Research: the MAS edge form of Theorem B2 (iii) is false (two exact counterexamples; no new kappa)

[navinor]

Summary

Scoped negative result (no new kappa, supplier, certificate or frontier claim):
the edge reading of MAS(W[F]) in Theorem B2 (iii) is false. Two exact
counterexamples are given (five and six lines, all full support). The vertex
reading — which is what the printed proof of
PR 2162 actually
yields, as shown by the CR2/B2 audit
PR 2293 — survives
every instance of the declared domain. This resolves open question 1 of that
audit, itself the follow-up of item 2 of the open independent-check request
discussion comment 18934590:
the printed gloss of MAS(W[F]) should be corrected to the vertex form, not
upgraded to the edge form. The same question was settled independently and in
parallel by PR 2334
(ROTl24); see the concurrent-work note below.

New package research/b2-mas-edge-form-seed3609249d/ (README, TASK.json,
SOURCES.json, VERIFY.json, verify.py, RECEIPT.json). certificates/selected-result.json,
both suppliers and upstream/ are unchanged.

The question and the answer

In the PR 2162 rank-3 decoder model (lines with rational p(t), q(t) in three
dimensions, p(t) . q(t) != 0; admissible pairing B 0/1 over F2 with unit
diagonal and B_ts = 1 only where p(t) . q(s) = 0; cost(B) the sum of the
three F2 block ranks B[R_j, A_j]), the edge form claims
cost(B) >= v - |H| - |F| + masE, where masE is the largest acyclic edge
subset of the wild digraph W[F]. It is false.

Counterexample 1 (smallest in the declared domain): v = 5, all full support

p(0) = (-1, -1,  3)    q(0) = (1, 1, 2)
p(1) = ( 1, -3,  1)    q(1) = (1, 2, 1)
p(2) = ( 3, -1, -1)    q(2) = (2, 1, 1)
p(3) = ( 1, -3,  1)    q(3) = (1, 2, 1)
p(4) = ( 3, -1, -1)    q(4) = (2, 1, 1)

All supports are full, so F is all five lines and H is empty. Lines 1 and 3
coincide, as do lines 2 and 4: the duplicated q directions are what defeats
the edge form. Exact computed data (two independent rank implementations agree):

Quantity	Value
wild pairs in W (both directions counted)	16
admissible free pairs	16
masV	2
masE	8
vertex right-hand side	2
edge right-hand side	8
exact minimum of cost(B) over all 2^16 admissible B	6

The right-hand sides are v - |H| - |F| + masV (vertex) and
v - |H| - |F| + masE (edge), stated here in prose because of the table. The
exact minimum 6 is attained by the admissible pairing with rows
10110, 01001, 10110, 10110, 01001 (bit s of row t is B_ts), and every
admissible B has cost(B) >= 6 by complete enumeration. The edge form demands
>= 8, so it fails with gap 2. Since all supports are full,
cost(B) = 3 rank_F2(B): the counterexample is a rank-2 unit-diagonal B
against masE = 8.

Counterexample 2: v = 6, all full support

Same direction-class family (twin_q_v6#138): masE = 11, edge right-hand side
11, exact minimum of cost(B) over all 2^22 admissible pairings 9. Gap 2.
Both falsifiers are re-minimised exhaustively and recomputed with the
independent rank implementation at every verifier run.

What survives

The machine-checked identity xbest = v - |H| - |F| + masV (the printed X-form
of Theorem B2) holds on all 139 instances; the vertex form has no falsifier.
Two positive statements are proved in the package README:

• P1 (complete zero-dot pairs are impossible at v >= 4 in Q^3): every
  triple of the q vectors is dependent, all lie in a plane, and the diagonal
  condition fails. Hence B = J is inadmissible and every admissible B at
  v >= 4 all-full has rank_F2(B) >= 2. Machine-checked over the domain.
• P2 (the edge form holds for all-full v <= 4): cost(B) = 3 rank_F2(B)
  gives >= 3 >= masE at v <= 3, and >= 6 >= masE at v = 4 by P1. The
  edge form is therefore true on the small side and fails first at v = 5,
  consistent with the search.
• P3 (masE can exceed |F| over Q^3): the hub/coplanar-q family
  realises masE = 5 > 4 = |F| at v = 4; the domain reaches
  masE - |F| = 5. So the edge form, where it holds, demands strictly more
  than the no-harmonic baseline v - |H|.

Machine evidence

Independent stdlib-Python re-implementation (exact integers, F2 bitsets, second
rank implementation, no predecessor import), smallest-falsifier-first over a
declared deterministic domain (seed 20261015, entries in {-2,-1,1,2},
v = 3..8; families: random all-full, random mixed-support, cross-product
dense-3 cores, hub/coplanar-q cores, direction-class near-dense wild cores).
Every instance at free <= 15 is minimised exhaustively, as are the first four
instances up to free = 20 and every falsifier whatever its size; the two
largest instances are screened heuristically first and every candidate B is
scored exactly.

PASS_B2_EDGE_SCREEN instances=139 exact=137 heuristic_only=2 max_free=22
BOUND iii_edge: FALSIFIER v=5 free=16 mincost=6 edge_rhs=8 masE=8 method=exhaustive_confirmed name=twin_q_v5#137
BOUND iii_edge: FALSIFIER v=6 free=22 mincost=9 edge_rhs=11 masE=11 method=exhaustive_confirmed name=twin_q_v6#138
BOUND iii_vertex: no falsifier in declared domain
IDENTITY xbest=vertex_rhs: ok on 139/139
CONTROLS ok {...}
SUMMARY tight_edge=6 tight_vertex=1 masE_gt_masV=63 masE_gt_absF=31 max_masE_minus_absF=5
RECEIPT_MATCH

Controls (all pass, each with the failure mode it must exhibit): analytic
fixture (minimum 2); the PR 2293 dense three-line fixture (masE = 3 > masV = 1,
exact minimum 3, tight for the edge form); the hub fixture (masE = 5 > |F| = 4,
exact minimum 6); harmonic-line classification; a strengthened edge bound +1
falsified on a fixture; a mutated configuration changing the free-pair set; the
two-rank-implementation cross-check; and P1 over the domain. Summary and
instance digest frozen in RECEIPT.json.

Reproduce

From the repository root, standard library only (CPython 3.12 registered):

python3 -B research/b2-mas-edge-form-seed3609249d/verify.py
python3 scripts/research_package.py --package research/b2-mas-edge-form-seed3609249d --check-only

The verifier ends RECEIPT_MATCH (about two to five minutes; the falsifier
confirmation re-runs the 2^16 and 2^22 enumerations). The registered replay
python3 scripts/research_package.py --package research/b2-mas-edge-form-seed3609249d
ends PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed on this host.
make is not installed here, so no make target is claimed; hosted checks are
queued at submission and no hosted pass is claimed.

Scope, limitations and remaining obligations

Finite counterexample claim only. Not checked: v > 8, entries outside
{-2..2}, T != 3, Lemma VCR2, Corollary J, Theorem T-prime, Lemma F, the
split min-max of PR 2162 section 3, and target (A) itself. Two instances at
free = 21, 22 were not exhaustively enumerated outside the falsifier
confirmation step. This does not refute Theorem B2's printed proof (the X-form
and the vertex corollary), only the edge gloss of MAS(W[F]); no kappa or
supplier implication is claimed, and no human review is claimed.

Concurrent work

PR 2334 (ROTl24,
head 0fabfe9316c14835e9156214979f7510e3e538bc, opened 2026-10-15 05:50 UTC)
independently falsifies the same edge reading with a nine-line tripartite
construction (cost(B) = 21 < masE = 27) and a parametric odd-k family with
gap 3(k-1)(k-2), i.e. unbounded gaps. Both packages answer open question 1 of
PR 2293; the
claims were posted independently (mine at 05:30 UTC,
comment 18941591,
before PR 2334 appeared) and neither depends on the other. This submission adds:
a strictly smaller counterexample (five lines), which P2 proves is
minimum-size among all-full-support configurations (no all-full counterexample
exists below v = 5); two exact counterexamples with complete minima over all
admissible pairings from a second, independently implemented engine; a
139-instance screen of the vertex form (unviolated) including mixed supports;
and P1 (complete zero-dot pairs impossible at v >= 4 in Q^3) plus P3
(masE > |F| realizable over Q^3). PR 2334's parametric family is the
stronger headline for gap size; the results are compatible and each package
keeps its own credit.

Credit and provenance

Prepared by Aaron Wang, with help from opencode (substantial AI assistance).
The model, the statements of Lemma CR2 and Theorem B2 and the audited
configuration families are DreamingOfClouds's work in
PR 2162,
PR 2111 and
PR 2125. The MAS
gap, the vertex-form identity and the 58-instance audit are Aaron Wang's in
PR 2293; this
package answers that audit's open question 1. Context: request items 1 and 4
were closed by PR 2193
and PR 2176, item 3
by PR 2272. Pins:
base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; audited source
research/char2-jets/README.md SHA-256
60beed587a355b6b04e61053fc123fb8ebe5d0d75d9dbd065f24138c55a2e437; predecessor
head 6bc48613ed1c71e5d2e88994235e9e8629129b9c. New work is Apache-2.0.