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Soft coherent radiation, resolution and inclusive cancellation — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Charged gauge data and SI-natural conventions
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Lp Spaces and Test-Function Conventions
- Complex Norms and Realification
- Complexification, Realification and Real Structures
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- Continuity, IVT, EVT, and Uniform Continuity
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
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- Countability Axioms and Cardinal Functions
- Density Separability and Convolution in Lᵖ
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- Distributions Test Functions and Differentiation
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- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
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- Soft coherent radiation, resolution and inclusive cancellation
- Soft-current angular norm and exact cutoff coherent model
- Spectral Measures and Borel Functional Calculus
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2 · Summary
3 · Logical flowchart
4 · Definitions, postulates, theorems and proofs
None yet.
5 · Examples, experiments and counterexamples
ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →
Detector resolution changes a finite inclusive event
Example
In the finite-cutoff coherent model, fix and . The probability of no resolved emission is For any two thresholds in , . This is an exact model law, not a reported detector count or an empirical estimate.
Example references
Facts & Assumptions
Given: and in the defined coherent model.
[F1]
Summing unresolved coherent emissions removes the lower cutoff: for every threshold , the no-resolved-emission probability is .
[F3]
The exponential function is strictly increasing: the real exponential is strictly increasing.
Verification
1.1givenF1F2algebra
By [F1], for . By [F2], , so because .
2.1F3step 1.1∎
Apply strict monotonicity [F3] to the two real exponents in step 1.1. It gives . This compares two predictions of the specified model and asserts no empirical observation.