LemmaStatement: AI-generatedProof: AI-generated

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Certified regular amplification excludes a PSD half-circle tail

Statement

Under the regular critical-PSD hypotheses of the mass lemma, suppose the graph is connected and put $C_{β} = {x : ∣ θ_{x} ∣ \geq β}$ . Then $C_{π / 2}$ is empty. The proof uses the scalar certificate with exact rational parameters $a_{0} = 0.0816$ , $A_{0} = 2.4093$ , $q = 2.1274$ , $h = 3.168$ . Define $u = \arcsin (0.3272)$ as an exact real angle using the principal inverse sine: $c_{i} = A_{0}^{i}$ and $β_{i} = π / 2 - i u$ for $0 \leq i \leq 3$ ; $c_{3 + k} = A_{0}^{3} q^{k}$ and $β_{i + 1} = β_{i} - \arcsin (h / c_{i})$ afterward. Rounded table entries never replace exact recurrence values.

Facts & Assumptions

[F1]

Use the graph, phase, algebraic and analytic interfaces of Regular critical PSD phases: a direct mass bootstrap at 0.0816, Regular critical PSD phases: kernel and half-circle, Critical PSD kernel phases: quantitative tail growth, Exact rational replay of the regular amplification constants, Two-sided spectral expander — Definition 1.9, Angular order and elementary inverse-trigonometric bounds. Every configuration assumed here is a critical point with positive-semidefinite Hessian; local minimality is never inferred from that condition.

[F2]

The norm is the induced operator norm on the finite-dimensional complex vertex space (The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖). For $f \neq 0$ , applying its definition to $f / ∥ f ∥$ gives $∥ T f ∥ \leq ∥ T ∥ ∥ f ∥$ ; this also holds for $f = 0$ . Cauchy–Schwarz (Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors) then gives $∣ f^{*} T f ∣ \leq ∥ f ∥ ∥ T f ∥ \leq ∥ T ∥ ∥ f ∥^{2}$ . The theorem's linear-first convention has the same norm and absolute pairing as the vertex convention.

[F3]

Sine is strictly increasing on $[- π / 2, π / 2]$ (Signs, monotonicity intervals, and ranges of sine and cosine, with integer parameter $m = 0$ ). Together with sine nonnegativity in [F1], this gives $\sin^{2} r \leq \sin^{2} t$ for $0 \leq r \leq t \leq π / 2$ .

Proof

Given: The hypotheses of the Statement.

Goal: Establish its conclusions directly for critical PSD phases.

1.1F1F2algebra

Regularity gives $Δ_{L} = - Δ_{A}$ . Applying [F2] to $T = Δ_{A}$ yields $∣ f^{*} Δ_{A} f ∣ \leq ∥ Δ_{A} ∥ ∥ f ∥^{2} \leq a_{0} d ∥ f ∥^{2}$ for every complex vertex vector $f$ . Thus $- a_{0} d I ⪯ Δ_{L} ⪯ a_{0} d I$ in the order defined in [F1], so the graph is an $(n, d, a_{0}, - a_{0}, a_{0})$ -expander. Suppose $m_{0} = ∣ C_{π / 2} ∣ > 0$ . The scalar replay proves every listed angle is positive, all are above $0.1356$ , $β_{3} \geq 0.570$ , and $c_{i} \to \infty$ . It also proves $2 (1 + 0.38) a_{0} / (1 - a_{0} - 0.23) < 0.3272$ , $1 + 0.23 / (2 a_{0}) > A_{0}$ , and $0.38 a_{0} > 0.031$ . Apply tail growth (a) with $ϵ = 0.23$ , $u = 0.38$ , and each of the first three exact angle decrements. Inductively this gives $∣ C_{β_{i + 1}} ∣ \geq \min {A_{0} ∣ C_{β_{i}} ∣, 0.031 n}$ . All inverse-sine domains and angle ordering are supplied by the replay.

2.1F1F3step 1.1algebra

A first-stage cap is impossible. Indeed $β_{i} \geq β_{3} \geq 0.570$ for $i = 1, 2, 3$ , and the scalar certificate proves $0.031 \sin^{2} (0.570) > 5 a_{0}^{2} / 4 = 0.0083232 > 0.00832$ . For these indices $0 < 0.570 \leq β_{i} < π / 2$ , so [F3] gives $\sin^{2} β_{i} \geq \sin^{2} (0.570)$ . Thus $∣ C_{β_{i}} ∣ \geq 0.031 n$ would imply $∣ C_{β_{i}} ∣ \sin^{2} β_{i} > 0.00832 n$ , contradicting the PSD mass lemma's tail bound. Hence the first three steps yield the uncapped induction $∣ C_{β_{i}} ∣ \geq c_{i} m_{0}$ .

3.1F1F3step 1.1step 2.1algebra∎

The scalar replay proves $2 (1 + 2 a_{0}) / (1 - a_{0} - 0.184) < h$ and $1 + 0.184 / (2 a_{0}) > q$ . Whenever $∣ C_{β_{i}} ∣ \geq c_{i} m_{0}$ , the exact decrement has sine $h / c_{i} > 2 (1 + 2 a_{0}) m_{0} / ((1 - a_{0} - 0.184) ∣ C_{β_{i}} ∣)$ . Tail growth (b) with $ϵ = 0.184$ therefore yields $∣ C_{β_{i + 1}} ∣ \geq \min {q ∣ C_{β_{i}} ∣, n / 2}$ . Each angle exceeds $0.1356 > 0.131$ , while the scalar replay gives $\sin^{2} (0.131) > 5 a_{0}^{2} / 2 > 2 (0.00832)$ . Since $0.131 < β_{i + 1} < π / 2$ , [F3] gives $\sin^{2} β_{i + 1} \geq \sin^{2} (0.131) > 2 (0.00832)$ . A cap $∣ C_{β_{i + 1}} ∣ \geq n / 2$ would therefore imply $∣ C_{β_{i + 1}} ∣ \sin^{2} β_{i + 1} > 0.00832 n$ , again contradicting the PSD mass lemma. Hence induction gives $∣ C_{β_{i}} ∣ \geq c_{i} m_{0}$ for every $i$ . Since $c_{i} \to \infty$ and $m_{0} \geq 1$ , this eventually exceeds $n$ , impossible. Consequently $C_{π / 2}$ is empty.

Depends on

Used by · 1

All critical PSD equilibria synchronize on positive regular expanders

Dependency tree · 2 levels

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