Galois Pairing–Trace Correspondence and the Omega Phase Energy Node (open, δ_open)_ A Structural Criterion for the Erdős Unit-Distance Problem (Norm Signature, the Anchor 5_4 and the Single-End Window
Year: 2026
Type: preprint
Abstract: The Erdős unit-distance conjecture (posed in 1946: among n points in the plane, the number of unit distances is u(n)=n^(1+o(1))) was disproved in May 2026: there is an absolute constant δ>0 such that u(n)≥n^(1+δ) holds for infinitely many n. The human-verifiable explicit lower bound advanced in two steps, through Sawin (δ≈0.014) and Naslund (δ≈0.043171, a pro-2 tower over Q(√241); Lean proves 1.04315 conditional on an explicit zeta inequality H_W, while 1.043171 also relies on non-formalised numerical computation and is starred); the current band is n^(1.043171) (conditional) ≤ u(n) ≤ O(n^(4/3-ε)), where the upper bound, frozen since Spencer–Szemerédi–Trotter (1984), was improved by OpenAI in 2026-09, with ε not made explicit. This paper does not claim to prove or improve any bound; it does three things inside the framework. First, using the Galois pairing–trace correspondence lemma (the v3/v5 unified paper), we give a structural explanation of eighty years of constructions: from the Erdős grid (Q(i), the degree-2 CM stagnation point) to the Golod–Shafarevich field families of OpenAI/Sawin (degree →∞), all known lower-bound constructions share a single norm signature—a unit-distance pair is a norm-1 element, and the gain of a construction is the maximisation of the count of low-norm elements in a number field. This disproof corresponds to the structural reading "the degree is freed from 2 to ∞"; the reading is falsifiable, but it is not an assertion of necessity. Second, we recast the remaining problem, the optimal exponent C84 = limsup(n→∞) log u(n) / log n, as a single-endpoint (upper-anchored) max problem, as opposed to the symmetric two-endpoint template used by the two min problems of v3/v5 (C1a, C1b): the limsup definition forces the open side downwards; the anchor 5/4=1+1/4 (the μ4 fundamental-domain share, with four same-framework readings: conductor(χ4)=4, |μ4|=4, |Units Z[i]|=4, and Tr(Φ^3)=L3=4) only marks the position and never corrects; the adjacent width on the open side is δ_open=6Φ^2 - 5π≈2.4066×10^-4; external corroboration is Sawin's method-family ceiling 1+1/4.116≈1.242954: anchor to ceiling ≈29.3 δ_open, window edge to ceiling (the family gap) ≈28.3 δ_open (§3.4). Third, we set a pre-registered verdict for C84 (the P-anchor and the P-window) and register a trap configuration: any construction reaching the open-side window [1.2497593, 1.25] must lie outside the Sawin method family—the tower core alone no longer suffices. We further give the exponent-ceiling theorem for the Q(i) scaling branch (the limsup exponent is always 1), turning "the grid branch cannot reach 1.25" into a provable statement rather than a qualitative judgement. All numerical values in this paper are reproducible in double precision (with c84_bounds_check.py checking each item), and the main structural claims carry falsification conditions. The external fact that "the conjecture has been disproved, and the disproof was produced by AI systems" is registered as it stands and used as the time origin of the verdict.
Authors: sagaistrue, Doub
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225321798