Coherence Invariants Across Physical Scales A Preliminary, Domain-by-Domain Assessment

We investigate a candidate, scale-free coherence invariantΠ ≡ℏ τmd2 ,built from an inertial (or effective modal) mass m, a characteristic coherence or separationlength d, and a coherence or correlation time τ . The original motivation was the possibilitythat a single dimensionless parameter might organize coherence behaviour across a widerange of physical regimes. In this work we take a more modest and technically explicitposition.First, P we present the theoretical framework that motivates Π: a phase–ledger C(t) =i Φi(t) whose conservation, ˙C = 0, leads to local quadratic invariant manifolds in native,dimensionless coordinates. We show, via Buckingham–Π analysis, that for systems whoserelevant variables are (m, d, τ ; ℏ) there is a unique dimensionless combination of this form.Second, we test the coherence invariant hypothesis on a set of heterogeneous butconcrete examples drawn from nine “rungs”: quantum decoherence and matter-waveinterferometry, nonlocal/Bell-type experiments, near- and far-field electromagnetism,astrophysical lines, condensates, cosmic expansion, and dynamic “poke-and-return” motionin atom interferometers and Bose–Einstein condensates. The goal is not to claim a fullyestablished universal law, but to examine where the proposed invariant is (and is not)consistent with data under clearly stated choices of (m, d, τ ).Third, we confront a key limitation: in many platforms there are multiple, physicallyplausible definitions of d and τ . Different choices can shift Π by many orders of magnitude.We therefore emphasize the risk of post-hoc scale selection (“curve fitting”) and propose acanonical scale protocol for future tests: (i) state all plausible definitions of (m, d, τ ) for agiven dataset, (ii) compute Π for each, and (iii) treat the choice that minimizes deviationfrom a hypothesized invariant surface as an empirical question, not a built-in assumption.With these caveats, we find that:• Near-field matter-wave interferometers analysed using the Talbot time as the native τadmit a natural mapping in which an onset band in Π near O(10−1) appears plausible.• Cold-atom and BEC transport experiments, when expressed in terms of their trapscales and transport times, yield very large Π (deep coherence), consistent with theintuitive classification of strongly regulated motion.• In other domains (Bell tests, far-field EM, astrophysical lines, cosmology) the presentevidence is mixed or inconclusive: sensible alternative choices of (m, d, τ ) cannot yetbe ruled out, and the existence of a single cross-domain onset constant is thereforeunproven.iThe main contribution of this paper is thus not a final claim of universality but aclearly defined hypothesis, a normalization and falsifiability protocol, and a set of workedexamples illustrating both the promise and the pitfalls of coherence invariants acrossphysical scales. We highlight specific experiments and re-analyses—especially in matterwaveinterferometry, collisional decoherence, condensates, and atom interferometer/BECdynamics—that could rapidly confirm or falsify key aspects of the framework.Contents