Bit word R685 (R687 lineage): L-copy inline + Bcw copy deletions, c = 1.323554·10⁻³ (+0.62% over #504's R687)

[DreamingOfClouds]

From a fresh git archive, verify.py --light and the default FULL run (with --discover-check, including #443's historical regressions) both pass. Stacked on #504 (head 35057a6).

Method + knowledge PR (no frontier claim). A new bit word, R685, from the R687 lineage of #504. It is priced at eleven stages with #442 whole-residual hosting and target mixing, with no
cross-family split, by the pipeline of #504 unchanged: c = 661777032472911750807/(5·10²³) ≈ 1.323554·10⁻³, +0.62% over #504's R687 (1.315433·10⁻³) and +2.99% over the R715 bit supplier (#463). With #442's complex
supplier κ is complex-bound as before (1.265849·10⁻³); the bit-only κ is 1.321805·10⁻³. Compositions with #471/#484 are not rerun here.

word	R	five-stage saving	s=11, greedy kernel	s=11, exact pivots
R687 (#504)	687	1.001418·10⁻³	1.303868·10⁻³	1.315433·10⁻³
R684: R687 + 3 word-level copy deletions	684	1.003262·10⁻³	1.303938·10⁻³	1.317462·10⁻³
inline only (28 L copies)	687	1.006976·10⁻³	1.309219·10⁻³	1.322341·10⁻³
R685	685	1.008233·10⁻³	1.309099·10⁻³	1.323554·10⁻³

What is new

1. D4(c) sums from their pairs (28 inlines). In R687 each D4(c) sum is built from two L's, so 28 L's have a third reader and need a third register chain. Building the sum from its four pairs
   gives every L two readers: 28 L copies fewer (raw R 811 → 782), contents and frames unchanged. 15 rules change kid lists (data/inline-kids.json, checked to be inlines of the old kids). This
   is the Bcw inline of our poset scout; the sections scout's exact per-block search over inline and re-pairing moves returns exactly these moves.
2. Optimal register chains. compile2's greedy chains are replaced by an optimum of an exact path-cover ILP (code/chain_model.py; pinned, --discover-check recomputes the objective). Equal-cost
   optima differ in c by several tenths of a percent, so the chains are part of the word.
3. Two copy deletions. code/copy_sweep.py replays the raw word exactly and finds the copies all of whose reads can take other registers at a legal frame (frames only climb, one frame per gate). Two
   D4(c) copies qualify and are deleted (reuse rebuilds the dirt compensation exactly): R = 685.
4. Exact kernel pivots on the new word. 233 candidates, upper bound 101 from the sum-rank lower bounds, attained by the pinned annealing selection (reproduced by code/select_pivots.py), so the maximum is exact
   without an exhaustive refutation; parity leaves 100.

Findings

• Raw −29 does not become R: the 28 L copies leave the dead pair slots that D5 births recycle, so reuse falls by as much. The gain is in climbs and the kernel (five-stage +0.56% for the inline, +0.125% for the deletions).
• R681 (all six D4(c) copies gone) is not reachable by these moves: the four hosts have no alternative source at any legal frame, and no single inline of a D4(c) sum raises the five-stage saving (73 moves).
• The clearing gap of the R687 raw word (floor 510 against 687) is structural: 196 of the 206 failed reuse births are L-copy births seeing one dead pair slot, whose clearing needs the complementary pair on a
  transverse chain or in sources whose next read frame does not contain the L frame. Exact twins are all chains (893 overlapping pairs, none mergeable).
  These counts come from exploratory scripts outside the package; verify.py reruns the copy sweep that selects the deletions.

Verification

python -B research/e8-r685-inline-deletions/verify.py /abs/out [--light]; CI workflow e8-r685-inline-deletions.

• --light (about 3 minutes, no C++): integrity (five pinned upstream packages, including Method + knowledge (no frontier claim): read-then-climb bit word R687 (c ≈ 1.315433·10⁻³), a copy lower bound, exact pivot certificates #504 at 35057a6); the word rebuilt from Method + knowledge (no frontier claim): read-then-climb bit word R687 (c ≈ 1.315433·10⁻³), a copy lower bound, exact pivot certificates #504's frozen hierarchy byte for byte; copy anatomy on the inlined DAG; κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's independent checker and
  mutants; the pivot bound, witness and annealing reproduction; the kernel and mixed words regenerated byte for byte; the relaxed all-column checker on both.
• FULL adds Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's parent verifier, the complex upstream, C++ cohort legality, five- and eleven-stage formal columns, record mutants, Gram, the mix check (4 mutants), eleven-stage geometry/template/audit,
  whole-residual hosting with the frozen selection (--discover-check recomputes it), the hosted audit and the assembly (47 strict inequalities), compared with EXPECTED.json.

Review points

• Chain override. One statement of κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's compile2 (the chain loop) is replaced at run time; each anchor must occur exactly once in the unchanged source. The pinned chains must cover the read events of every symbol exactly.
• Word-level deletion. A copy is deleted only if every read has an alternative set whose x-parts XOR to its content, legal at that read's own frame; Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's reuse step rebuilds D and C, and κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's checker verifies the F2 semantics on all 925 columns.
• Kernel selection. A witness at the exact maximum chosen among annealing seeds (other seeds move c by about ±0.05%); Method + knowledge (no frontier claim): read-then-climb bit word R687 (c ≈ 1.315433·10⁻³), a copy lower bound, exact pivot certificates #504's frozen selection for R687 is the same kind of object.
• Inherited. The usual compiler, dirty-helper and analytic interfaces; from Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448) #450 only Lemma M. No unconditional theorem or Lean proof is claimed.

Credits: the posets and sections scouts' inline and chain model, and the sweep, are by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; the R687 lineage (read-then-climb, exact pivots, mixing generalisation) is #504's;
#463 (R715, carrier bound); chafreaky (#443/#451/#454/#458/#472), eumemic (#436 reuse/retime, #493), DaysSky (#442), Dylan Crow (#370 bit word, producer and checkers), Dugongue (#471), and all source-pinned authors.

🤖 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: R685 bit word through inlining and copy deletion.

Review and limits: Supplier and method improvement in the R687 lineage, not a new complex-side or final-frontier claim. Inherited workflows must not be mistaken for replay of the new inline-deletion package.

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: 6590606e733768f6a7004b3f55ae35f988ba515f. Publication: 459f9dbdf41ea86f17fa21a6c71df00010a62c9b.