LemmaStatement: AI-adaptedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Numerical amplification argument for alpha 0.0816

PDF p. 29: Proof of Theorem 1.4 and Table 1; PDF p. 30: Proof of Theorem 1.4; equations (5.2), (5.3)

Statement

Assume countable choice. Let $G = (V, E)$ be a finite connected undirected $d$ -regular graph, with $n = ∣ V ∣ > 0$ , symmetric binary adjacency matrix $A$ , and loops counted once in degrees. Suppose $∥ A - (d / n) J ∥ \leq 0.0816 d$ , where $J$ is the all-ones matrix and the norm is the Euclidean operator norm. Let $θ$ be a stable state, meaning a local minimum of the energy $\frac{1}{2} \sum_{x, y \in V} A_{x y} (1 - \cos (θ_{x} - θ_{y}))$ . Rotate its phases so that $n^{- 1} \sum_{x} e^{i θ_{x}}$ is real and nonnegative. Use representatives $θ_{x} \in (- π, π]$ and define $C_{β} = {x \in V : ∣ θ_{x} ∣ \geq β}$ for $0 \leq β \leq π$ . Then $C_{π / 2}$ is empty.

More precisely, if $r = ∣ C_{π / 2} ∣ > 0$ , the amplification construction starts with $a = 2.4093$ , $q = 2.1274$ , $u = \arcsin (0.3272)$ , $c_{i} = a^{i}$ and $β_{i} = π / 2 - i u$ for $0 \leq i \leq 3$ ; subsequently $c_{3 + k} = a^{3} q^{k}$ and $β_{i + 1} = β_{i} - \arcsin (3.168 / c_{i})$ . The three first transitions have cap $0.031 n$ , and subsequent transitions have cap $n / 2$ . All angles exceed $0.1356$ , while the uncapped lower bounds $c_{i} r$ grow without bound. Both cap alternatives contradict the stated regular-graph threshold.

Statement references

Adjacency, degrees, graph Laplacian, and spectral notation

Adjacency spectral expander — Definition 1.3

Equilibria, stable states, synchronized states, and spurious minima

Phase tail sets for amplification

The Axiom of Countable Choice (AC_ω)

Facts & Assumptions

[F1]

Under the stated graph, state, normalization and $r > 0$ hypotheses, the table supplier gives the exact recurrence above, the lower bound $β_{i} > 0.1356$ , the first three inequalities $∣ C_{β_{i + 1}} ∣ \geq \min {a ∣ C_{β_{i}} ∣, 0.031 n}$ , and the later inequalities $∣ C_{β_{i + 1}} ∣ \geq \min {q ∣ C_{β_{i}} ∣, n / 2}$ whenever $∣ C_{β_{i}} ∣ \geq c_{i} r$ . Before a cap occurs, the lower bounds are $∣ C_{β_{i}} ∣ \geq c_{i} r$ , and $c_{i} \to \infty$ (Published amplification parameters and bounds — Table 1).

[F2]

The same exact recurrence has $β_{3} \geq 0.570$ and $c_{3} = 13.985327532357 > 13.985$ . Its numerical replay proves $0.031 \sin^{2} (0.570) > 5 (0.0816)^{2} / 4$ and $\sin^{2} (0.131) > 5 (0.0816)^{2} / 2$ (Exact rational replay of the regular amplification constants).

[F3]

Under the stated graph and state hypotheses, no $ξ \in (0, π / 2)$ can satisfy $∣ C_{ξ} ∣ \geq 0.00832 n / \sin^{2} ξ$ (Regular-graph numerical contradiction threshold).

[F4]

Sine is nonnegative on $[0, π / 2]$ ; the principal inverse sine takes values in $[- π / 2, π / 2]$ (Angular order and elementary inverse-trigonometric bounds). Sine is increasing on $[0, π / 2]$ (Signs, monotonicity intervals, and ranges of sine and cosine, with $m = 0$ and restriction to this subinterval). The tail sets use the principal phase representatives stated above (Phase tail sets for amplification).

Proof

Given: The graph, expansion bound, normalized stable state and countable choice in the statement.

Goal: Prove that $r = ∣ C_{π / 2} ∣$ is zero.

1.1givenF1F2F3F4algebraassume-contra

Suppose $r > 0$ . Apply [F1]. The first three angles after $β_{0} = π / 2$ lie in $(0, π / 2)$ and are at least $β_{3} \geq 0.570$ : their decrements are positive and [F2] gives the lower bound. Since $5 (0.0816)^{2} / 4 = 0.0083232 > 0.00832$ , [F2] and [F4] imply $0.031 \sin^{2} β_{i} > 0.00832$ for $i = 1, 2, 3$ . Thus, if a first-stage cap occurs, its tail cardinality is at least $0.031 n > 0.00832 n / \sin^{2} β_{i}$ , contrary to [F3]. Consequently all three first-stage updates give the uncapped lower bound, including $∣ C_{β_{3}} ∣ \geq c_{3} r$ .

2.1F1F2F3F4step 1.1algebradischarge-contradiction∎

For each later update, [F1] gives an angle in $(0, π / 2)$ exceeding $0.1356 > 0.131$ . By [F2] and [F4], $\sin^{2} β_{i + 1} \geq \sin^{2} (0.131) > 5 (0.0816)^{2} / 2 > 2 (0.00832)$ . A cap $∣ C_{β_{i + 1}} ∣ \geq n / 2$ therefore violates [F3]. With no cap at any stage, [F1] yields $∣ C_{β_{i}} ∣ \geq c_{i} r$ for every $i$ , while $c_{i} \to \infty$ and $r \geq 1$ . This eventually exceeds $n$ , impossible for a subset of $V$ . The supposition $r > 0$ is false, so $C_{π / 2}$ is empty.

Source notes

The article’s proof uses $ε = 0.23$ and $ρ = 0.38$ for the first stage, and $ε = 0.184$ for the second. It records the lower-bound entries $0.570$ , $13.985$ , $0.131$ and $63.295$ , and bounds the later decrements by $3.326 / c_{i}$ . The exact recurrence is not restarted at a rounded table entry. The article’s displayed tail estimate is at least $0.131$ , and it uses $\sin^{2} (0.131) \geq 0.016646$ ; [F2] gives a stronger certified comparison. The prose immediately before that display says the sequence will never produce an angle with $β_{M} \geq 0.131$ , reversing the direction of the displayed lower bound. The source attribution is to the proof of Theorem 1.4 on PDF pp. 29–30; the argument above uses the stated suppliers.

Depends on

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Expander graphs are globally synchronizing · PDF pp. 29–30, Table 1 and proof of Theorem 1.4, equations (5.1)–(5.3)