ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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.

Detector resolution changes a finite inclusive event

Example

In the finite-cutoff coherent model, fix $C > 0$ and $0 < ε < Δ < Λ$ . The probability of no resolved emission is $P (Δ) = \exp ⁣ (- C \log \frac{Λ}{Δ}) .$ For any two thresholds $Δ_{1} < Δ_{2}$ in $(ε, Λ)$ , $P (Δ_{1}) < P (Δ_{2})$ . This is an exact model law, not a reported detector count or an empirical estimate.

Example references

Summing unresolved coherent emissions removes the lower cutoff

Dereziński, §§5.3–5.6

PTB Quantum Optics, Lecture 5

Facts & Assumptions

Given: $C > 0$ and $0 < ε < Δ_{1} < Δ_{2} < Λ$ in the defined coherent model.

[F1]

Summing unresolved coherent emissions removes the lower cutoff: for every threshold $Δ$ , the no-resolved-emission probability is $\exp (- C \log (Λ / Δ))$ .

[F2]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log (b / a) = \int_{a}^{b} d ν / ν$ for $0 < a < b$ .

[F3]

The exponential function is strictly increasing: the real exponential is strictly increasing.

Verification

1.1givenF1F2algebra

By [F1], $P (Δ_{j}) = \exp (- C \log (Λ / Δ_{j}))$ for $j = 1, 2$ . By [F2], $\log (Λ / Δ_{1}) - \log (Λ / Δ_{2}) = \int_{Δ_{1}}^{Δ_{2}} d ν / ν > 0$ , so $- C \log (Λ / Δ_{2}) > - C \log (Λ / Δ_{1})$ because $C > 0$ .

2.1F3step 1.1∎

Apply strict monotonicity [F3] to the two real exponents in step 1.1. It gives $P (Δ_{2}) > P (Δ_{1})$ . This compares two predictions of the specified model and asserts no empirical observation.

Detector resolution changes a finite inclusive event

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources