LemmaStatement: AI-generatedProof: 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.

Strict numerical budgets for the two-stage PSD amplification

Statement

Let $0 < a \leq 1 / 5$ , $- 1 < c_{-} \leq 0 \leq c_{+}$ , $s = 1 + c_{-}$ , $w = c_{+} - c_{-} > 0$ , $B = 1 + c_{+} + a$ , $T = 1 + 4 c_{+} - 3 c_{-} = s + 4 w$ , and $L_{0} = \log (B / (2 a))$ . Suppose $32 a T L_{0} / s^{2} < 1$ . Put $δ = s / (2 w)$ , $q = 1 + δ$ , and $K = ⌈ L_{0} / \log q ⌉$ . Then $a / s < 1 / 32$ , $a < 1 / 32$ , $q^{K} \geq B / (2 a)$ , and $K \arcsin (8 a / s) < π / 8$ , $\sum_{j = K}^{\infty} (π / 2) 4 B / (s q^{j}) \leq 4 π a (s + 2 w) / s^{2} < π / 8$ .

Facts & Assumptions

[F1]

The principal inverse sine and the bounds $\arcsin v \leq π v / 2$ for $0 \leq v \leq 1$ and $\log (1 + v) \geq 2 v / (2 + v)$ for $v \geq 0$ are supplied by Angular order and elementary inverse-trigonometric bounds. The logarithmic bound is strict when $v > 0$ .

[F2]

The ceiling $⌈ x ⌉$ is the least integer at least $x$ . For $q > 1$ , $b > 0$ and an integer $k \geq 0$ , $\sum_{j = k}^{\infty} b q^{- j} = b q^{- k} q / (q - 1)$ (Geometric growth, truncation and total decrement budgets).

[F3]

The natural logarithm is defined on the positive reals, is strictly increasing, satisfies $\log 1 = 0$ , and obeys $\log (x y) = \log x + \log y$ for $x, y > 0$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

Proof

Given: The data and hypotheses of the Statement.

Goal: Establish the two parameter bounds, the lower bound on $q^{K}$ , and both strict angle budgets in the Statement.

1.1givenF1F3algebra

Since $B \geq 1 + a$ and $a \leq 1 / 5$ , $B / (2 a) \geq 3$ . Applying the strict logarithmic bound of [F1] at $v = 2$ gives $\log 3 > 1$ , and [F3] gives $L_{0} \geq \log 3 > 1$ . Also $0 < s \leq 1$ , $w > 0$ , and $T = s + 4 w > s$ . Hence $32 a / s < 32 a T L_{0} / s^{2} < 1$ , so $a / s < 1 / 32$ and $a < 1 / 32$ . In particular $0 < 8 a / s < 1 / 4$ .

2.1F2F3step 1.1algebra

We have $δ > 0$ , $q > 1$ , and $\log q > 0$ by [F3]. Put $x = L_{0} / \log q > 0$ . By [F2], $K = ⌈ x ⌉$ is a positive integer and $K \geq x$ . Applying the product law in [F3] repeatedly to the $K$ factors of $q^{K}$ gives $\log (q^{K}) = K \log q \geq L_{0} = \log (B / (2 a))$ . Strict logarithmic monotonicity therefore gives $q^{K} \geq B / (2 a)$ .

3.1givenF1F2step 1.1step 2.1algebra

By [F1] at $v = δ$ , $\log q \geq 2 δ / (2 + δ)$ . If $0 < x \leq 1$ , the least-integer characterization gives $K = 1$ , so $32 a K / s < 1$ by step 1.1. If $x > 1$ , minimality of $K$ implies $K - 1 < x$ , hence $K < x + 1 \leq 2 x$ . Therefore $K \leq 2 L_{0} / \log q \leq (2 + δ) L_{0} / δ = T L_{0} / s$ , and the stated strict hypothesis again gives $32 a K / s < 1$ . Since $0 < 8 a / s < 1 / 4$ , the inverse-sine estimate of [F1] applies and yields $K \arcsin (8 a / s) \leq 4 π a K / s < π / 8$ .

4.1givenF2step 1.1step 2.1step 3.1algebra∎

Apply the geometric-series identity in [F2] with $b = 2 π B / s > 0$ and $k = K \geq 1$ . It gives $\sum_{j = K}^{\infty} (π / 2) 4 B / (s q^{j}) = 2 π B q / (s δ q^{K}) \leq 4 π a q / (s δ) = 4 π a (s + 2 w) / s^{2}$ , using $q - 1 = δ$ and step 2.1. Since $w > 0$ and $L_{0} > 1$ , $s + 2 w < T < T L_{0}$ . Thus the last expression is strictly less than $4 π a T L_{0} / s^{2} < π / 8$ . Together with steps 1.1, 2.1 and 3.1 this proves every stated conclusion.

Depends on

Used by · 1

Positive-reference expanders have only synchronized critical PSD phases

Dependency tree · 2 levels

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