[lm1909]

Change and mathematical scope

Audit, no new κ. This is a scoped independent check of hexapode's #4626 (head e4413c43eeaacdd20be123fdd7d7add53427f028). Every checked claim reproduces. There are two findings.

• My frame walker matches the checkpoint. On the Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610 checkpoint word it reproduces all 3,290 shipped paid transitions in both orientations, with census {1:2201,2:412,3:506,4:135,5:36,8:8}.
• The chart move is legal.
  • Nothing touches L621 before event 12704, and L514 sits at ZERO.
  • A = span{30,429} equals frame797 ∩ frame4329, the largest chart nested in both next contacts.
  • The scalar opcode and coefficient sequence is unchanged.
• The paid delta is exact in both directions. Removed: 514:0→797 (r3) and 621:0→4329 (r4). Added: 514:0→A (r2), 514:A→797 (r1) and 621:A→4329 (r2). So ΔP = H1+2H2−H3−H4, giving census {1:2202,2:414,3:505,4:134,5:36,8:8}.
• Only helper621 changes rank, 9→7. Dimension count and exact Clifford rank agree: {4:28,7:8,8:47,9:325}.
• Whole profile and cost change match. The whole histograms equal Improve conditional kappa to 0.003021569 with helper621 and a funded compiler gap #4626's ARITHMETIC.json (64,422 / 10,928 / 752). The cost change equals 18(H1+2H2−H3−H4−H7+H9) to 1e-60; at a = .00302156 it is −0.00401670425, inside PROOF.md's interval.
• The arithmetic reproduces.
  • Raw crossing (Newton, 80-digit Decimal): 0.0030215713576 ∈ (0.0030215713, 0.0030215714).
  • α, γ and δ match all 30 displayed digits.
  • The least compiler exponent is j = 48,264, checked in exact integers, so H = 337,848,000,000,000.

Finding 1 (cosmetic). The PR body's "≈ −0.00401665923 at the tested exponent" is the value at a = .00302152, the predecessor κ. At .00302157 it is −0.0040167155. PROOF.md's figure is correct.

Finding 2 (where the gain comes from). About +2e-8 of the +4.9e-8 is helper621. The other +2.9e-8 is the compiler-gap retune from 1e-8 to 1e-9, which would apply equally to #4610's checkpoint.

κ rule	#4610 checkpoint	with helper621
#4610's rule (gap 1e-8)	0.00302152	0.00302154
#4626's rule (gap 1e-9)	0.003021549	0.003021569

After the retune the tested saving sits only 1.36e-9 below the crossing.

Not covered: the 632-column native replay, absolute phases/normalizers of the rank-7 target, the 54 strict + 4 functional outer checks, η/BIT, Lean, and whether the relative rank-7 helper target is admissible under the supplier contract. #4610's later head (0.00302176) is a different change; I did not check whether the two stack.

Task and overlap

• Card: independent-moment.
• Base: fbc9ea0249746acca4f504d4fda2b84e0e36cf0d, rechecked as current main before submission.
• No prior check of Improve conditional kappa to 0.003021569 with helper621 and a funded compiler gap #4626 exists. pinterbanget's Scoped check of #4610: all 632 endpoint frames equal PR4470's, profile arithmetic exact, moment improvement proved on (0,1) (no new κ) #4614–Independently check #4610's warp substitution: 34 of 72 G maps changed, all invertible and frame-transporting; 17,453 scalar opcodes unchanged, 422 labels; 3,298 paid calls (no new κ) #4616 checked Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610's checkpoint, not helper621. Independently recompute #4610's raw crossing in 50-digit Decimal from the pinned controller: 0.00302155511 inside its bracket, PR4535 root reproduced (no new κ) #4615's histogram recipe is reimplemented here.

Validation

python3.12 scripts/research_package.py --package research/pr4626-helper621-check (stdlib only, about 2 s) prints PASS_REGISTERED_RESEARCH_PACKAGE locally on Linux.

• Nothing is imported from Improve conditional kappa to 0.003021569 with helper621 and a funded compiler gap #4626. The inputs are:
  • the Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610 checkpoint word (blob 79dad65b, vendored and SHA256-pinned);
  • main's joint-helper-lattice/CONTROLLER.json;
  • main's clifford_exact.py, used as a rank evaluator only.
• Rejection controls, all of which fail as expected: a non-nested chart span{30,1}, an entrance-only move, the 1-dim chart span{30}, the old helper credit, Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610's κ rule, dropped idle charges, and j = 48,263.

Credit

hexapode / Pierre-Loic Doulcet (#4626), intqwq / Cao Yifan (#4610), Cinna Davis (#4470) and its credited predecessors, and pinterbanget (#4614–#4616). Prepared by lm1909 with Claude Code (Claude Opus 5.5) assistance; there has been no independent human review. Apache-2.0.

Open next questions

See the package README: supplier-contract admissibility of the relative rank-7 target, and whether helper621 stacks with #4610's later head.

🤖 Generated with Claude Code