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research: R439 sub-10k six-way composition (9,974 ADDs), whole-A star duals, and UT4/dyadic floors - #3183

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research: R439 sub-10k six-way composition (9,974 ADDs), whole-A star duals, and UT4/dyadic floors#3183
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@dcharb78 dcharb78 commented Oct 17, 2026

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[bit-helper-contract / bit-new-register-model] Sub-10,000-ADD Six-Way r439 Composition (9,974 ADDs) and Exact Dual Witnesses across Star-Edge, UT_4(F_2), and Dyadic Borel Models

Summary of Contributions

This package directly addresses the Track 1 (bit-mechanisms, 20% combined portfolio weight in docs/research/current-priorities.md across bit-helper-contract and bit-new-register-model) priority question:

"Can one precisely specified helper-mixing word satisfy a replacement for the inherited H2/K interface?"

Through four coupled, deterministically verified pillars (verify.py -> RESULT.json), this package establishes a constructive sub-10,000-ADD breakthrough (9,974 ADDs) on the actual r439 (4-4-20) 661-column witness ladder while certifying three unconditional structural impossibility barriers across the star-edge, F_2[UT_4(F_2)], and dyadic block-Borel register models:

  1. Pillar 1 — First Sub-10,000-ADD Constructive Witness on the r439 (4-4-20) Ladder (9,974 ADDs, -518 vs. PR Bit construction: exact five-wire windows reduce composed R439 to 10,492 ADDs #1188, -74 vs. PR Construction: R439 one-slot multi-column interleaving reduces deferred scatter 88 -> 84 ADDs (no new kappa) #2440):
    We verify from pinned verification inputs that PR Finite bit construction: two nonpositive helper moves save 11 ADDs on R439 #1543 (research/r439-neutral-helper-chain-seed17, -11 ADDs) and PR Finite bit construction: five-helper cycle saves 12 R439 ADDs (no new kappa) #1709 (research/r439-five-cycle-seed9f01, -12 ADDs) commute with PR Research: twisted helper contract absorbs three R439 helper-star restorations into A (10,492 -> 10,489 on PR1188's word; no new kappa) #1868 (research/r439-twisted-star-absorption-seed02, -3 ADDs) and PR Research: deferred target basis — V R = I replaces H5 with the H2 target clause; scatter 527 -> 97 on every committed R439 target-basis word (PR1868's PR1188 word 10,489 -> 10,059) (no new kappa) #2115 (research/r439-deferred-target-basis-seed06, -433 ADDs) on PR Bit construction: exact five-wire windows reduce composed R439 to 10,492 ADDs #1188 (research/r439-gl5-shortest, 10,492 ADDs) to form 4-Way Composition A (10,045 / 10045 ADDs) and 4-Way Composition B (10,044 / 10044 ADDs), and that combining PR Finite bit construction: five-helper cycle saves 12 R439 ADDs (no new kappa) #1709 + PR Research: twisted helper contract absorbs three R439 helper-star restorations into A (10,492 -> 10,489 on PR1188's word; no new kappa) #1868 + PR Research: deferred target basis — V R = I replaces H5 with the H2 target clause; scatter 527 -> 97 on every committed R439 target-basis word (PR1868's PR1188 word 10,489 -> 10,059) (no new kappa) #2115 with PR Construction: R439 one-slot multi-column interleaving reduces deferred scatter 88 -> 84 ADDs (no new kappa) #2440 (research/r439-one-slot-multicolumn-interleave, 10-pair target-445 scatter reducing scatter_ADDs from 94 to 84) yields a 5-way composition at 10,034 ADDs (10034). We then identify 6 new pairwise-disjoint middle-column helper shears ((a_local, b_local) = (127, 240), (16, 239), (121, 128), (422, 371), (3, 235), (238, 29)) that wrap the middle-column section-D window [lo_m=472 .. hi_m=569] with 12 setup/restore gates and save 14 + 11 + 10 + 9 + 8 + 8 = 60 ADDs (w_b' = popcount(col[b] ^ col[a]) = w_b + w_a - 2k, net_saving = w_b - w_b' - 2, reducing total b-weight across the 6 pairs from 85 to 13). The resulting 6-way witness achieves 9,974 ADDs (9974, F = 15,539 / 15539, B = 2,830,702 / 2830702, scatter_ADDs = 84, paid_rank_mass = 5,129 / 5129) and passes 100% of the 661-column forward/reverse replays, the 91-bit local evaluator, and the 174-bit 9-stage evaluator.
  2. Pillar 2 — Whole-Section-A and Coupled (a, c) Star-Edge Absorption Theorem (693 + 675 Explicit RREF Dual Hyperplane Witnesses):
    Across all three canonical r439 star-factored words (PR Finite bit construction: phase-B block programs on the R439 helper-star D, 11,783 -> 11,500 ADDs (no new kappa) #1062 research/r439-bsection-stars-seed01 at 11,500 ADDs, PR Finite bit construction: compose R439 A/B/star D to 10,652 ADDs #1122 research/r439-abc-star-composition at 10,652 ADDs, and PR Bit construction: exact five-wire windows reduce composed R439 to 10,492 ADDs #1188 research/r439-gl5-shortest at 10,492 ADDs), each word has 236 B-section star edges (5 solvable edges (506, 374), (507, 355), (514, 367), (264, 536), (326, 530) and 231 obstructed edges). Removing the reader-horizon (r_0) barrier completely—allowing arbitrary monotone a-writes across the entire section A ([0, |A|]) and arbitrary coupled (a, c) writes across both section A and section C—yields 0 additional solvable star edges (5 / 236 solvable, 231 / 236 obstructed). verify.py constructs and verifies all 693 (3 * 231) explicit 661-bit RREF dual hyperplane witnesses and all 675 (3 * 225) explicit 1,322-bit coupled (a, c) RREF dual witnesses.
  3. Pillar 3 — Unconditional All-Length, All-Role n_b >= 6 Theorem in F_2[UT_4(F_2)]:
    Prior PR One-sided P112 fitting rigidity from the support census #1803 / PR Research: complete PR1122 nine-stage hosted invoice (no new kappa) #1806 (research/ut4-fifteen-call-five-b) and PR research: exclude 15-call five-b UT4 splits across dirty workspaces #2828 (research/bit-ut4-mixed-targets-barrier, "merged_for_archive_only", "partially_reviewed") report (author-checked) that n_b = 5 middle calls fail at fixed total lengths 13 ((4, 5, 4)) and 15 ((5, 5, 5)). By proving across all 256 (256 / 256) endpoint pairs (S_a, S_c) that the reachable central accumulator fiber in each gap is the exact 3-dimensional affine trace hyperplane {q in M_2(F_2) : Tr(q) = Tr(S_a S_c)}, we evaluate all 3,600 (3600) zero-sum 5-b middles across all unconstrained prefix-sum profiles (p_0, ..., p_4) in M_2(F_2)^5. Across all 3,240 (3240) non-degenerate middles (rank X_b = rank Y_b = 2), 0 (ut4_nondegenerate_compatible_pairs = 0) length-unconstrained (a, c) profile pairs exist at any finite word length L and any role count m >= 3. Across the 360 degenerate middles (rank X_b = 1 or rank Y_b = 1), only 16 middles (3,145,728 / 3145728 profile pairs across minimal realizable lengths L in {13, ..., 27}) survive the trace screen, and 100% (3,145,728 / 3,145,728) identically annihilate ACB[0, 0] = 0 or BAC[0, 0] = 0.
  4. Pillar 4 — Full 4x4 Block-Borel Schubert-Cell Invariance and 3-Call Parity Floor (mW = 4):
    Prior PR DONE: exact J3 gate/positive-envelope frontier (PR1520) #1507, PR Cycle 17 P104 correction with paired dirty-query baseline cancellation #2107, PR Scoped corollary (no κ): PR288 open item 3 (full-Borel bit register model) on unit targets is closed for 2-integral data — mW floor via PR288's FO-on-Λ/2Λ with P1's ν; free maps singular mod 2 allowed #2613 (research/bit-full-borel-2adic-floor), and PR Audit selected scatter parity across word and record checks #2977 (research/paired-diagonal-borel-gauge, "merged_for_archive_only", "partially_reviewed") report (author-checked) a 2-call trace obstruction, a restricted dyadic Gram/Schubert FAQ, a 2-adic valuation floor, and a paired-diagonal Borel gauge analysis. Lifting these to the full 4x4 block-Borel group (B_full, including arbitrary strictly lower-block cleanup shears L_{21} in M_2(F_2)), verify.py verifies exact Schubert-cell invariance across 40,593 (40593) free block-Borel checks (13,531 planes across O_0, O_1, O_2), verifies Delta c <= 1 across 54,124 (54124) rank-1 shear checks, and exhaustively tests all 20,224 (20224) 3-call (1 + 1 + 2) parity configurations, finding 0 survivors (dyadic_3call_parity_survivors = 0) achieving E_2 = I_2 and E_1 = 0, certifying the paid-rank floor mW = 4 (dyadic_paid_rank_floor = 4) at m = 2, W = 2.

Deterministically Verified Summary Metrics (RESULT.json)

Metric Key (RESULT.json["summary"]) Exact Value Formatted Meaning / Verification Scope
r439_sub10k_sixway_adds 9974 9,974 Total ADDs of the new 6-way sub-10k r439 (4-4-20) straight-line witness
r439_fiveway_adds 10034 10,034 Total ADDs of the 5-way composition (PR #1709 + PR #1868 + PR #2115 + PR #2440 on PR #1188)
r439_fourway_b_adds 10044 10,044 Total ADDs of 4-Way Composition B (PR #1709 + PR #1868 + PR #2115 on PR #1188)
r439_fourway_a_adds 10045 10,045 Total ADDs of 4-Way Composition A (PR #1543 + PR #1868 + PR #2115 on PR #1188)
r439_delta_vs_pr1188 -518 -518 Net ADD reduction of the 9,974-ADD witness relative to PR #1188 (10,492 ADDs)
r439_delta_vs_pr2440 -74 -74 Net ADD reduction of the 9,974-ADD witness relative to PR #2440 (10,048 ADDs)
r439_disjoint_shears_saving 60 60 Net ADD reduction from the 6 new pairwise-disjoint middle-column helper shears (14+11+10+9+8+8)
r439_scatter_adds 84 84 Remaining middle-column deferred scatter ADDs after target-445 10-pair scatter (94 -> 84)
r439_sub10k_F 15539 15,539 Exact F cost metric (ADDs + SUBs + 2*NEGs) of the 9,974-ADD witness
r439_sub10k_B 2830702 2,830,702 Exact bit-level operation metric B of the 9,974-ADD witness
r439_local_bits 91 91 Verified bit width in the local r439 evaluator (local_ok = True)
r439_nine_stage_bits 174 174 Verified bit width in the 9-stage r439 composition evaluator (nine_ok = True)
r439_paid_rank_mass 5129 5,129 Invariant paid-rank mass across the r439 star-factored witness ladder
star_edges_per_word 236 236 Total B-section star edges per word in PR #1062, PR #1122, and PR #1188
star_solvable_edges_per_word 5 5 Solvable star edges per word ((506,374), (507,355), (514,367), (264,536), (326,530))
star_obstructed_edges_per_word 231 231 Obstructed star edges per word under both whole-section-A and coupled (a, c) absorption
star_whole_a_dual_witnesses_total 693 693 Verified 661-bit RREF dual hyperplane witnesses (3 * 231) across PR #1062, PR #1122, PR #1188
star_coupled_ac_dual_witnesses_total 675 675 Verified 1,322-bit coupled (a, c) RREF dual witnesses (3 * 225) across A-reader edges
ut4_trace_hyperplane_matches 256 256 Exact matches (256 / 256) for the affine trace-hyperplane fiber theorem Tr(q) = Tr(S_a S_c)
ut4_zero_sum_five_b_middles 3600 3,600 Total zero-sum 5-b middle sequences (n_0, n_1, n_2, n_3, n_4) in F_2[UT_4(F_2)]
ut4_nondegenerate_middles 3240 3,240 Non-degenerate zero-sum 5-b middles (rank X_b = rank Y_b = 2)
ut4_nondegenerate_compatible_pairs 0 0 Compatible (a, c) profile pairs at any word length L across all 3,240 non-degenerate middles
ut4_degenerate_surviving_middles 16 16 Degenerate zero-sum 5-b middles (rank X_b = 1 or rank Y_b = 1) surviving the trace screen
ut4_degenerate_compatible_pairs 3145728 3,145,728 Degenerate compatible profile pairs (100% annihilating ACB[0,0] = 0 or BAC[0,0] = 0)
dyadic_borel_invariance_checks 40593 40,593 Verified full 4x4 block-Borel Schubert-cell invariance checks (13,531 planes * 3 generators)
dyadic_rank1_shear_checks 54124 54,124 Verified rank-1 upper-shear Schubert-cell step checks (13,531 planes * 4 rank-1 shears)
dyadic_3call_parity_configs 20224 20,224 Exhaustively tested 3-call (1 + 1 + 2) parity configurations in the dyadic block-Borel model
dyadic_3call_parity_survivors 0 0 Surviving 3-call (1 + 1 + 2) configurations achieving E_2 = I_2 and E_1 = 0
dyadic_paid_rank_floor 4 4 Certified paid-rank lower bound mW = 4 at m = 2, W = 2 under full block-Borel cleanup

PR Landscape Comparison and Provenance (SOURCES.json)

Single-Move Behavioral Negative Controls (controls = 6)

verify.py enforces 6 explicit behavioral negative controls (all recorded in RESULT.json["controls"]):

  1. control_1_overlapping_seventh_shear_corrupts_661_semantics: Appending an overlapping 7th middle-column shear (349 ^= 461, reusing a_wire = 349 from Shear 1) without updating coupled middle-column target supports corrupts 2 output registers in the 661-column forward replay.
  2. control_2_omitting_shear1_prep_or_undo_gate_breaks_targets: Omitting the 2 setup/restore gates of Shear 1 (349 ^= 462) while retaining the sheared middle-column reads breaks 62 output registers across the 16 shared middle-column target wires.
  3. control_3_dual_hyperplane_witness_validation_and_corrupted_dual_rejection: All 693 whole-section-A 661-bit RREF dual hyperplane witnesses and 675 coupled (a, c) 1,322-bit RREF dual witnesses pass exact orthogonality (u^T M = 0, u^T target = 1), all 693 target-flipped dual witnesses are rejected, and 0 of the 5 solvable edges per word are falsely obstructed.
  4. control_4_ut4_degenerate_rank1_middle_annihilates_acb00_or_bac00: Relaxing the non-degeneracy hypothesis (rank X_b = rank Y_b = 2) in F_2[UT_4(F_2)] admits 16 degenerate zero-sum 5-b middles (3,145,728 profile pairs), 100% (3,145,728 / 3,145,728) of which identically annihilate ACB[0, 0] = 0 or BAC[0, 0] = 0.
  5. control_5_ut4_omitting_any_call_from_direct_464_word_fails: Omitting any 1 of the 14 calls from the direct (4, 6, 4) formal UT_4(F_2) witness fails the 192-column formal target identity (14 / 14 rejected).
  6. control_6_dyadic_borel_single_block_and_illegal_upper_shear: A single rank-2 call (paid_rank = 2) has maximum F_2-rank 1 < 2 on E_2, whereas an illegal free upper shear H -> D jumps V_D directly to Schubert cell O_2 at paid_rank = 0.

Open next questions

  1. Joint Coupled Multi-Shear Basis Optimization Below 9,974 ADDs: Can a non-disjoint (triangular/cascaded) middle-column helper-shear sequence in [lo_m, hi_m] absorb overlapping B-wires whose residual weights (b_weight_after in {2, 3, 6}) are currently non-zero, pushing the r439 (4-4-20) witness below 9,900 ADDs?
  2. Deferred-Scatter Absorption Beyond the Remaining 84 Scatter ADDs: Can simultaneous basis retargeting across section D and section C absorb additional high-fanout middle-column targets beyond target 445 (research/r439-one-slot-multicolumn-interleave) without increasing B-section star reads?
  3. Six-b (n_b = 6) Minimum Total Length Below 14 Calls in F_2[UT_4(F_2)]: With n_b <= 5 unconditionally excluded at all finite word lengths L and all role counts m >= 3, does a 13-call word with n_b = 6 ((3, 6, 4) or (4, 6, 3)) exist in F_2[UT_4(F_2)], or is L = 14 ((4, 6, 4)) the unconditional all-length minimum across all n_b >= 6?

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Mister-Meeseeks merged commit 7db27e8 into CrocSwap:main Oct 17, 2026
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@dcharb78 @Mister-Meeseeks