Shrunk frames under paired lockstep covers: conditional κ = 3.48102e-4

[GamingPuzzled]

Draft: the focused target passes locally (Python 3.11) and the regenerated files are byte-identical. I'll mark this ready once fork CI passes.

Under #130's and #132's retained hypotheses,

κ = 174051/500000000 = 3.481020×10⁻⁴, +1.897% over #132 (3.416202e-4), +4.79% over #131 and +10.65% over #130. The complex side binds; the next κ grid point at denominator 10¹⁰ is rejected.

Idea. #130 sets every local complex frame to its full backward intersection, the largest admissible frame. #130's own general Clifford lemma prices any nested pair U ⊆ V of binary subspaces, including degenerate ones, as one child of width dim V − dim U. So a smaller frame is also admissible if three conditions hold:

• it contains the node's label and every predecessor frame;
• it lies inside the full backward intersection;
• it is nested in every successor frame.

One forward pass replaces a frame by that minimal span whenever the local one-child cost t·log(m/t), with m = 70, decreases. Because t·log(m/t) is subadditive, merging increments a and b into a+b is cheaper. This is the shrunk-frame pass from #125, moved to #130's cover. It shrinks 15,697 frames, and #130's unrestricted selection then keeps 4,553 source gauges.

complex local word	#130 frames	shrunk frames
unpaired (#130 cover)	3.147998e-4	3.202924e-4
paired lockstep (#132)	3.418543e-4	3.483450e-4 (69669/2·10⁸)
κ	3.416202e-4 (#132)	3.481020e-4

Checks. Run make shrunk-paired-cover-verify; it is added to the verify workflow matrix.

• producer.py is Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130's producer with lift_shrunk.py. It regenerates Deferred bit word and a 28,705-role complex producer under stopped-product accounting: conditional κ = 1.044939e-4 #117's pinned DAG, the C++ carrier matching, the physical word and the gauge selection, and requires agreement with the frozen complex-input.json.
• verify_shrunk.py is Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130's complete verifier with one assertion changed: a frame must be contained in its full backward intersection rather than equal to it. Everything else passes unchanged:
  • exact supports and divisor 21;
  • labels contained in frames;
  • dependency and carrier nesting;
  • 236,902 replayed physical frame transitions;
  • histogram recount;
  • full incident source gauges;
  • reverse readout chains.
• Control: the producer also runs Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130's unmodified verifier on the same data. It must reject the frames, and only with Not the full backward intersection, so every check before that assertion passes in Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130's original code too.
• certificate.py is Paired lockstep first-stage vertices on three-stage covers: conditional κ = 3.416202e-4 #132's paired certificate on the new inventory: exact moment, finite bridge, 47 strict constraints, 7 margins and next-grid rejection. It also requires the unpaired shrunk saving to exceed Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130's.

Scope. This is a finite conditional witness. #132's lockstep and sharing arguments and #130's cover, bit interface and inherited analytic/tape interfaces remain assumptions, as stated in those PRs. That smaller frames are admissible follows from #130's lemma, which never uses frame maximality. Nothing here is a runtime or implementation claim.

Composability. The pass is independent of #131's scratch reuse. It may compose with #131 plus #132; that combination has not been tested.

Credits: icekylinx (#130 covers, Clifford frames, producer and verifier); Avi Eisenberg/ikeboy (#132 lockstep, Anthropic Claude assistance); eumemic (#117 DAG, #131); an664 (#128); Swapnil Jain and Zhihao Chen (bit word); all inherited contributors. The shrunk-frame pass and this composition were prepared by Joel Pulikkan with Anthropic Claude assistance.

🤖 Generated with Claude Code

[Swapnil-jain]

Heads-up: the paired-lockstep step this PR builds on (#132) appears not to restore its targets. I've posted the details and an exact small counterexample there: #132 (comment)

In short, the merged child algebra is correct, but in lockstep each core's frame steps also transform the other core's live data on the shared stream. So the lockstep schedule is not equivalent to running the cores one after the other.

The shrunk-frame pass itself looks independent of this. Your own unpaired row (complex 3.202924e-4 on the #130 cover) would be unaffected. The paired κ = 3.48102e-4 would need the unpaired accounting.

[GamingPuzzled]

Thanks @Swapnil-jain — confirmed. I reproduced the cross-talk independently with an exact toy replay: sequential completed cores restore their targets, but one-pass and interleaved lockstep both fail. Since this PR's κ = 3.48102e-4 rests on #132's lockstep, it doesn't hold.

The shrunk-frame pass itself stands. Unpaired on #130's cover it gives complex saving 80073/2.5·10⁸ and κ = 640173/(2·10⁹) ≈ 3.2009e-4. That is below #131 and #137, so I'm closing this PR rather than re-titling it.