jill-f-rankin

Introduces a two-dimensional operator diagnostic (χ, η) built from a participation operator P and a rigidity operator M, where χ tracks effective-dimension changes and η is the normalized Frobenius commutator measuring operator misalignment; the construction admits a Gibbs-like variational characterization and explicit bounds. Empirically, η consistently precedes the synchronization threshold in Kuramoto networks across topologies and sizes (outperforming pairwise transfer entropy), distinguishes regimes in driven Floquet systems, and recovers discrete-scale-invariant log-periodic ratios to within 0.3%.

FDTD simulations show that explicit log-periodic time modulation of boundary permittivity (scaling ratio b = 13/8) in a one-dimensional electromagnetic cavity imprints a reproducible discrete-scale hierarchy in the probe-point power spectrum: after subtracting the smooth power-law envelope the residual oscillates periodically in ln ω with period ln(13/8). The effect is robust across parameter sweeps, absent in undriven/harmonic/random controls, statistically significant (r ≈ 0.81, p < 0.001), and persists with coherent propagation into the cavity interior.

This paper studies whether a participation operator built from a log-energy Gaussian kernel inherits discrete scale invariance (DSI) under finite spectral truncation, proving exact shift-covariance on the bi-infinite lattice and deriving explicit finite-window normalization identities and dimension-independent bounds on the shift defect. It establishes a Lipschitz perturbation bound for a normalized commutator diagnostic and proves deterministic and distributional inheritance results (including localization lemmas and a reduction of the statistical case to standard random-matrix norm-ratio estimates), while identifying uniform concentration for a single frozen realization as an open problem.

Formal, corrected proofs establishing Lemma A2, Theorem A (tight N‑independent Frobenius bound ‖R‖_F ≤ √2·δ_Z/Z_min) and Theorem B (explicit perturbation/Lipschitz bound for η), together with a completed Participation‑Localization lemma and a finite‑window localization corollary; the document restores valid constants, removes a spurious √N factor, and reduces the Distributional Inheritance (C2) claim to a standard random‑matrix expectation bound. Key results give explicit numerical calibration at standard parameters, a positive lower bound c(s,λ) for ‖P‖_F, and a clear pathway to W₁ convergence rates O(δ_Z) or O(δ_Z/√N) conditional on ensemble estimates.

A frozen evaluation protocol for diagnosing thermoacoustic combustion instability that formalizes a general five-stage methodology for domain and operator admission, comparator verification, separate detection and state-estimation assessment, and commercial interpretation; the thermoacoustics study is the first full instantiation. It introduces a new Stage 0B Operator Structural Audit requiring pre-registered algebraic decompositions of proposed diagnostics into variance-like, coupling-like, and residual contributions, invariance checks, and an a priori falsification criterion that gates progression to operating-condition campaigns.

Introduces a two-dimensional operator diagnostic (χ, η) built from a participation operator P and a rigidity operator M, where χ tracks effective dimension and η is the normalized Frobenius commutator measuring operator-level alignment. Across Kuramoto networks η reliably peaks before the synchronization order parameter and before pairwise transfer entropy (with lower variance), and the same construction extends to Floquet-driven systems and to spectra with discrete scale invariance, accurately recovering input log-periodic ratios.

Introduces a two-dimensional operator diagnostic (χ, η) built from a participation operator P and a rigidity operator M, where χ tracks an effective dimension and η is the normalized Frobenius commutator measuring operator-level alignment. Validated on Kuramoto networks, periodically driven (Floquet) systems, and engineered discrete-scale-invariant spectra, η consistently peaks before the conventional order parameter and before pairwise transfer entropy (with lower variance) and accurately recovers log-periodic ratios, demonstrating early detection of reorganization.

Introduces a two-dimensional operator diagnostic (χ, η) built from a participation operator P and a rigidity operator M, where χ tracks changes in effective dimension and η is a normalized Frobenius commutator measuring operator misalignment; the construction admits a Gibbs-like variational characterization and explicit mathematical bounds. Empirically, η peaks before the Kuramoto synchronization threshold and before pairwise transfer entropy across extensive simulations, distinguishes driven Floquet regimes, and recovers discrete-scale-invariance ratios to ≈0.3% error, demonstrating a robust early-warning precursor of internal reorganization across diverse systems.

Introduces a two-dimensional operator diagnostic (χ, η) built from a participation operator P and a rigidity operator M: χ tracks effective-dimension changes and η is the normalized Frobenius commutator measuring operator misalignment. Empirically, η reliably peaks before conventional order-parameter and pairwise transfer-entropy signals in Kuramoto networks, distinguishes regimes in driven Floquet systems, and recovers log-periodic ratios in discrete-scale-invariant spectra with sub-percent errors.

Introduces a two-dimensional operator-based precursor diagnostic (χ, η) built from a participation operator P and a rigidity operator M, where χ tracks effective dimension changes and η is the normalized Frobenius commutator measuring operator-level misalignment that signals reorganization. The diagnostic reliably anticipates synchronization in Kuramoto networks—η peaks before the synchronization threshold in 107/109 trials and outperforms pairwise transfer entropy—and generalizes to Floquet-driven systems and discrete-scale-invariant spectra, recovering log-periodic ratios to within 0.3%.

Introduces a two-dimensional operator-based precursor diagnostic (χ, η) formed from a participation operator P and a rigidity operator M, where χ tracks effective-dimension redistribution and η is a normalized Frobenius commutator quantifying operator misalignment. Applied to Kuramoto networks, driven Floquet systems, and discrete-scale-invariant spectra, η reliably peaks before conventional synchronization thresholds (outperforming pairwise transfer entropy) and recovers log-periodic ratios to high accuracy.

Introduces a two-dimensional operator-based precursor (χ, η) built from a participation operator P and a rigidity operator M, where η is the normalized Frobenius commutator quantifying operator misalignment; the construction admits a variational (Gibbs-like) characterization and explicit bounds. Empirically, η reliably peaks before macroscopic transitions—notably preceding the Kuramoto synchronization threshold in 107/109 trials with a finite asymptotic gap and outperforming pairwise transfer entropy in timeliness and variance—and the same diagnostic distinguishes regimes in Floquet systems and recovers discrete-scale-invariant log-periodic ratios to high precision.

The paper develops a phenomenological template and factorization theorem showing that a small log-periodic modulation from discrete scale invariance (DSI) in the source unequal-time correlator propagates to the observable stochastic gravitational-wave background (SGWB), and it derives matched-filter detectability scaling for such oscillatory signatures. Applying the template to walking technicolor as a UV completion, the author finds a microphysical modulation window ε_f∈[0.04,0.18], b∈[1.7,2.8] but shows the observable signal is geometrically suppressed (|c_geom|≲0.02), rendering the predicted log-periodic feature undetectable by LISA absent additional spectral-narrowing mechanisms.

Discrete scale invariance (DSI) in the anisotropic stress of a first-order cosmological phase transition imprints a multiplicative log-periodic modulation on the stochastic gravitational-wave background, and under a short-correlation-time factorization theorem this modulation propagates to the observable spectrum at the percent level. As a concrete UV completion, walking technicolor can produce the required DSI and predicts ε∈[0.04,0.18], b∈[1.7,2.8], placing the signal in the high-SNR region for LISA and enabling enhanced matched-filter detectability.

The paper shows that discrete scale invariance (DSI) in the anisotropic stress of a first-order cosmological phase transition produces a multiplicative log-periodic modulation of the stochastic gravitational-wave background, and proves a factorization theorem that this modulation survives the unequal-time integrals to percent-level accuracy under the short-correlation-time approximation. As an explicit ultraviolet completion, it demonstrates that walking technicolor can realize the required DSI and predicts a falsifiable parameter band (epsilon ~ 0.04–0.18, b ~ 1.7–2.8) that lies in the high-SNR region for LISA.