Exact verifier
PASS 7 fronts
1ec126887cd1f09b8cdc074872567dd4962c2a399c1b14db8b035a8ea50dc591
Exact finite graph theory · representation restriction · solver architecture
Borel Fourier anatomy and the star-complement firewall
Passes 3528–3534 execute five requested continuations and two outside-box constructions. Exact claims are frozen in a deterministic certificate; the missing degree-57 Moore graph remains open.
Perkel Borel group
19:9 · order 171
Every map (i,j)→(i+m,4^m j+b) is verified directly against the 57-vertex graph.
Rational Fourier split
Q⁵⁷ = 1 + 3V₁₈ + V₂
The golden 36-space is 2V18; the rational −3 twenty-plane is V18+V2.
The 19:9 restriction
20¹ · 1¹⁸ · 2³⁸ · (−1)¹¹⁴
The exact −3 character distribution is obtained from 171E = −A³+9A²−19A+6I.
2 · 2 · 8 · 0
Fixed dimensions for C19, C9, C3, and the full Borel group.
The 57-vertex nonexistence mechanism
Independent candidate census
22 → 784 → 157,349 → 3,720
Canonical extension and spectral filtering reproduce the exact star-complement candidate count and ledger hash.
Compatibility obstruction
31 < 38
The published compatibility graphs never contain the 38-clique required to reconstruct SRG(57,14,1,4). The margin is seven.
Landscape gap
17…28 absent
No candidate compatibility graph has maximum clique in this interval.
Staged Moore search
Global spectral gate
n ∈ {1,2,6,56}
Only C5, Petersen, Hoffman–Singleton, and the open M57 fibre size survive. Every even stage 8 through 54 fails globally before CSP.
Unrestricted track
172,480 integer variables
Double fibrations, inverse channels, vertex stars, and lazy triangle and μ=1 separators.
Involutive track
86,240 extra equations
Every pair matching is forced into cycle type 2²⁸; holonomy constraints are separated without claiming necessity.
Two exact no-go boundaries
Typed descendants
240/120/40 ≠ 100/77/56
W33 uses uniform fibre quotients. Higman–Sims → M22 → Gewirtz uses induced restriction and point avoidance. One uniform quotient functor cannot cover both.
Spectral transplantation
Q[x]/(x²+2x−8)
Only polynomial identities auto-port between W33 and Gewirtz. Multiplicities, incidence, codes, automorphisms, and descendants require new evidence.
Outside-box one-factorization theorem
56 coupled factorizations of K₅₆
Inside the involutive branch, the vertex-star all-different law makes the 55 fixed-point-free involutions at each source row a complete 1-factorization. Each pencil partitions all 1,540 edges of K56. Hoffman–Singleton supplies the exact six-symbol control: six coupled non-group pencils of K6.
Reproduce
python analysis/bt3528_3534_borel_star_moore_functor_transplant.py
pytest -q tests/test_bt3528_bt3534_borel_star_moore_functor_transplant.py
python analysis/bt3529_star_complement_census.py --quick
python analysis/bt3530_m57_dual_tracks.py --track stats
python analysis/bt3532_spectral_transplant_repo_audit.py