LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Deterministic ratio error and Taylor remainder

Statement

Define $R : R \times (0, \infty) \to R$ by $R (a, c) : = 1 + a / c$ . Here $c$ is only the positive denominator variable in this lemma, not the speed of light. For real $δ a, δ c$ and nonnegative bounds $ϵ_{a}, ϵ_{c}$ satisfying $∣ δ a ∣ \leq ϵ_{a}$ and $∣ δ c ∣ \leq ϵ_{c} < c$ , both denominators are positive and

$∣ R (a + δ a, c + δ c) - R (a, c) ∣ \leq \frac{c ϵ_{a} + ∣ a ∣ ϵ_{c}}{c (c - ϵ_{c})} .$

For the first-order statement, write $x = (a, c)$ , $h = (u, v)$ , and $∥ h ∥_{2} : = \sqrt{u^{2} + v^{2}}$ . Suppose the segment ${x + t h : 0 \leq t \leq 1}$ lies in the closed rectangle $Q = [A_{-}, A_{+}] \times [c_{-}, c_{+}]$ , where $A_{-} \leq A_{+}$ and $0 < c_{-} \leq c_{+}$ . Define $A_{*} : = \max (∣ A_{-} ∣, ∣ A_{+} ∣)$ , $K_{Q} : = (A_{*} + c_{+} / 2) / c_{-}^{3}$ , and the real-linear map $L_{x} (u, v) : = u / c - a v / c^{2}$ . The total derivative of $R$ at $x$ means a real-linear map $L$ such that $\lim_{h \to 0} ∣ R (x + h) - R (x) - L (h) ∣ / ∥ h ∥_{2} = 0$ . Then $L_{x}$ is the total derivative, and

$∣ R (x + h) - R (x) - L_{x} (h) ∣ \leq K_{Q} ∥ h ∥_{2}^{2} .$

Facts & Assumptions

Given: Let $c > 0$ , $ϵ_{a}, ϵ_{c} \geq 0$ , and $x, h, Q$ satisfy the conditions in the statement. All variables are real; the inequalities are deterministic interval conditions, with no probability space or sampling assumption.

[A1]

$R (a, c) = 1 + a / c$ , $c > 0$ , $∣ δ a ∣ \leq ϵ_{a}$ , $∣ δ c ∣ \leq ϵ_{c} < c$ , and $ϵ_{a}, ϵ_{c} \geq 0$ .

[A2]

For the derivative bound, $x = (a, c)$ , $h = (u, v)$ , $∥ h ∥_{2}^{2} = u^{2} + v^{2}$ , the segment $x + t h$ lies in $Q = [A_{-}, A_{+}] \times [c_{-}, c_{+}]$ , $A_{-} \leq A_{+}$ , and $0 < c_{-} \leq c_{+}$ . Put $A_{*} = \max (∣ A_{-} ∣, ∣ A_{+} ∣)$ and $K_{Q} = (A_{*} + c_{+} / 2) / c_{-}^{3}$ .

Proof

1.1A1algebra

From [A1], $c + δ c \geq c - ϵ_{c} > 0$ . Thus both ratio values are defined, and subtracting their fractions gives $R (a + δ a, c + δ c) - R (a, c) = \frac{c δ a - a δ c}{c (c + δ c)}$ .

1.2A2algebra

For $η \to 0$ with $c + η > 0$ , the coordinate difference quotients are $\frac{R (a + η, c) - R (a, c)}{η} = \frac{1}{c}$ and $\frac{R (a, c + η) - R (a, c)}{η} = - \frac{a}{c (c + η)}$ . Their limits give the candidate linear map $L_{x} (u, v) = u / c - a v / c^{2}$ ; the total-derivative conclusion follows from the full remainder estimate below.

1.3A2algebra

Direct subtraction gives $R (x + h) - R (x) - L_{x} (h) = \frac{a v^{2} - c u v}{c^{2} (c + v)}$ .

2.1A1step 1.1algebra

The triangle inequality gives $∣ c δ a - a δ c ∣ \leq c ϵ_{a} + ∣ a ∣ ϵ_{c}$ , while [A1] gives $c (c + δ c) \geq c (c - ϵ_{c}) > 0$ . Dividing these positive-denominator bounds proves the displayed deterministic ratio-error inequality.

2.2A2step 1.3algebra

Since the segment lies in $Q$ , $c, c + v \geq c_{-}$ , $c \leq c_{+}$ , and $∣ a ∣ \leq A_{*}$ . Also $2 ∣ u v ∣ \leq u^{2} + v^{2}$ , because $(∣ u ∣ - ∣ v ∣)^{2} \geq 0$ . Therefore $∣ a v^{2} - c u v ∣ \leq A_{*} v^{2} + c_{+} ∣ u v ∣ \leq (A_{*} + c_{+} / 2) (u^{2} + v^{2})$ , and $c^{2} (c + v) \geq c_{-}^{3}$ . Combining these estimates with the exact remainder in step 1.3 gives $∣ R (x + h) - R (x) - L_{x} (h) ∣ \leq K_{Q} ∥ h ∥_{2}^{2}$ .

3.1A2step 2.2algebra∎

For fixed $x = (a, c)$ with $c > 0$ , choose $Q = [a - 1, a + 1] \times [c / 2, 3 c / 2]$ , so $x$ is interior and its lower second-coordinate bound is positive. For every sufficiently small $h$ , both endpoints lie in $Q$ , and convexity places the segment there. Step 2.2 then gives $\frac{∣ R (x + h) - R (x) - L_{x} (h) ∣}{∥ h ∥_{2}} \leq K_{Q} ∥ h ∥_{2} \to 0$ . By the stated definition, $D R (x) = L_{x}$ .

Remarks

The two local bounds are deterministic consequences of positive denominators and algebra. The proof uses no statistical uncertainty model, independence assumption, Taylor theorem, chain rule, compactness theorem, or probability supplier. Penning-trap frequency ratios motivate the application; the exact finite-error and quadratic-remainder bounds are proved here and are not attributed to an external source.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.