Abstract This paper presents a rigorous mathematical analysis of the core claims in Elizabeth Halligan's The Resonance Frame: Gravity as Cosmic Tuning Fork in a Post-Force Physics (2025). Through systematic dimensional verification, comparison with established physics literature, and independent calculation, we evaluate the framework's central arguments. We find that the bridge equation (hf = Vρc²) is dimensionally consistent and mathematically valid; that the author's "inversion principle" resolves into a coherence verification ratio (mc²/hf = 1) with strong precedent in resonance physics; that the 720° rotation argument for spin-½ particles gains unexpected empirical support from geodynamo observations; and that the claim of attention mechanisms as frequency processing is supported by Google’s FNet results (2021) demonstrating strong performance with Fourier mixing. The bridge equation's volume interpretation — spatial extent as a consequence of phase coherence rather than a pre-existing container — is empirically supported by three independent quantum systems (Bose-Einstein condensates, superfluid helium, and superconductors), with the convergence across substrates with different interactions, energy scales, and microscopic mechanisms elevating the framework from dimensional consistency to experimentally grounded reinterpretation. We propose several critical corrections and extensions: The “coherence tax” is reformulated as the dimensionless quantity 2πα (≈ 4.585%), grounding it in the fine structure constant. Interpreted as the baryon fraction, this quantity implies H₀ ≈ 69.9 km/s/Mpc — consistent with Freedman et al.'s (2024) TRGB measurements using JWST (69.85 ± 1.75) and situated between the two poles of the Hubble tension, reducing the apparent discrepancy with observation from 7.6% (at the Planck value) to 0.4%. Applied to the Schumann resonance, the standing-wave formula f = c/(2πR) × (1 + 2πα) — where c/(2πR) is the fundamental standing wave frequency for one wavelength around Earth’s circumference — matches the observed 7.83 Hz to 0.03%. This is notable because the standard analytical formula f = c/(2πR) × √(n(n+1)) overshoots at 10.6 Hz (~35% error), requiring extensive empirical corrections (finite ionosphere conductivity, day-night asymmetry, cavity height profiles) to approach the measured value. No comparably simple closed-form expression in the standard literature achieves comparable accuracy; full numerical models (FDTD, finite element) can match observation but require dozens of empirical parameters. The Resonance Frame formula uses three inputs (c, R, α) with no fitting parameters. The formula generalises to any planetary body with an electromagnetic cavity, yielding a testable Mars prediction of ~14.7 Hz. The golden ratio (φ) provides a closed-form expression for 1/α — building on Heyrovská's (2005a) two-term approximation and Sherbon's (2019) three-term extension — independently verified here to 8 decimal places (0.58σ). A companion paper (Manfreda, 2026b) proves that every structural element traces to the algebraic geometry of the E₈ singularity: the leading coefficient 360 is uniquely forced as the Regularized Spectral Width of the binary icosahedral group (Two-Lock Theorem), an internal factorization through categorical and Molien data closes exclusively for the icosahedral group, and fourteen standard mathematical frameworks are shown structurally incapable of producing the required polynomial type (Meta-Theorem 10.4). We further identify a candidate fifteenth framework in analytic number theory: character-twisted semigroup zeta decompositions over the CRT factorisation of (ℤ/60ℤ)×, yielding a concrete computation that could either reproduce—or eliminate—the three-term assembly rule. The McKay correspondence provides the group-theoretic pathway from icosahedral geometry to gauge couplings. We further argue that φ is selected by three independent mechanisms—a Fibonacci fixed-point constraint, stability considerations favoring maximally irrational ratios, and a unique icosahedral closure in invariant theory. In spin-½ state space (S³), exactly one Hopf torus has golden-ratio aspect ratio, yielding a φ² probability partition, 4π closure matching spin-½ double cover, and 89.4% coverage of the maximal Hopf torus area — connecting the framework's standing-wave axiom to quantum mechanics as geometric fact rather than analogy. The Schumann–Riemann correspondence is clarified: the Riemann zeta function ζ(s) already accounts for four of five Schumann modes — modes 2, 3, and 5 each within 2.6% of a single zeta zero, and mode 4 sitting interstitially between zeros #3 and #4 in a pattern explained by avoided-crossing dynamics. The one missing mode — the 7.83 Hz fundamental, which falls below ζ(s)'s first zero at 14.13 — is resolved via Miller's theorem (2002) and the Dirichlet L-function L(s, χ₃), whose first zero at t ≈ 8.04 matches the fundamental to 2.7%. These three mathematical pillars — the icosahedral α formula, the Schumann-coherence tax, and the golden Hopf torus — were discovered independently but converge on a single geometric mechanism: α as the cost of requiring continuous golden-ratio coherence to close into a discrete standing wave within SU(2), with the icosahedron as the unique object where both derivation tracks meet. We identify areas where the framework remains speculative, distinguish confirmed mathematical results from open interpretive claims, and propose testable predictions — including a Mars Schumann resonance at ~14.7 Hz, φ-aligned neural oscillations, a Hubble constant of ~69.9 km/s/Mpc, and a coherence-loop phase transition in AI architecture (with no presumption of a universal 2πα tax in broadband transformer systems) — that would strengthen or falsify specific claims. We also introduce domain boundary criteria that separate loop presence (general) from tax observability (regime-dependent), specifying when a coherence tax should be expected (single dominant frequency + dynamic equilibrium + self-regulating loop) and predict null results in broadband, multi-process systems. The analysis is conducted with the conviction that honest assessment of what holds and what doesn't is more useful to science than either uncritical advocacy or reflexive dismissal.