Conditional κ = 7.09424406659975e-4: five-stage completed banks with #210’s 493-source helper

[rohanarun]

Conditional κ = 28376976266399/40000000000000000 = 0.000709424406659975: +0.119% over #237 (0.000708582410781108) and +0.813% over #234 (0.000703701743496697). A full local run passed: --full runs #234's own verifier, then the steps below.

Idea

#234 and #237 build the five-stage cover from #210's older bit helper (df95878, 471 borrowed sources). #210's head 1331149 is stronger. It adds:

• 22 more source entrances;
• 82 equal-response target aggregation groups;
• 7 rank-compression groups;
• 4 signed echelon target transforms;
• κ = 6.94787513005285e-4: plateau retiming on PR210's 471-source word #230's frames.

That leaves 16,621 physical registers instead of 16,643. This PR runs that helper through #234's five-stage pipeline and then applies #237's width-120 completed banks.

• Helper. κ = 7.10069340338651e-4: five-stage banks and F₂ movement fusion #210's own code runs unchanged:
  • all 20,141 F2 columns;
  • the literal majorant (24-bit digits);
  • both defining-integer signs;
  • all 17 corruption controls.
• Telescope. Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234's frame-tag telescope is adapted to κ = 7.10069340338651e-4: five-stage banks and F₂ movement fusion #210's newer replay. It records 2,569,922 tagged events of 22 kinds. An independent literal projection and its inverse reproduce the F2 endpoint and the event digest.
• Five-stage lowering, exact geometry, primes and finite bill. These are Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234's code with only the helper constants rebound to fresh values.
  • Profile: m = 120, W = 23,661, 494,909 calls, deficit 4,400.
  • All 26,464 used bases have prime-witnessed determinants.
  • The payload-prefix lift needs 101 bits.
• Banks. The same 231 charts. 176,915 width-120 banks with a proper 60-colouring; W = 261,395 (12 copies), deficit 52,800.
• Arithmetic. Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234's interval engine gives coarse 709928047185833/10^18. Three leaf levels and Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234's outer-47 assembly follow; the next 10^-18 grid point is rejected.

The bit side binds; the complex side (7.4745e-4) is unchanged.

	helper	bit coarse	κ
#234	471-source, paid exteriors	7.041973e-4	0.000703701743496697
#237	471-source, banks	7.090849e-4	0.000708582410781108
this	493-source + target transforms, banks	7.099280e-4	0.000709424406659975

Reproduce

git worktree add --detach ../pr210 13311491eb74a3a9a443ba4e318c68c263f064f0
git worktree add --detach ../pr234 af3fe331ca60c936229e060681ffccbc1c208678
python3 -m pip install sympy==1.14.0
python3 -B research/five-stage-banks-493/verify.py --pr210-root ../pr210 --pr234-root ../pr234 --full

It takes about 10 minutes. A workflow is included.

Scope

This is a conditional result, with the same hypotheses as #234 and #237. #234's helper-specific checks are reused with recomputed constants. The branch sits on top of #237.

Credits:

• eumemic (κ = 7.10069340338651e-4: five-stage banks and F₂ movement fusion #210), with Dugongue (Conditional kappa 6.838643131376e-4: composed cascade frames, global cuts and four finite bootstraps #216) and sennemmi (κ = 6.94787513005285e-4: plateau retiming on PR210's 471-source word #230).
• hcg890 (Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234) and Jacob Sussman.
• Evan McKinney (Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197).
• Chafik Boukhalfa (A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200).
• Avi Eisenberg (Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193).

Prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy

[Mister-Meeseeks]

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

Contribution: Five-stage completed banks with source527-era helper changes.

Review and limits: Prior exact-head all changed-word checks, 17 controls, charts, bank geometry, finite invoice and 47 assembly constraints passed; pinned parents reused.

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: 1068237fdb977f45e68107d118d0eaf0c7a4b516. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.