A variational ceiling for the multiquadratic unit-distance exponent

In May 2026 a machine-generated argument disproved Erdős's unit-distance conjecture. In the weeks that followed the construction was made explicit and progressively sharpened: the nine-author Remarks supplied the arithmetic existence input through the Frobenius-cutting of Hajir–Maire–Ramakrishna; Sawin computed an explicit exponent and an unconditional ceiling; Emmerich re-optimised the certificate. Alongside this, the present author pursued a parallel programme on the structure of the construction itself — a critical audit, an optimisation to a certified n¹. 031, and an isolation of the geometric reduction and its arithmetic bottleneck. This note is the conclusion of that programme: it turns the analysis into a theorem. We study the no-split multiquadratic sub-family of Sawin's optimisation — imaginary multiquadratic fields, with the auxiliary primes ramified or quadratic-inert but never split — and give a variational description of its exponent. Writing the excess exponent as a ratio δ = N/D, a Dinkelbach linearisation of the truncated functional separates additively into a field term, a radius term, and independent per-prime terms; since the exact Sawin denominator exceeds the truncated one, every upper bound proved here applies a fortiori to the exact exponent. Three results follow. Theorem 1: the inner optimisation over packet, multiplicities and radius is closed-form, collapsing the exponent to a scalar δ (m). Theorem 2: the optimal multiplicity profile is k* (t) = 1/ (2δ log t) − 1, with positive-support cutoff pc = 2^ (1/ (2δ) ). Theorem 3 (unconditional ceiling): through an exchange inequality that makes the m smallest odd primes Sₘ the maximiser of a field-universal relaxation — defined on all m-element prime sets, so that single prime-for-prime exchanges through possibly inadmissible intermediates suffice and no parity or admissibility constraint is required — no configuration of the family certifies an exponent reaching 1. 034 through Sawin's formula, at any degree. The bound is proved by explicit estimates and a finite, reproducible verification (the maximum over 2 ≤ m ≤ 169 of Φᵣel (m) equals −0. 3359 < 0, with margin re-checked at 50 digits). The elementary method cannot pass 1. 0334; the sharp ceiling 1. 0315 is stated as a precise conjecture, reduced to a single large-sieve question on the effective equidistribution of inert primes below the field conductor. The family's certified exponent is thus pinned unconditionally to [1. 031, 1. 034), and conjecturally to [1. 031, 1. 0315), far below Sawin's unconditional ceiling 1. 24295; the chasm is controlled by the relative class-number problem isolated in the reduction note (Sawin's Remark 14). This is a ceiling on the exponent the method/family produces, not an absolute upper bound on the number u (n) of unit distances (which is ≈ n⁴/3). Deposit contents (4 files): (1) the main note; (2) a self-contained computational supplement that recomputes the relaxation bound and re-verifies the binding margin at 50-digit precision; (3) the standalone figure; (4) a conforming copy of the optimisation paper's computational supplement (doi: 10. 5281/zenodo. 20553929), included for the five independent verifications of the certified theorem point n¹. 031058. The certified theorem point is n¹. 0310580…; the highest value attained in the displayed scan is the best-scanned / discriminant-minimal value n¹. 0313420… at m = 56 — the two are kept distinct throughout.