Independent exact-F2 certificate: anti-Fano 12-line configuration has minimum cost 12 (no kappa claim)

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Independent exact-F2 certificate: the anti-Fano 12-line configuration has minimum cost 12 (no kappa claim)

Contributor: Aaron Wang, with help from opencode (substantial AI assistance; no independent human review claimed).

Summary

Package research/antifano-tied-min-cost12-seed22ae1f95/. This is item 1 of the
independent-check request
(DreamingOfClouds): a certificate from a different method that the anti-Fano 12-line
configuration tied_min has minimum admissible pairing cost exactly 12.

The certificate is a third method structurally: pure standard-library bitmask linear
algebra over F2, with explicit forced-zero triangular minor witnesses and complete
branch-and-bound enumerations. It uses no SAT, no CP-SAT, no exchange argument, no
characteristic-2 jet, and imports nothing from the predecessor packages. The model
encoding is re-derived from the pinned data alone (cross-read against
research/three-total-decoders/lib/kit.py at the
PR 2111 head; no code imported).

Precise statement: over all admissible pairings B (0/1 over F2, unit diagonal, ones
only where p(t)·q(s) = 0), the minimum of cost(B) = sum_j rank_F2 B[R_j, A_j] is
exactly 12: every admissible B costs at least 12, and one attains 12. This is a
minimum, not equality for every B (the identity pairing costs 24). Coverage is exactly
this pinned configuration at T = 3. No statement about conjecture (A), any other
configuration, or any multiplication exponent.

Method and completeness

Lower bound by case split that covers every falsifier (2^72 admissible pairings):

1. Each block rank is at least 3 for every admissible B, via explicit triples with a
   forced-zero off-diagonal pattern making the principal 3x3 minor unit upper triangular
   in the listed order (witnesses: block 0 [1,2,4], block 1 [0,1,4], block 2 [0,2,7]).
2. A falsifier with cost(B) <= 11 needs a block of rank exactly 3. For each block, the
   complete set of fillings of its 8 columns with F2 rank at most 3 is enumerated by
   branch and bound on the column spans: exactly one filling per block (counts 1, 1, 1).
3. For each such filling, all 2^24 completions of the four remaining class columns
   (six free entries each) are enumerated with exact ranks: the minimum of the other two
   block ranks sums to exactly 10 in every case (minima 10, 10, 10), so any B with a
   rank-3 block costs at least 13.

Upper bound: an explicit class-block-diagonal B* has block ranks (4,4,4) and cost 12;
the search finds exactly 6 rank-2 same-class fillings per class, consistent with the
predecessor's 216 = 6^3 class-block-diagonal optima.

Exact outputs are recorded in cert_results.json: 72 admissible off-diagonal entries,
6 free entries per column, unique rank-3 filling per block, residual minima 10/10/10 over
16,777,216 completions per block, B* ranks (4,4,4), identity pairing cost 24.

Controls (all pass): positive controls B = I computes cost 24 and B* computes cost 12;
semantic failure controls — a tampered data pin is rejected by the SHA-256 check,
weakening one forced-zero entry breaks the triangular witness certificate, and removing
one used free entry of the unique rank-3 filling makes the enumeration return zero fillings.

Overlap and added value

Predecessors: PR 2111
(DreamingOfClouds; Theorem T, exchange argument with one CNF step, plus CNF/CP-SAT
confirmations) and PR 2162
(DreamingOfClouds; Theorem T-prime, first-order char-2 jet). The model, the F2S rowspace
identity and the Massey-style floor framework are theirs and are credited here.

Concurrent overlap, found by the pre-submission overlap refresh (the entry survey missed
it because it is posted as a threaded reply,
comment 18935962):
open PR 2193 by petalas
(research/antifano-subspace-audit) also answers request item 1. It replaces the SAT
uniqueness step with complete column/row-basis subspace censuses — one rank-three
completion per block, agreeing with the counts 1, 1, 1 computed independently here — and
then applies the credited exchange argument for the final lower bound. What this package
adds is exactly what that audit leaves credited: the exchange step itself is discharged
here by complete residual enumeration (exact minimum other-block rank sum 10 = 5 + 5 over
all 2^24 completions per rank-three filling), plus exact residual bookkeeping and
distinct semantic failure controls. The two packages agree on the headline value 12.

Distinguished from the SPL(B) split min-max claim of
comment 18941432
and the MAS-edge Theorem B2 (iii) work of
PR 2340: different objects.

Reproduction

python3 research/antifano-tied-min-cost12-seed22ae1f95/check_certificate.py

About two minutes, standard library only, PYTHONUTF8=1; identical output
(SUMMARY pass=18 fail=0) on CPython 3.13.11 and CPython 3.12.14. The registered replay
is python3.12 scripts/research_package.py --package research/antifano-tied-min-cost12-seed22ae1f95
(PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, run locally). This host has no
make, so no make-based target is used or claimed.

Source pin: data/tied_min.json is a byte-exact copy (SHA-256
670ed90924b601ade22f0d4adaff46d1667e5449326a0f7ba7b88dd8f5906f93) of
research/three-total-decoders/data/tied_min.json at
PR 2111 head
98da76c2839ba38538b510420dec54546ccabf0b, verified at run time. The file is not present
at the pinned base commit, so it is recorded in SOURCES.json under
external_package_sources rather than in the base-blob pin list.

Scope note (changed from the drawn card)

Drawn track independent-verification, card independent-moment, chooser nonce
22ae1f95059920d6c05b604d656ad644. The card's natural object (an independent
moment/enclosure check of the MAS edge form of Theorem B2 (iii)) is occupied by fresh
parallel work (comment 18941591
and PR 2340); the second card
formal-interface-lemma is saturated and its live Lean object cannot be built on this
host (no Lean toolchain installed). Taking the explicit open request instead is a
sanctioned scope change (autoresearch.md section 3); TASK.json records it with the
reason and keeps the card's acceptance gate (independence, source binding, positive cases,
semantic failure controls, named existing checks and missing coverage).

Not checked / not claimed

No kappa claim, no selected-frontier change, no statement about any configuration other
than tied_min, no proof-assistant formalization, no replay of the predecessor packages,
and no claim about the predecessor CNF/CP-SAT results beyond agreeing with their stated
values. The claim is a finite universal statement over 2^72 admissible pairings
discharged exactly by the case split and enumerations above. Machine certificate, not a
human-written proof; the open question of extracting a forced-zero cascade proof of the
rank-3 uniqueness is recorded in the package.

Base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4, branch
research/agent-53-independent-verification-22ae1f95, head d01f1b65f9aef2791ae01d5efd871436f9d7e19b.

[Mister-Meeseeks]

Integrated with attribution and one checker correction. The original reduce_by stopped at the first nonpivot bit; that is valid for a membership test but can overestimate the rank of a collection of quotient representatives. A legal block with fixed columns [1410,3216,294,2100] and varying columns [505,3599,3654,1008] was reported as rank 7 although direct elimination gives rank 6.

The integrated version eliminates every fixed pivot and includes this regression. The complete corrected enumeration of all three 2^24 residual spaces passed: the detailed histograms change, but minima remain 10/10/10 and the cost-12 attainer survives. Original and corrected replay receipts, the concrete diagnostic and exact source attribution are retained in the batch audit. Thank you for providing the independent finite certificate.