Cross-Domain Convergence of Lower Bounds Near the Planck Scale

Papers 54, 55, and 56 of this series applied a uniform inversion method to established physical equations: demanding that each equation produce a finite output under conditions that would otherwise yield a divergence, and solving for the minimum time, mass, or length required. Each paper was independently self-contained and submitted separately. The present paper synthesises and compares the three results.Twenty-one independent physical equations were inverted across the three papers, drawn from six domains: general relativity, quantum mechanics, wave mechanics, special relativity, thermodynamics, and information theory. Seventeen of the twenty-one (81%) yield results in the range 0.17–2.00 times the corresponding Planck unit — without any Planck unit being assumed as input. The convergence is not uniform across domains: quantum mechanics (5/5 equations, 100%), general relativity (3/3, 100%), GR+QM boundary (2/2, 100%), special relativity (1/1, 100%), and information theory (2/2, 100%) all converge completely. Thermodynamics converges partially (1/3), and electromagnetism does not converge (0/2), for reasons that are physically explained: the electromagnetic results are modulated by the fine structure constant α ≈ 1/137, and the Hawking evaporation timescale involves a cubic power of mass that amplifies deviations.The pattern of convergence and non-convergence is itself informative. The domains that converge completely (QM, GR, GR+QM, SR, information theory) are those in which the fundamental coupling is gravitational or quantum mechanical. The domains that deviate (EM, partial thermodynamics) involve couplings modulated by dimensionless constants (α, 8π) that set their natural scales far from the Planck scale. This distinction supports a physical interpretation of the convergence: the Planck units emerge as a candidate lower-bound scale precisely for those physical processes governed by the quantum-gravitational coupling — the combination of ℏ, G, and c — and not for processes governed by electromagnetic or higher-order couplings.The central conclusion of the trilogy — that the Planck units (tₚ, mₚ, lₚ) emerge as strong candidates for a characteristic lower-bound scale suggested by the observed convergence pattern, derived from the requirement of physical finiteness without assuming them as input — is substantially strengthened by the cross-domain comparison: the convergence is unlikely to be explained by a random distribution across scales, and is consistent with a systematic pattern across multiple physical domains. The analysis does not establish a unique or fundamental minimum scale. The results do not constitute a proof of a unique minimum scale.