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Ideal linear Penning mode-frequency invariant — Examples

1 · Prerequisites

Ideal linear Penning mode-frequency invariant

2 · Summary

This companion gives one hypothetical deterministic ratio-error calculation. Its interval inputs are stipulated and are not experimental observations or statistical uncertainty estimates.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Hypothetical calibration error interval

Example

Take the stipulated nominal values $a = 1 H z$ , $c = 100 H z$ , and interval bounds $ϵ_{a} = 0.1 H z$ , $ϵ_{c} = 1 H z$ . The nominal ratio $R = 1 + a / c$ is dimensionless, and the exact absolute error bound is $11 / 9900$ . These inputs are hypothetical interval values: they are not measurements, error bars, a standard error, or a confidence interval, and no reported Fan et al. data are used.

Facts & Assumptions

Given: The four numerical values in the example are stipulated. Fix the formal unit $u = 1 H z$ and model frequency quantities as $F_{u} = {x u : x \in R}$ , with addition $(x u) + (y u) = (x + y) u$ and real scalar multiplication $t (x u) = (t x) u$ . The numerical-coordinate map is $n_{u} : F_{u} \to R$ , $n_{u} (x u) = x$ . Define $x u \leq y u$ to mean $x \leq y$ , $∣ x u ∣ : = ∣ x ∣ u$ , and the dimensionless quotient $(x u) / (y u) : = x / y$ for $y > 0$ . Thus the quantity expression $1 + a / c$ means $R (n_{u} (a), n_{u} (c))$ , where $R$ is the real function in [F1].

[A1]

$a = 1 u$ , $c = 100 u$ , $ϵ_{a} = \frac{1}{10} u$ , and $ϵ_{c} = 1 u$ . Write $α = n_{u} (a) = 1$ , $γ = n_{u} (c) = 100$ , $e_{a} = n_{u} (ϵ_{a}) = 1 / 10$ , and $e_{c} = n_{u} (ϵ_{c}) = 1$ . Allowed perturbations are $δ a = s u$ , $δ c = t u$ , with $s, t \in R$ , $∣ s ∣ \leq e_{a}$ , and $∣ t ∣ \leq e_{c} < γ$ ; these are exactly the stipulated quantity interval conditions under the given coordinate map.

[F1]

Deterministic ratio error and Taylor remainder (Statement): the current carrier is a draft lemma with a complete local proof but no independent acceptance stamp. Its exact bound is conditional input to this numerical verification, not a result certified by this example. It defines $R : R \times (0, \infty) \to R$ by $R (x, y) = 1 + x / y$ and gives, for real $α, s, t$ , $γ > 0$ , and nonnegative $e_{a}, e_{c}$ , the bound $∣ R (α + s, γ + t) - R (α, γ) ∣ \leq (γ e_{a} + ∣ α ∣ e_{c}) / (γ (γ - e_{c}))$ whenever $∣ s ∣ \leq e_{a}$ and $∣ t ∣ \leq e_{c} < γ$ .

Verification

1.1givenA1F1algebra

Apply [F1] to the real coordinates in [A1]: $γ + t \geq 100 - 1 = 99 > 0$ , $γ e_{a} + ∣ α ∣ e_{c} = 100 \cdot (1 / 10) + 1 \cdot 1 = 11$ , and $γ (γ - e_{c}) = 100 \cdot 99 = 9900$ . Hence every allowed perturbation satisfies $∣ R (α + s, γ + t) - R (α, γ) ∣ \leq 11 / 9900$ . By the given quotient definition, these are precisely the perturbed and nominal quantity ratios $1 + (a + δ a) / (c + δ c)$ and $1 + a / c$ .

2.1givenA1step 1.1algebra∎

The nominal value is $R (α, γ) = 1 + 1 / 100 = 101 / 100 = 1.01$ . The coordinate and quotient definitions make both the ratio and its bound dimensionless. The bound is a deterministic calculation for the stated intervals only, with no empirical or statistical interpretation.