[hexapode]

Summary

This package (research/p104-commuting-contact/) resolves the open reordered-contact and fixed-contact-ceiling questions of PR #5580 (research/p104-event-contact-step) and PR #5590 (research/p104-fixed-contact-ceiling) on the P104 word and certifies conditional κ = 0.00308534 (tested at 0.0030853405), up from 0.00308189 in PR #5580 and 0.00308181 in PR #5569.

• Headline word (WORD.json.gz, Stage A + Stage B + Stage C): Exact raw crossing lies in (0.00308534541177, 0.00308534541178), a raw gain of +3.45044310e-6 to +3.45044312e-6 over PR Conditional κ = 0.00308189 (+7.6453e-8 raw over #5569) #5580's (0.00308189496866, 0.00308189496867). Complete paid calls drop by 108 (66,852 -> 66,744), local MOVE instructions drop by 6 per call and scalar ADD operations drop by 1 per call (25,140 -> 25,133 events), while all 208 main endpoints, 12,814 productive fields, rank-mass surplus 760, and all eight owned lifetimes remain unchanged.
• Fixed-endpoint supporting word (WORD-FIXED-ENDPOINTS.json.gz, Stage A only): Retains all 735 initial and final frames and all 527 helper targets identical to PR Conditional κ = 0.00308189 (+7.6453e-8 raw over #5569) #5580 (local['helpers'] == parent_local['helpers']), reducing complete paid calls by 72 (66,852 -> 66,780) with local paid delta -4 H1 - 4 H4 + 4 H5 and raising the raw crossing to (0.00308415555080, 0.00308415555081) (+2.26058214e-6 over PR Conditional κ = 0.00308189 (+7.6453e-8 raw over #5569) #5580, certifying κ = 0.00308415).

Proof · Complete receipt · Source pins · Construction · Validation

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Mathematical Mechanism

PR #5561, PR #5569, PR #5580, and PR #5590 kept the linear execution order and instruction set of the 21,686 scalar operations from PR #5551 fixed. Reordering commuting ADD writes (Stages A and B) and commuting a triangular uncompute addition across a nested subspace chain (Stage C) yields three composable improvements:

1. Stage A (WORD-FIXED-ENDPOINTS.json.gz, fixed helper endpoints, -4 H1 - 4 H4 + 4 H5 locally):

   • Between 13528 and 13634, main port 198 receives only ADD writes (13529: ADD 198, 710, 2 and 13625, 13628, 13631, 13634: ADD 198, 648/649/650/652, ±1/2), and helpers 648, 649, 650, 652 are in their initial read-only epoch at chart 3 (0D).
   • Commuting 13625, 13628, 13631, 13634 before 13528: ['MOVE', 198, 3, 597, False] to chart 3 (0D) vacates chart 597 (1D) on all four helpers 648, 649, 650, 652, collapsing each itinerary 3 (0D) -> 597 (1D) -> 120 (5D) (H1 + H4) into a single direct transport 3 (0D) -> 120 (5D) (H5).

2. Stage B (Domino cascade on 113, 531, 197, 319, 507, -4 H1 - H3 + 2 H4 paid and -H8 + H9 target locally):

   • Commuting 13806: ['ADD', 113, 531, '1'] before 13524: ['MOVE', 113, 3, 597, False] to chart 3 (0D) allows all three initial read contacts of helper 531 (13806 with 113, 13843 with 197, and 14404 with 507) to execute at chart 3 (0D) with initial[531] = 3.
   • Because 197 stays at 3 (0D) after 13843 instead of climbing to 597 ([82]), its next contact 13846: ['ADD', 197, 319, '1'] demotes from 3-plane 617 ([30, 44, 76]) to 2-plane 355 ([30, 44]), vacating 617 on both 197 (3 -> 355 -> 788, H2 + H3 instead of H1 + 2 H2) and 319 (355 -> 728, H2 instead of 2 H1), while 507 climbs directly 3 -> 728 (H4 instead of H1 + H3) and 531 climbs 3 -> 741 (H4 instead of H3, with helper credit H8 -> H9).

3. Stage C (WORD.json.gz, triangular uncompute commutation on 341, 345, 712, +2 H1 - H7 paid and +H4 - H9 target locally):

   • Helper 341 completes all lower-dimensional work at event 11161 in 4-plane 240 ([13, 19, 70, 416]), where row(341)|_11161 = - 341 + row(345)|_11066, and is never read again.
   • In PR Conditional κ = 0.00308189 (+7.6453e-8 raw over #5569) #5580, 341 jumped 240 (4D) -> 127 (9D) (16652) only to execute 16654: 341 += -345, 16655: 341 *= -1, and 16915: 341 += 4 * 712, because 345 added -4 * row(712) at 13569 (for readers 170 at 13809 and 347 at 14444 in 3-plane 739) before uncomputing 16920: 345 += 4 * 712 after 16654.
   • Because span(391) (2D, [30, 491]) ⊂ span(739) (3D, [30, 70, 429]) ⊂ span(240) (4D, [13, 19, 70, 416]) ⊂ span(127) (9D), commuting 16920: 345 += 4 * 712 before 16654: 341 += -345 to chart 739 (3D) immediately after 14444 cancels 16915: 341 += 4 * 712 identically by transitivity of ADD (-1 scalar ADD), demotes 16654 and 16655 to chart 240 (4D), deletes 16652: MOVE 341: 240 -> 127 (-H5), and retires 341 at final[341] = 240 (4D instead of 9D, target +H4 - H9), while 345 climbs 739 -> 240 -> 127 (H1 + H5 - H6) and 712 climbs 391 -> 739 -> 127 (H1 + H6 - H7).

Word	Paid calls	Local paid delta vs #5580	Local target delta	Raw crossing interval	Claimed κ
PR #5580 (p104-event-contact-step)	66,852	baseline	baseline	(0.00308189496866, 0.00308189496867)	0.00308189
Stage A (WORD-FIXED-ENDPOINTS.json.gz)	66,780	-4 H1 - 4 H4 + 4 H5	0 (all 735 endpoints fixed)	(0.00308415555080, 0.00308415555081)	0.00308415
Stage A+B (intermediate cascade)	66,726	-8 H1 - H3 - 2 H4 + 4 H5	-H8 + H9 (initial[531]: 597 -> 3)	(0.00308521055201, 0.00308521055202)	0.00308520
Stage A+B+C (WORD.json.gz, headline)	66,744	-6 H1 - H3 - 2 H4 + 4 H5 - H7	+H4 - H8 (final[341]: 127 -> 240)	(0.00308534541177, 0.00308534541178)	0.00308534

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Verification

Standard-library Python 3.10+ suffices; no external solver is required:

python3 -B scripts/research_package.py --package research/p104-commuting-contact

Expected markers are PASS_P104_COMMUTING_CONTACT_KAPPA_00308534 in both normal and -O modes and PASS_REGISTERED_RESEARCH_PACKAGE.

Credits

Credit to @aliromman (PR #5580, PR #5544, PR #5545), @GamingPuzzled (PR #5569), @DreamingOfClouds (PR #5561, PR #5568, PR #5590), @thu-nmrc (PR #5551), and the retained P104 contributors. Apache-2.0.