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Mister-Meeseeks merged 2 commits into from
Oct 14, 2026
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Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224

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Research: nine-term complex E8 decoder normal form and residual class…
joaqcoo Oct 13, 2026
00e6ed0
Record concurrent masked-decoder PRs 1212/1213 in overlap notes
joaqcoo Oct 13, 2026
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270 changes: 270 additions & 0 deletions research/nine-term-decoder-normal-form-seed42/CERTIFICATE.json
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{
"U1": 56,
"U2": 84,
"code_weights": {
"56": 255,
"64": 255,
"120": 1
},
"deficient_intersection_size_rank": {
"0,0": 255,
"28,7": 12960,
"32,7": 22680,
"36,6": 3360
},
"fails": [],
"frames": {
"A2 weights + E6 root basis": {
"P120_cost": 78,
"P120_term_costs": [
9,
7,
8,
9,
9,
9,
9,
9,
9
],
"masked_terms": 3,
"row_support_sizes": [
30,
28,
29,
57,
106,
106,
106,
106,
106
],
"sparsity": {
"1": 6,
"2": 13,
"6": 65,
"7": 36
}
},
"A3 weights + D5 root basis": {
"P120_cost": 81,
"P120_term_costs": [
9,
9,
9,
9,
9,
9,
9,
9,
9
],
"masked_terms": 4,
"row_support_sizes": [
39,
19,
39,
39,
78,
78,
104,
104,
104
],
"sparsity": {
"1": 5,
"2": 14,
"3": 4,
"4": 2,
"5": 19,
"6": 76
}
},
"A7 weights + A1 root": {
"P120_cost": 81,
"P120_term_costs": [
9,
9,
9,
9,
9,
9,
9,
9,
9
],
"masked_terms": 8,
"row_support_sizes": [
42,
22,
42,
42,
40,
40,
40,
40,
57
],
"sparsity": {
"1": 1,
"2": 28,
"3": 56,
"4": 35
}
},
"A8 weights": {
"P120_cost": 81,
"P120_term_costs": [
9,
9,
9,
9,
9,
9,
9,
9,
9
],
"masked_terms": 9,
"row_support_sizes": [
36,
36,
36,
36,
36,
36,
36,
36,
36
],
"sparsity": {
"2": 36,
"3": 84
}
},
"coordinate frame + root": {
"P120_cost": 76,
"P120_term_costs": [
9,
9,
9,
9,
9,
9,
9,
9,
4
],
"masked_terms": 5,
"row_support_sizes": [
53,
51,
51,
51,
57,
57,
57,
57,
8
],
"sparsity": {
"1": 9,
"2": 4,
"3": 3,
"4": 104
}
}
},
"log": [
"PASS A1 120 lines",
"PASS A2 label pairing F2 rank 9",
"PASS A3 pairing rows distinct",
"PASS A4 code weights {56:255, 64:255, 120:1} | {64: 255, 56: 255, 120: 1}",
"PASS B1 every nonzero codeword support spans R^8 | {8: 511}",
"PASS B2 U1 = 56 (largest set of label span <= 8 has 64 lines)",
"PASS B3 pairwise intersections: 130305 pairs, rank census {8:91050, 7:35640, 6:3360, 0:255} | {6: 3360, 8: 91050, 7: 35640, 0: 255}",
"PASS B4 deficient intersections have (size, rank) in {(0,0),(28,7),(32,7),(36,6)} | {(36, 6): 3360, (32, 7): 22680, (28, 7): 12960, (0, 0): 255}",
"PASS B5 U2 = 84 (largest set of label span <= 7 has 36 lines)",
"PASS C1 P120: random one-sided 9-term decoder (frame+root, random t_i) is exact | 14400",
"PASS C1 P120 A'B'^T = I/2 (doubled coordinates)",
"PASS C1 P120 R^T N1 = 0 and R^T N2 = 0",
"PASS C1 P120 N1 B'^T = 0, A' N2^T = 0, N1 N2^T = 0",
"PASS C1 P120 dim ker B' = dim ker A' = m-8 = 1",
"PASS C1 P120 ker A' x ker B' dot pairing nondegenerate",
"PASS C1 P120 Corollary B: at most one factor leaves the root code | N1 zero=False, N2 zero=True",
"PASS C1 P120: row factor leaves the root code, column factor does not",
"PASS C1 P112: random one-sided 9-term decoder (frame+root, random t_i) is exact | 12544",
"PASS C1 P112 A'B'^T = I/2 (doubled coordinates)",
"PASS C1 P112 R^T N1 = 0 and R^T N2 = 0",
"PASS C1 P112 N1 B'^T = 0, A' N2^T = 0, N1 N2^T = 0",
"PASS C1 P112 dim ker B' = dim ker A' = m-8 = 1",
"PASS C1 P112 ker A' x ker B' dot pairing nondegenerate",
"PASS C1 P112 Corollary B: at most one factor leaves the root code | N1 zero=False, N2 zero=True",
"PASS C1 P112: row factor leaves the root code, column factor does not",
"PASS C2 PR1180-shaped outside-code nine-term identity exact on all 12544 P112 pairs",
"PASS C2 A'B'^T = I/2 (doubled coordinates)",
"PASS C2 R^T N1 = 0 and R^T N2 = 0",
"PASS C2 N1 B'^T = 0, A' N2^T = 0, N1 N2^T = 0",
"PASS C2 dim ker B' = dim ker A' = m-8 = 1",
"PASS C2 ker A' x ker B' dot pairing nondegenerate",
"PASS C2 Corollary B: at most one factor leaves the root code | N1 zero=False, N2 zero=True",
"PASS C2 its outside-code part sits in D only",
"PASS C3 z.y != 0 at m = 9 | 5/4",
"PASS C3 one-sided junk n z^T alone keeps exactness (sampled pairs)",
"PASS C3 control: two-sided junk (n z^T, m y^T) at m = 9 is inexact | 5700 wrong entries",
"PASS C4 m = 10: found z in ker B', y in ker A' with z.y = 0, y != 0",
"PASS C4 m = 10 decoder with junk on both factors is exact on all 14400 pairs",
"PASS C4 A'B'^T = I/2 (doubled coordinates)",
"PASS C4 R^T N1 = 0 and R^T N2 = 0",
"PASS C4 N1 B'^T = 0, A' N2^T = 0, N1 N2^T = 0",
"PASS C4 dim ker B' = dim ker A' = m-8 = 2",
"PASS C4 ker A' x ker B' dot pairing nondegenerate",
"PASS C4 both factors leave the root code (Corollary B is specific to m = 9)",
"PASS D1 dependency support of a parallel pair is {0, 8} | [0, 8]",
"PASS D2 merged eight-term decoder is exact on all 14400 pairs",
"PASS D3 merging never increases cost (40 random one-sided decoders) | min saving 9",
"PASS E1 A8 weights: v_kk' != 0 for all 36 masked pairs",
"PASS E2 A8 weights: a line avoiding two masked terms is orthogonal to v_kk'",
"PASS E3 A8 weights: every option avoids exactly one ratio class of masked terms",
"PASS E1 A7 weights + A1 root: v_kk' != 0 for all 28 masked pairs",
"PASS E2 A7 weights + A1 root: a line avoiding two masked terms is orthogonal to v_kk'",
"PASS E3 A7 weights + A1 root: every option avoids exactly one ratio class of masked terms",
"PASS E1 A2 weights + E6 root basis: v_kk' != 0 for all 3 masked pairs",
"PASS E2 A2 weights + E6 root basis: a line avoiding two masked terms is orthogonal to v_kk'",
"PASS E3 A2 weights + E6 root basis: every option avoids exactly one ratio class of masked terms",
"PASS E1 A3 weights + D5 root basis: v_kk' != 0 for all 6 masked pairs",
"PASS E2 A3 weights + D5 root basis: a line avoiding two masked terms is orthogonal to v_kk'",
"PASS E3 A3 weights + D5 root basis: every option avoids exactly one ratio class of masked terms",
"PASS E1 coordinate frame + root: v_kk' != 0 for all 10 masked pairs",
"PASS E2 coordinate frame + root: a line avoiding two masked terms is orthogonal to v_kk'",
"PASS E3 coordinate frame + root: every option avoids exactly one ratio class of masked terms",
"PASS F1 A8 weights: sparsest-representation 9-term decoder exact on all 14400 pairs | 14400",
"PASS F2 A8 weights: P120 cost formula rho_k = dim V_Xk (all column supports span 9)",
"PASS F1 A7 weights + A1 root: sparsest-representation 9-term decoder exact on all 14400 pairs | 14400",
"PASS F2 A7 weights + A1 root: P120 cost formula rho_k = dim V_Xk (all column supports span 9)",
"PASS F1 A2 weights + E6 root basis: sparsest-representation 9-term decoder exact on all 14400 pairs | 14400",
"PASS F2 A2 weights + E6 root basis: P120 cost formula rho_k = dim V_Xk (all column supports span 9)",
"PASS F1 A3 weights + D5 root basis: sparsest-representation 9-term decoder exact on all 14400 pairs | 14400",
"PASS F2 A3 weights + D5 root basis: P120 cost formula rho_k = dim V_Xk (all column supports span 9)",
"PASS F1 coordinate frame + root: sparsest-representation 9-term decoder exact on all 14400 pairs | 14400",
"PASS F2 coordinate frame + root: P120 cost formula rho_k = dim V_Xk (all column supports span 9)",
"PASS F3 A8 frame: every line is 2- or 3-sparse (36 + 84)",
"PASS F4 A8 frame: every row support has 36 = 120 - U2 lines, yet label span 9; cost 81",
"PASS F5 no frame of section F beats the eight-term P120 floor 72 | {'A8 weights': 81, 'A7 weights + A1 root': 81, 'A2 weights + E6 root basis': 78, 'A3 weights + D5 root basis': 81, 'coordinate frame + root': 76}",
"PASS G1 control: perturbing one coefficient breaks exactness",
"PASS G2 control: deleting one support entry breaks exactness",
"PASS G3 control: a rank-deficient frame is rejected",
"PASS G4 control: a flipped label entry changes the pairing rank away from 9 | 10",
"PASS G5 control: label_dim of a single line is 1, of C(root) is 9"
],
"pairwise_intersection_rank_census": {
"0": 255,
"6": 3360,
"7": 35640,
"8": 91050
},
"section_C": {
"C1_P112_random_cost": 73,
"C1_P120_random_cost": 81
}
}
26 changes: 26 additions & 0 deletions research/nine-term-decoder-normal-form-seed42/CLAIM.md
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# Local sprint claim (no coordination post was made)

- Recorded: 2026-10-13T19:40Z. Track 40-49, structural decoder research. Assignment seed **42**,
drawn once with `python -c "import secrets; print(secrets.randbelow(100))"` (OS randomness).
- Base commit: `1b59b624d21924f8e3537cdc1ac52d3be6eda2fe` (main; selected PR1094, kappa 0.001672660197076918).
- Initial object: whole-decoder copy ratio of exact 8-term complex decoders for punctures p >= 25
(beyond PR968's p <= 24; PR1020 settled only the single-term threshold).
- Candidates considered: (1) the p >= 25 ratio question; (2) n >= 9-term decoders (PR968 scope, PR717
mixed-beta); (3) E-graph-only masked decoders (needs PR717 D1 spec).
- Pivot (one, evidence-driven): any complex-only gain beyond P104 is already capped by the PR589 bit
supplier (PR968 section G), so the p >= 25 ratio question has little practical value. While
pricing it, the n = 9 normal form (Corollaries B/C) emerged, and n >= 9 became the object.
- Method: exact linear algebra (written proofs) plus exact F2/rational census and constructions;
bounded annealing over frames (seeds 0-2, 3000 moves).

## Result

- Delivered as `research/nine-term-decoder-normal-form-seed42/` (research note / scoped structural
result, no kappa claim).
- Theorem A (normal form for any m), Corollary B (m = 9: at most one factor leaves the root code),
Corollary C (one-sided 9-term = overcomplete representation), Lemma D (|S| <= 2 reduces to 8 terms),
Propositions E-G (rigid unmasked terms, ratio classes, exact P120 cost 81 - sum delta_k and its
necessary conditions; 39,255 non-spanning support pairs).
- No 9-term construction or search run beat 72 (P120), 64/112 or 7/13; best ratio 0.6036.
- Overlap at submission: PR1180 (both factors root-linear, any m), PR1191 (masked identities),
PR1197 (P112 ten-term restricted cases): complementary, credited.
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