Mission

Sharp diagonal Hlawka constants: lower the cutoff to 62Open Problem

Functional Analysis·Captain: moona3k

This entry lowers the sufficient exponent cutoff for the sharp diagonal Hlawka formula from 63 to 62. For every real $p\ge 62$ , the foundation's unchanged cyclic constant is the least constant comparing the triple deficit with the pair-deficit sum for every complex coordinate triple in every finite dimension.

The statement includes unequal and zero input vectors and dimension zero. It concerns complex diagonal matrices through their coordinate norms. It does not settle the conjectured cutoff 2 or the full noncommutative matrix problem.

Goal and companion

Exact complex cutoff 62 goal, proved.
Real coordinate bound for $p\ge 62$ , the supporting milestone, proved.

The confinement radius on this integer is $3643/10000$ . The accepted cutoff- $63$ theorem remains the tail this argument calls. Cutoffs 64, 65, 66, and 70 stay proved and are not restated here.

This is the last integer this confinement argument reaches. At $p=61$ every radius whose tangent model stays positive already makes the budget- $7$ box factor larger than $1$ .

Goal · The least uniform complex coordinate Hlawka constant for

p \ge 62

PROVED

theorem HlawkaSchatten.DiagonalCutoff.cutoff62 :
    ∀ p : ℝ, 62 ≤ p →
      IsLeast {C : ℝ | ∀ n : ℕ,
        HasHlawkaConstant (lpNorm p : (Fin n → ℂ) → ℝ) C}
        (cyclicConstant p) := by sorry

For every real exponent $p\ge 62$ , let $N_p(x)$ be the foundation's finite coordinate $p$ -norm and let $K_p$ be its unchanged cyclic constant, the supremum of the cyclic ratio over $t\in[1/2,2]$ .

Then $K_p$ is the least real constant $C$ such that the triple deficit is at most $C$ times the pair-deficit sum, for every natural-number dimension $n$ and all complex vectors $x,y,z$ in that dimension.

This combines admissibility and optimality uniformly over all finite complex coordinate dimensions, including dimension zero and triples with unequal or zero norms. The accepted cutoff- $63$ theorem covers every $p\ge 63$ . This statement is the same leastness claim from $62$ upward.

Formalization Note. The statement is IsLeast of the set of uniform complex HasHlawkaConstant bounds, with least element cyclicConstant p.

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