Conditional kappa 6.838479e-4: cascade-optimized frames on PR200's banked bit word

[rohanarun]

Draft, pending our full rerun.

Conditional κ = 683847872495777/10^18 = 0.000683847872495777:

• +0.0962% over Conditional kappa 6.831904550365e-4: coordinated frame optimization and completed entrance banks #207 (0.0006831904550365);
• +0.1152% over Pack PR200 diagonal bit residuals into completed banks (conditional κ = 0.000683061299399923) #205 (0.000683061299399923).

The bit supplier binds. PR193's complex saving 700918443859411/10^18 is unchanged.

Idea

#207 showed that PR200's operation frames are not optimal for the paid moment, and admitted 302 frame changes. This PR pushes the same lever much further.

• Chain model. Every physical chain of PR200's word is modeled exactly: splices, root frames and the full endpoint. Each operation frame is shared by its two chains. A frame is feasible when it:

  • contains the value span;
  • is G-nondegenerate;
  • is nested between its neighbours on both chains.

  Each chain costs Σ t·ln(72/t) over its rank increments.

• Cascading moves.

  • Raise a frame to a later neighbour (up to four slots ahead) or to the global feasible maximum, joining forward to repair nesting.
  • Lower it to an earlier neighbour or the global minimum, meeting backward.
  • Root frames, gauge frames, start frames and alias endpoints stay fixed.
  • Greedy plus short annealing reaches an optimum where single-op moves stall.
  • Result: 6,191 frames change, and internal chain cost falls 0.135%.

• Admission. Only frames change, and PR200's own modules re-check:

  • value spans, nondegeneracy, all chain nestings and the 1,760 alias containments;
  • the 34-sink terminal word on every source, target and dirty column, over F2 and over the integers;
  • prime witnesses for all 24,611 used frames.

  Unpacked coarse saving: 6.785484e-4 (PR200: 6.77774e-4).

• Banks and composition. Banks are as in Pack PR200 diagonal bit residuals into completed banks (conditional κ = 0.000683061299399923) #205: W = 56,402, deficit 5,808, maximum child 22. Composition uses three finite leaf levels and the balanced 47-constraint assembly with η = β = 10⁻²⁴. The next 10⁻¹⁸ grid point fails.

	Packed bit coarse	κ
#205	6.835282e-4	0.000683061299399923
#207		0.0006831904550365
this	6.843158e-4	0.000683847872495777

Reproduce

python -B research/cascade-frames-bit/verify.py --complex-root <PR193 @187e101> --bit-root <PR200 @a117544> --full

About 6 minutes on two cores. The run reproduces certificate.json byte for byte. A workflow is included.

Scope

This is a conditional finite supplier certificate with the same inherited all-size hypotheses as #193, #197, #200, #205 and #207. The frame search is heuristic, with no optimality claim. The branch sits on top of #205's commit.

Credits:

• PR200 (Chafik Boukhalfa): word and checkers.
• PR193: complex supplier.
• PR197 (Evan McKinney): banks.
• PR185/187: finite composition.
• Conditional kappa 6.831904550365e-4: coordinated frame optimization and completed entrance banks #207 (Dugongue): first frame re-optimization and its admission route.

Prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy