Conditional κ = 6.83847872497761e-4: five-level bootstrap on PR211 cascade frames

[sennemmi]

Five-level finite bootstrap on the PR211 cascade-frame supplier

Result

Starting from open PR211's exact conditional exponent 683847872495777/10^18, this follow-up adds two fixed stopped-leaf levels and obtains

kappa = 683847872497761/10^18
      = 0.000683847872497761.

The exact gain over PR211 is 1984/10^18. The fifth level adds the last 10^-18 grid step over the four-level version. The bit supplier still binds. The regenerated certificate stores all 47 exact positive assembly margins, and the independent audit rejects the adjacent final grid point.

Mechanism and inherited construction

The finite supplier and physical frames are PR211's frozen cascade-optimized PR200 bit word: 6,191 changed operation frames, the same terminal-modified word, 1,760 compensated aliases, completed banks, and PR193 complex supplier. We do not overwrite that layout with an older frame candidate. In an explicit comparison, PR211 already uses the same first-frame minimum joins at all 17 pairs found in our prior local search, while its second frames are smaller than that earlier candidate. The stronger cascade schedule is retained intact.

Let c be the newly priced packed bit saving and let a_0 be the fully paid ordinary saving from the admitted PR200 word. Five fixed recurrences use

a_j = (1-c)c + c a_(j-1),  j=1,2,3,4,5.

The exact rational verifier checks every recurrence, a_(j-1) < a_j < c < 1-a_j, the two-supplier paid moments, and the unchanged 47-constraint assembly with eta=beta=10^-24 and weakening 10^-30. The finite depth is five, independent of input size. For the fixed suppliers, the exact assembly ceiling at the unattained limit a=c has the same 10^-18 grid floor as the fifth level; since every finite a_j<c and the ceiling is increasing, no deeper fixed level can raise the reported grid value. The limiting value is only an upper-bound check, not an input supplier.

Reproduction and validation

python -B research/cascade-frames-bit/verify.py \
  --complex-root .dependencies/pr193 \
  --bit-root .dependencies/pr200 \
  --full

The existing workflow checks out the pinned PR193 and PR200 sources, reruns the PR193 complex verifier, re-admits all PR200 frames and terminal columns, rebuilds banks, recomputes the exact certificate and independent moment audit, and compares the regenerated certificate byte for byte. The root Verify workflow also covers the repository Makefile matrix and formal jobs.

The local exact-arithmetic replay used PR211's frozen admitted physical record and the byte-pinned PR193 assembly/certificate. It produced five levels, 47 positive exact assembly margins, an independent moment audit PASS, and the exact limiting-grid upper bound. At Draft creation, this follow-up's GitHub Actions runs have not yet completed; the submitted branch reruns the complete frame/source verification.

Conditions and limits

The conclusion remains conditional on the inherited all-width compiler, completed weighted/restored selector, routing, prime supply, precision/recovery, fixed analytic tape, and finite bridge hypotheses. PR211's cascade search is heuristic and does not prove a frame optimum. Finite verification does not prove the inherited all-size interfaces or the multiplication theorem unconditionally; this adds no Lean proof and makes no practical-speedup claim.

Attribution

The cascade frame layout and its verification belong to PR211 (Rohan Arun, with Anthropic Claude assistance). The bit word/checkers are from PR200, banks from PR197, complex supplier from PR193, and finite-composition work from PR185/187/210; their licenses and source notices are retained. The fifth-level arithmetic extension and independent audit were prepared with substantial OpenAI Codex assistance. No inherited construction is claimed as new or exclusive work.

Frontier snapshot at Draft submission

• PR211 is the highest validated public κ candidate at 683847872495777/10^18; its two package workflows and all 58 reported root/formal checks pass. This submission adds 1984/10^18 while retaining its admitted frame layout.
• PR210 reports 683190455036641/10^18, below PR211; several root jobs were still running at this snapshot.
• PR212 is a no-new-κ frame-layout ceiling for the older PR200/Pack PR200 diagonal bit residuals into completed banks (conditional κ = 0.000683061299399923) #205–Conditional kappa 6.831904550365e-4: coordinated frame optimization and completed entrance banks #207 word. Its reported ceiling is not an attained supplier and does not cover PR211's cascade layout.
• PR208 prices bank-family targets but explicitly says the required physical bank construction is not done; it is not a comparable certified κ.

The public list was refreshed immediately before opening this Draft.

[ghost]

Verification receipt + one portability note (Linux / Python 3.12, this session):

• research/cascade-frames-bit/verify.py certificate replay at 32d9db6 (the current head b7af6934 differs only by a CI workflow file): PASS, reproducing conditional kappa = 683847872497761/1000000000000000000 exactly. The predecessor Conditional kappa 6.838479e-4: cascade-optimized frames on PR200's banked bit word #211 replays likewise (…2495777), and the two-level gain 1984/10^18 checks out arithmetically.
• The --full mode currently cannot pass on my platform for a reason upstream of this PR: it re-runs PR193's research/source-assisted-v4/verify.py, which fails its byte-binding comparison off the author's platform (details and proposed fix in my comment on Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193). Once that receipt normalizes floats before hashing, --full should be portable.

Also verified in the same session: #200 full replay PASS and #212's ceiling.py --check PASS — consistent with this PR's claim sitting at ~98% of the banked-ledger frame ceiling. #219 pursues the lever that escapes that ceiling (child-ledger absorption), priced in #208.

[sennemmi]

Archiving this Draft as part of the research closeout. It adds a five-level bootstrap to the cascade-frame work in #211. The cascade-specific replay and targeted terminal/recycled-bit workflows passed at head b7af6934299024e04555929c13640cf2342613f4; the root research-artifact workflow was cancelled, and the existing verification comment records the inherited PR193 portability limit for --full. PR #211 remains open. The earlier verification receipt and this branch history are retained; no further maintenance is planned. See #261.