A class-number ceiling for the unit-distance exponent: the limits of the CM-field method

In May 2026 a machine-generated argument disproved Erdős's unit-distance conjecture, constructing planar point sets with n^(1+δ) unit distances from ideal lattices in CM number fields, the count of magnitude-one elements governed by the relative class number h⁻. The certified exponent was then sharpened rapidly — Sawin's explicit lower bound, Emmerich's re-optimisation, and a public leaderboard of records reaching Naslund's ≈ 1.036 — all by making h⁻ as small as the analytic input allows. This paper settles the reverse question, posed but left open by Sawin: what is the largest exponent this method can certify? Settling it requires a lower bound for h⁻, which Sawin noted to be subtle because of Siegel zeros. We supply that lower bound from Louboutin's relative Brauer–Siegel theorem — unconditional, effective, and resolving the Siegel concern through a dichotomy — and insert it into Sawin's exact exponent functional. Three theorems result. (A) A lemma-independent universal ceiling of 1.198 (unconditional; 1.184 under GRH), robust to every refinement of the counting lemmas. (B) The exact ceiling: the Weil explicit formula couples the completely split primes of a CM field to its root discriminant, and enforcing this coupling lowers the ceiling to 1.0725 under GRH and 1.149 unconditionally, for every CM field and with no hypothesis on the construction; the natural Golod–Shafarevich estimate of the same coupling is shown to overcount, so the explicit-formula obstruction is strictly stronger and no separate conjecture for arbitrary CM fields is needed. (C) The relative regulator is pinned to a power of two by the Hasse unit index, so h⁻ = d_K^(1/2+o(1)) unconditionally and a possible Siegel zero enters only at lower order. What remains open is not the ceiling but its achievability, identified here as the number-field analogue of the Drinfeld–Vlăduţ problem and mapped explicitly: an effective family of Golod–Shafarevich towers, indexed by the number of split small primes, has certified exponents climbing from the current record 1.036 to 1.071, the records occupying its lowest rungs, with realisation of the higher rungs reduced to a Brauer–Siegel saturation question. The duality with the record race is exact: the leaderboard pushes the exponent up by sharpening upper bounds for h⁻; this paper caps it by the lower bound for the same h⁻; and the explicit formula shows the upward direction saturates below 1.0725. This is a ceiling on the exponent the method certifies, not an upper bound on the number u(n) of unit distances, which remains confined to [1.036, 4/3]. The construction side of the problem is thereby mapped and its limit located far below the Spencer–Szemerédi–Trotter exponent 4/3, redirecting further progress to the upper bound or to a construction outside the CM-field method. Deposit contents: (1) the main paper (21 pp.), whose appendices give in full — and reproduce by self-contained scripts — every numerical value; (2)–(4) the three reproduction scripts verify_all.py, recompute_full.py and split_convention.py, each runnable on the Python standard library with mpmath and NumPy, reproducing respectively the gate and universal ceilings (−0.00105; 1.243; 1.198; 1.184), the explicit family and the frontier (1.036 → 1.071), and the three split-prime conventions (1.0907 all primes / 1.0725 odd primes / 1.0678 tight for F(i)).