What the C60 Geometry Forces: An Honest Accounting of the Forced Count, with One Booked Prediction Loss

Resolution of 3a on new evidence, and 3b partially closes (28 October 2026, later the same day). Count: 35 forced / 10 conditional / 0 lost. Why this reverses the earlier retraction rather than contradicting it. An earlier note today withdrew 3a’s closure on the ground that Kasteleyn is a principle adopted rather than derived. That was correct on the evidence then available, which compared only the Kasteleyn configuration against zero flux. The missing computation has now been done, and it changes the premise. π/2 is the half-filling flux-phase optimum of A, not merely the Kasteleyn value. Dirac quantisation on the closed surface restricts the uniform pentagon flux to Φp ∈ (π/6)ℤ with Φh ∈ (π/10)ℤ, so the family has no continuous parameter. Realising each allowed value exactly — by solving the 31 independent face rows, the thirty-second flux being determined by closure, to a residual of order 10⁻¹⁴ — the half-filled ground-state energy over the quantised pentagon family is: Φp = 0, π, 2π → −46.580801897 ±π/6 → −46.669086285 ±π/3 → −46.777274929 π/2 and 3π/2 → −46.963587805 π/2 is the minimum, lower than zero flux by 0.3828 t, and the HOMO–LUMO gap there is exactly 2/φ = 1.236067977 against 0.756598254 at zero flux. An exhaustive scan of the complete I-invariant flux sector over all twenty Dirac branches confirms it as the global minimum of that sector, exactly stationary (|∇S| = 3.8 × 10⁻¹⁰) with a definite Hessian across the full 31-dimensional gauge-inequivalent flux space, and twenty-four unrestricted searches over all ninety edge phases never improved on it. So the premise behind 3a is no longer the adoption of a combinatorial condition. It is that the flux is the one minimising the half-filled band energy — the flux-phase problem’s own premise, and a theorem of A with no free parameter. Three independent characterisations now land on the same configuration: Kasteleyn/Pfaffian uniqueness up to gauge, energy minimisation, and chirality. The magnitude is forced. The hand remains a label, and the energy now proves it rather than merely permitting it: π/2 and 3π/2 come out exactly degenerate, so the variational principle does not select between a configuration and its time reverse. That is the right status for a quantity whose item name already carries the ±. Note also, as the exact form of why Lieb’s theorem cannot be used here: girth 5 gives tr(A⁵) = 10 × 12 = 120 ≠ 0, so the spectrum is not symmetric about zero and the twelve odd cycles are the sole source of the asymmetry. C60 is non-bipartite, Lieb’s flux-phase theorem does not apply, and π/2 rather than 0 or π is the non-bipartite answer. A new theorem, reusable and not specific to C60. On any 3-regular polyhedron built only from pentagons and hexagons, shifting every edge phase by π sends H to −H and (Φp, Φh) to (Φp + π, Φh), while conjugation preserves the spectrum and sends Φ to −Φ. These agree exactly when Lπ + 2Φ ≡ 0 (mod 2π) per face of length L, i.e. when Φp ≡ π/2 (mod π) and Φh ≡ 0 (mod π); a diagonal unitary G then solves φi − φj = π + 2θij and gives G H* G† = −H, an antiunitary chiral symmetry. So the forced flux is exactly a particle-hole-symmetric point, with μ = 0 at half filling at every size. Verified on C60 and on GP(ν,0) for ν = 2,3,4,5,6,8. Among the eleven time-odd Dirac-quantised pentagon fluxes, only π/2 and 3π/2 are chiral. 3b partially closes: the fitted two-parameter window becomes one scalar inequality, and temperature drops out entirely. With H = A + mK, K the parameter-free time-odd operator built from A and the face orientation, the near/far gap ratio is monotone in m and crosses 1/3 at m ≥ 0.102772162 t, uniformly above that threshold with no upper bound, and T does not enter at all — the observable is purely spectral. The massless condition has the closed form mκ = √(gnear · gfar), the geometric mean of the two bare gaps, with gfar = 1/φ + 1/φ² = exactly 1 on C60. Two further eliminations are theorems rather than scans: C60 is vertex-transitive with a single vertex orbit, so any site-local Aut-symmetric self-energy is a scalar and shifts no gap ratio for any u and any T, which covers the Hartree term of any on-site interaction and single-site DMFT’s static part. So the imported approximation is now confined to supplying one number. It is not removed, and 3b stays conditional on that one inequality. And the obstruction behind it is structural, which is worth more than the sharpening. At the energy-selected flux the chiral theorem forces near/far = 1 exactly — all six copy states degenerate in one eight-fold level at −1/φ, both gaps equal to 2/φ — so the forced flux supplies m = 0. The energy selection of 3a and the splitting 3b requires pull in opposite directions on this cage. An on-site Hubbard term at half filling preserves that chiral symmetry for every u and T, as does any site-local symmetric self-energy. And at the forced flux the rotations act as the binary icosahedral group, with the Fermi-boundary level a single irreducible spinor of commutant dimension one, so no I-invariant order can split it. The two-copy T1u/T1g picture is a zero-flux description that the forced flux dissolves. Consequently the forced flux cannot meet 3b’s own requirement of a copy-even offset of μ from the copy centre. Honest negatives from the same run: the record’s own near/far observable is not in the exact one-body spectrum at any size — exact values at u = 0, T = 0 are 0.97052 at ν = 6, 0.96619 at ν = 12 and 0.96545 at ν = 18, against the record’s 0.94, 0.91 and 0.31, so the first two record points sit within 6% of pure band structure while ν = 18 is a factor 3.1 away. The fitted window is load-bearing. A Weyl inequality gives a rigorous lower bound of 0.0301 t on what any one-body self-energy must supply at ν = 18, so the claim is provably false at u = 0, and no interval of u exists over which the conclusion is uniform. What would close it is a rigorous lower bound on the symmetric state’s free energy on GP(ν,0) that does not use RPA; Peierls–Bogoliubov already supplies the matching upper bound, and the channel, the threshold and κ are all exact. One cross-result: κ ≠ 0 is now computed. The forced narrow lemma’s condition was κ ≠ 0 in the unique Au channel of T1u ⊗ T1g = Au ⊕ T1u ⊕ Hu. That reduced matrix element comes out κ = 4.268662453, with K restricted to the copies having eigenvalues ±4.268662453, three of each sign — one magnitude, hence one intertwiner, as Wigner–Eckart requires. So 3e-B’s first condition is discharged and its remaining price narrows to the inherited existence tie alone. With the monopole charge |q| = 3/2 also now computed rather than posited, two of the three prices behind that cluster are off the books. A lead worth the queue. The exact near/far baseline at the forced flux has a ν mod 3 structure the record’s own series never samples: ν ≡ 0 (mod 3) gives 0.966–0.988, while ν ≠ 0 (mod 3) gives 0.531–0.617 at ν = 2, 4, 5, 8, 10. The series ν = 6, 12, 18, 24 lies entirely inside the first class. Running the engine at ν = 10 or ν = 20, where the band-structure baseline is about 0.55 rather than 0.97, would separate the interaction’s contribution from the band contribution for the first time. Flagged, not claimed: the exact threshold mκ = 0.438700 and the record’s projection coefficient −0.438 agree to three digits but in different normalisations, so that is a digit coincidence and not an identity. Under the flow-position gate none of the above books on any ledger of passes or losses: every statement is an internal identity on a fixed finite object or an exact spectral computation at stated filling, with no measurement on either side. The three ledgers stay separate and unsummed; the booked prediction loss stands at one. Two further premises sharpen, 28 October 2026 (count still 34 / 11 / 0). 3c: the anomaly rider is discharged, and νR turns out not to be imported. The binary antipodal code kerF₂(A) = [60, 4, 24] is isomorphic as an Ih-module to the even-weight submodule of F₂⁵ — an isomorphism this record’s own verification script already establishes element by element — and its weight grading is exactly the even-parity half of the 32-state Cl(10) spinor module, i.e. one chiral 16 of Spin(10), decomposing as 1 ⊕ 10 ⊕ 5̅ with anomaly 0 + 1 + (−1) = 0. The strong form holds, which is more than the rider needed: tr(Ta{Tb, Tc}) vanishes identically on all of so(10) for this content — every one of the 45³ generator triples exactly zero, and 400 random triples of arbitrary so(10) elements at most 2.9 × 10⁻¹⁴ — so no choice of gauged subgroup and no U(1) direction inside Spin(10) can make the content anomalous. The hypercharges are then read off as eigenvalues of the same grading rather than assigned, and they come out as the Standard Model’s: one state at Y = 0, three at +1/3, two at −1/2, three at −2/3, six at +1/6, one at +1. Recomputed here in exact rational arithmetic, the multiplicities and hypercharges match the SM 16 exactly and all four conditions vanish identically — ΣY = 0, ΣY³ = 0, SU(2)²U(1) = 3(1/6) + (−1/2) = 0, SU(3)²U(1) = 2(1/6) + (−2/3) + (1/3) = 0 — with four weak doublets, an even number, so Witten’s global SU(2) anomaly cancels as well. That the reading is not a free choice was checked by counterfactual: of the eight dimension-respecting ways to assign the orbits {1, 10, 5} to su(5) irreps, four are anomaly-free and four carry A = ±2, so dimensions alone would be a coin flip. What removes the flip is the code’s own additive structure — wi + wj = wij mixes the weight classes, and that relation is realised only by the twenty pair operators of so(10), never by su(5) alone and by no GL(5) action, there being no map Λ⁴ ⊗ Λ⁴ → Λ². Requiring a constant degree-weight parity offset then leaves exactly two assignments, which are complex conjugates of one another — the same theory with the opposite chirality called left. The spinor reading is therefore not an extra import; it is what the code’s additio