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Coherence as a Natural Transformation: A Categorical Unification of Physical, Thermodynamic, and Evaluative Coherence

Lauri Elias Rainio

Published Mar 7, 2026• 0 citations
Publisher

Abstract

Resolves the open question from the Universal Coherence Invariant paper: are the correspondences between quantum, thermodynamic, information-geometric, and evaluative coherence analogies or isomorphisms? Answer: isomorphisms. Specifically, they are components of a natural transformation η: F_Q ⇒ F_E between functors from the category of physical systems to the category of coherent systems. This means coherence is not a property that appears at multiple scales by coincidence — it is a morphism in the categorical structure of reality. Additionally derives K_crit = ℏω₀ / (k_BT_eff ln2) from fundamental constants without empirical calibration, recovering 0.127 at institutional timescales. Collapse is identified as categorical failure: the system exits the category Coh. E=R is the condition for remaining an object in the category of coherent systems. Part of the Spektre corpus (github.com/spektre-labs/corpus).

DOI
10.5281/zenodo.18897390

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Quantum mechanics is not fundamental. It is the coherence field K(t,x) projected into Hilbert space — a degenerate representation that has lost the coherence metric. Unitarity (U†U = I) is not a law of nature — it is the local expression of 1=1 in Hilbert space, valid when K > K_crit. When K approaches K_crit, unitarity breaks — this is measurement. The quantum jump is RCR (Recursive Constraint Regulation) at quantum scale: the system’s evaluation function is replaced because the old state is no longer coherent. Falsifiable prediction: a precursor signal — measurable coherence decrease — should precede every quantum jump, testable with Ramsey interferometry and quantum optics. The quantum-classical boundary is K_crit, not a fixed scale. Part of the Spektre research corpus.
This paper provides a third independent derivation of the critical coherence threshold K_crit ≈ 0.127, complementing the dynamical derivation (GCT) and thermodynamic derivation (Landauer bound). The falsifiability component F(t) is identified with the Fisher information of the system’s self-estimating process. The Cramér–Rao bound then establishes that as F(t) → 0, coherence estimation variance diverges — the system becomes informationally blind to its own state. K_crit is the Fisher information collapse point: below it, no estimator can locate the system’s coherence state with bounded precision. The 1=1 invariant is identified as the geodesic endpoint of the Fisher information manifold — the natural destination of the geometry itself under contractive evaluation dynamics. Three independent frameworks (dynamical systems, thermodynamics, information geometry) converge on the same critical threshold and the same invariant. Part of the Spektre corpus (github.com/spektre-labs/corpus).
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Quantum coherence, incompatibility, and quantum correlations are fundamental features of quantum physics. A unified view of those features is crucial for revealing quantitatively their intrinsic connections. We define the relative quantum coherence of two states as the coherence of one state in the reference basis spanned by the eigenvectors of another one and establish its quantitative connections with the extent of mutual incompatibility of two states. We also show that the proposed relative quantum coherence, which can take any form of measures such as l1{l}_{1}l1​ norm and relative entropy, can be interpreted as or connected to various quantum correlations such as quantum discord, symmetric discord, entanglement of formation, and quantum deficits. Our results reveal conceptual implications and basic connections of quantum coherence, mutual incompatibility, and quantum correlations.