Pre-registered spectral predictions for the bond-alternation family of the truncated icosahedron: parameter-free ratios, an exact 1/√5 response, and a spectroscopic determination of the bond-alternation ratio

This is a pre-registration, not a report of agreement. Every number in it was computed from the graph before any comparison with measurement, and no measured quantity is used as an input anywhere. The comparisons are named but not yet made. The timestamp is the evidence: a prediction recorded after the data is seen is not a prediction. Let Γ be the truncated icosahedron, the graph of the C60 cage. Aut(Γ) ≅ Ih acts with exactly two edge orbits — the 60 pentagon–hexagon bonds and the 30 hexagon–hexagon bonds — so the weighted adjacency A(r) = r·A66 + A65 carries a single parameter r = β(6:6)/β(6:5). Six predictions follow, each labelled FORCED, FORCED-GIVEN[X], IMPORTED or OPEN, and each with the condition that would falsify it. P1. The nine-fold level at λ = 1 is an accidental degeneracy and splits 4 + 5; no other level splits, and the level count at any r ≠ 1 is exactly 16. P2. An exact first-order slope table dλ/dr for all 15 levels, including the result that the HOMO (hu) response is exactly 1/√5 on all five states, and the nine-fold slopes are exactly 2/3 (×4) and 1/3 (×5). P3. The HOMO–LUMO gap is strictly increasing in r across 0.8 ≤ r ≤ 1.4, so equal bonds sit at the minimum. P4. Sealed values at r = 1.145. P5. Five parameter-free ratios in which the energy scale β cancels, leaving nothing to fit; the g9/h9 splitting is predicted at 116–136 meV. P6. The ratio M1 is strictly increasing in r and therefore inverts, so a measured splitting-to-gap ratio determines r spectroscopically with no further input, to about ±0.01 from a 5% measurement. P6 is the point of the set. The bond lengths give one determination of r; the spectrum gives an independent one. If they agree, r is overdetermined and the length-to-hopping import is validated rather than assumed. If they disagree, the single-β alternation picture fails and the failure is located rather than merely recorded. The deciding measurement. The predicted ~120 meV splitting sits inside the 0.4–0.5 eV linewidth of C60 film photoemission, so every existing film spectrum is consistent with it and none constrains it. Gas-phase or high-resolution valence photoemission below about 50 meV resolution, looking for a doublet in the HOMO−1 band, decides P4, P5 and P6. A clean singlet at that resolution falsifies the prediction. P1 is a consistency check against the known C60 Hückel literature, not a new result. The novel content is the exact slope table, the parameter-free ratios, and the inversion. Nothing here bears on any claim about fundamental physics; the subject is the π spectrum of one molecule, and Hückel theory describes the π manifold only. The accompanying script rebuilds the cage from coordinates, verifies the two edge orbits, and checks every printed number; it prints ALL CHECKS PASS. The only input is r.