Retimed complex word (#495) on cross-copy hosting with #493's bit word: conditional κ 1.452948511251494e-3 (stacked on #494)

[rosswaycaster]

Requested by Ross · project thread

Change and mathematical scope

Before: the strongest open comparable witness was PR #494, conditional κ = 358867191156973/2.5·10¹⁷ =
0.001435468764627892, bound by its complex supplier (PR #448's cross-copy hosting of PR #433's word, b =
718766148669433/5·10¹⁷).

After: conditional κ = 726474255625747/500000000000000000 = 0.001452948511251494, complex-bound, 1.22 % above PR #494, with the same bit
supplier (PR #493's R489 word, unchanged) and a new complex supplier.

The complex word is PR #495's retimed R560 word (Ali Romman), which combines PR #449's shared-donor kernel (DreamingOfClouds) with PR #458's paid two-sided residual adapters (chafreaky). It is rebuilt from its frozen FULL-WORD.json.gz by PR #495's own word-level scripts, vendored byte for byte (SOURCE.json). It is placed in PR #433's nine-stage template (m = 73, eight replicas), the template PR #494 prices. Every helper residual is hosted on idle data coordinates, so no banks are needed:

• the 79 entering (width-8) residuals per stage go whole onto pattern copies, with PR Add combined E8 witness at κ = 0.001307044366925301 #495's route chart and adapters, each alone on its family in its call, as in PR Add combined E8 witness at κ = 0.001307044366925301 #495;
• the 481 width-9 residuals per stage go on PR 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's cross-copy tape, as in PR Under review (split-residual interface): #493's R489 bit word with #448's cross-copy complex supplier; computed κ = 0.001435468764627892 not established #494.

Pattern copies and tape copies are different families. The complex lower bound rises from b = 1.437532·10⁻³ (PR #494) to b = 1455062642351477/10¹⁸ = 1.455063·10⁻³. The bit side is unchanged (c = 1.565442·10⁻³). PROOF.md gives the construction and the compatibility argument.

How: research/r489-retimed-cross-copy-complex/verify.py runs PR #494's own verify.py unchanged, then this package's steps, then the exact
assembly (code/assemble_complex.py, PR #494's assemble_489.py with the complex word directory as an argument): two
rational moment engines, six bootstrap levels, all 47 strict outer inequalities, next grid point and zero-η, zero-β,
unit-β controls rejected. The result is compared with EXPECTED.json and must exceed PR #494's κ.

Stacked on PR #494 (head 419907c71246c050f1c3fd2a504469b74bf7798e); only the commits after that head are new. Every assumption of
PR #494 stays an assumption. PR #448's cross-copy slot is disputed in the discussions on PR #451 and PR #485, and this package uses it only where PR #494 does. This package adds PR #495's assumptions for its word: PR #449's expanded scalar interface, PR #458's paid two-sided residual adapters, and PR #495's pending mutable-target, idle-hosting and Gaussian-adapter interfaces. It also adds one new composition point: PR #495's whole entering-residual hosting is used in the nine-stage m = 73 template, on families kept separate from PR #448's tape. The audit checks that separation in every call; PROOF.md gives the argument. A finite certificate is not a formal proof of the asymptotic theorem, no Lean
proof is claimed, and κ is an exponent saving, not a measured runtime speedup. Prepared with Anthropic Claude via
Claude Code for Ross (rosswaycaster); full credits are in research/r489-retimed-cross-copy-complex/NOTICE.md.

Credits: Ali Romman (PR #495: the retimed word and the vendored word scripts), chafreaky (PR #458: paid two-sided residual adapters), DreamingOfClouds (PR #449: shared-donor kernel; PR #484: entering-helper invoice form), higherbeing (PR #494: the composition this stacks on; PR #448: cross-copy hosting), eumemic (PR #493: the R489 bit word), DaysSky (PR #442: idle-data hosting and its verifier), icekylinx (PR #433: the nine-stage template and split centres), and every author in the source-pinned notices.

Validation

• Full clean rerun passed. A fresh clone of this branch ran verify.py in full mode. It ran PR Under review (split-residual interface): #493's R489 bit word with #448's cross-copy complex supplier; computed κ = 0.001435468764627892 not established #494 from scratch, including PR Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's verify.sh and PR Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's complete verifier, and reproduced every PR Under review (split-residual interface): #493's R489 bit word with #448's cross-copy complex supplier; computed κ = 0.001435468764627892 not established #494 expected value. It then ran this package's five steps and printed PASS_FULL_R489_BIT_RETIMED_CROSS_COPY_COMPLEX_COMPOSITION with κ = 726474255625747/500000000000000000 (Python 3.13.16). The tracked tree was unchanged.
• Component certification passed against a verified, complete PR Under review (split-residual interface): #493's R489 bit word with #448's cross-copy complex supplier; computed κ = 0.001435468764627892 not established #494 output, with the same result.
• Receipts, step logs and digests of every output file are in research/r489-retimed-cross-copy-complex-validation/.
• make verify, run target by target on this branch, will be recorded in the same directory.

🤖 Generated with Claude Code

https://claude.ai/code/session_01HGosS7FJ68HAdQbYMuqrxZ

[higherbeing]

Heads-up, since this stacks on #494: hipotures' single-stream register model on #448 (#448 (comment)) shows a split residual is a program of the priced word only with one extra helper per split, which costs more than the remainders it removes; I've acknowledged that and retitled #448/#480/#494 as under review. The cross-copy complex supplier used here inherits the same status. The retiming itself may well stand on a whole-residual complex word.

[JorgeCepeda]

Missing parenthesis: 718766148669433/5·10¹⁷

[Mister-Meeseeks]

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

Covered: Retimed complex supplier composed with cross-copy hosting. The claimed composition depends on the disputed cross-family split interface. A finite price or replay of that invoice does not supply the missing physical operator realization.

Keeping this PR open: The proposal prices a cross-copy/cross-family split. PR451 supplies a trace obstruction to treating that split as the present physical operator. Retain the proposed ledger, but supply a realizable operator/interface and its complete charge before accepting the claimed improvement.

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: 0f3e55dbf6b29a28bf462b4b3ec2a3ef06a28f5d.