Alphabeta MathRadiation, spin and code interfaces
Supply the regulated photon model, thermal moments, spectral kernels and conditional ULE contracts together with angular-momentum and dipole identities. Use the same spin conventions for absorption-emission code criteria, and specify the bosonic, inner/outer encoding and continuous-monitoring interfaces needed for platform comparisons.
1 · Prerequisites
- ULE kernel adjoint symmetry and rate identities
- Stationary correlation spectra and ULE kernel regularity
- The Axiom of Countable Choice (AC_ω)
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions
- The L1 transform is bounded and uniformly continuous
- Integrable real and complex functions, and their integrals
- The Lebesgue integral is linear on L¹(μ)
- The modulus of an integral is bounded by the integral of the modulus
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- A measurable function between measurable spaces
- The product sigma-algebra and its finite iterates
- Arithmetic and lattice operations preserve measurability whenever they are defined
- Complex Lp classes and Euclidean test-function conventions
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system
- Integral invariance under measure-preserving maps
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0
- A nonnegative integral over a null set vanishes
- Additivity of the nonnegative Lebesgue integral
- The Borel product of ℝᵐ and ℝⁿ is the Borel sigma-algebra of ℝᵐ⁺ⁿ
- Order-one absorption–emission tensor error span
- Exact recovery implies the Knill–Laflamme condition
- An explicit syndrome recovery corrects a Knill–Laflamme error family
- Tensor-family components equal coherent Clebsch–Gordan transition maps
- Finite spanning-list compression and channel qualification
- Real coefficient arrays for the signed spin-four isometry
- Signed mixed binomial sums cancel at order one
- Signed diagonal binomial moments agree at order one
- Class of codes correcting absorptions and emissions · §II, PDF p. 2, Eqs. (1)–(3); §III A, Theorem III.1, conditions (C1)–(C4), PDF pp. 3–4
- Finite-dimensional matrix exponential
- Unitarity and differentiation of the finite Hermitian matrix exponential
- Every finite-dimensional normed space is Banach
- Strongly measurable Banach-valued function
- Bochner-integrable function
- Bochner integrability criterion
- Bochner integral norm inequality
- Density operator
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Convergent jump integrals and Hermitian Lamb operators
Lemma
A first-moment bound for the bath correlation convolution
Lemma
Mass and first moment of the ULE scalar envelope
Lemma
Finite signed-code KL product test
Lemma
Static Bohr-frequency evaluation of the jump integral
Lemma