Logarithmic Spectral Structure from the Chronoflux Continuity Law

This paper replaces my 15 April 2025 publication, Logarithmic Spectral Structure from the Chronoflux Continuity Law. The earlier paper identified the local spectral obstruction and the prime generated logarithmic lattice correctly, but its derivation of the amplitude law and its treatment of the infinite remainder were not mathematically secure. Version 2.0 rebuilds the argument from the beginning and records those corrections openly.

I begin with the covariant Chronoflux continuity law and a declared barotropic closure. Linearisation about a smooth positive background produces the local scalar perturbation equation. Its normalised quadratic action determines the local acoustic principal tensor and the corresponding effective metric seen by the perturbation. On a compact spatial section, Weyl growth then proves that purely local differential closure produces polynomial spectral growth rather than frequencies of the form log n.

The logarithmic sector is introduced separately through the scale coordinate

σ = log x

Translation in σ is generated by a self adjoint scale operator. Arithmetic support is then declared on the prime generated additive semigroup

Λ = {log n : n ∈ N}

The equality between this semigroup and the integer logarithmic lattice follows directly from unique prime factorisation. I do not claim that covariant continuity alone generates the integers. Continuity supplies the transport structure, while arithmetic admissibility selects the discrete scale support.

The amplitude law is reconstructed from balance under the logarithmic measure. Since

dx = x dσ

equal reconstruction weight per logarithmic interval gives the critical coefficient

a_n ∝ n^(-1/2)

This replaces the incorrect Hilbert normalisation argument used in the earlier paper.

The logarithmic phases are generated by the positive number operator N through

A = log N

so that

exp(-itA)e_n = n^(-it)e_n

The phase law therefore follows from a defined self adjoint generator rather than from identifying the logarithmic coordinate with its Fourier dual.

The unwindowed critical series is not absolutely convergent and is not assigned a false decaying tail. The admissible reconstruction is instead finite or windowed,

Z_W(t) = Σ W_n n^(-1/2) exp(-it log n)

with the window chosen inside a declared summability class.

The central new theorem concerns finite recoverability. Distinct logarithmic frequencies produce a Vandermonde observation map. A finite coefficient vector is therefore uniquely recoverable from a sufficient set of exact temporal samples. Stability is controlled by the smallest singular value of that observation matrix. I define a recoverability ratio which distinguishes algebraic invertibility from a reconstruction that is experimentally or numerically well conditioned.

The result is organised through the latest Chronoflux order of Admissibility, Recoverability and Persistence. Admissibility determines which scale modes, windows and observation maps belong to the active branch. Recoverability determines whether the hidden coefficients can be reconstructed from the available observations. Persistence records which modes survive repeated contractive refinement without falling below the recoverable threshold.

The paper keeps the local acoustic metric and the arithmetic scale spectrum separate. The recovered spacetime perturbation metric does not acquire prime or integer structure merely because the reconstruction coefficients are supported on log n. No claim concerning the Riemann hypothesis, the zero distribution of the zeta function or a proof of any external number theoretic conjecture is made.

Version 2.0 contains the complete corrected manuscript, seven original figures, the improvement ledger, the pre-submission checklist, reproducible figure data and the source used to generate the numerical illustrations.

This version supersedes the 2025 paper as the active mathematical authority for logarithmic spectral reconstruction within the Chronoflux framework.