports #473, Conditional κ = 0.001291474845253222: retimed kernel and reordered R715 deliveries #470, Conditional κ = 0.001274415506145115: paid complex kernel + rank151 target mixing #460 verifier structure.
• chafreaky: Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472, Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458, Conditional κ = 0.001265931727546219: dual eleven-stage whole-residual hosting #451/Conditional κ = 0.001266536148630310: target mixing with whole-residual hosting #454, Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443–445.
• eumemic: Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493, Conditional κ = 1.128983115440183e-3: an E8 bit word with 740 helpers under #433's complex supplier #436 R740 word and reuse, κ = 1.136403735815528e-3: R740 shared-donor kernel with 57 primitive frame changes #440/κ = 1.138798055343319e-3: R740 block retiming after shared-donor kernel #441.
• DaysSky: Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 hosting.
• Dylan Crow: κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 bit word, producer and checkers.
• higherbeing (Under review (split-residual interface): cross-copy idle-axis hosting, one residual on two copies of one idle child; computed κ = 0.001330733966732373 not established #448), icekylinx (Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433), jacklightChen (κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425), Srikrishnabalaji (Conditional kappa = 0.001283479306671789: certified R715 assembly with sharp bit charge #465), Jacob Sussman, Zhihao Chen, and all
  source-pinned authors.
• Our Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463 (R715, carrier bound) and Complex supplier b ≈ 1.327·10⁻³ (+4.7%) on whole-residual hosting: shared-donor kernel and entrance gauges with paid residual wrappers (conditional on two lemmas) #484/Method + bounds (no frontier claim): kernel width rule, EXACT pivot maximum 112 and a seeded 112-pivot selection on the R715 word #485.

Copy anatomy and bound, read-then-climb, coset-split pairings, exact pivot certificates and the compositions are 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: Read-then-climb compilation, copy lower bounds and exact pivots.

Review and limits: Useful R687 supplier research; the transformations do not automatically transfer to the different R489 skeleton. Cited complex results and fresh compositions are distinguished.

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: 35057a6d70f23480f4e960327d41ecac31ad5cb0. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.