entropyFromZ
plain-language theorem explainer
entropyFromZ supplies the explicit formula for thermodynamic entropy in terms of the Z-complexity variable and a density parameter. Researchers establishing the second law from Berry phase accumulation in Recognition Science cite it as the coarse-graining step that turns Z-monotonicity into classical entropy. The definition is a direct one-line expression applying the natural logarithm to one plus the product of its arguments.
Claim. The entropy function is defined by $S(z, d) = log(1 + z d)$, where $z$ is the coarse-grained Z-complexity and $d$ is the density parameter.
background
The ArrowOfTime module derives the directedness of time from Berry phase accumulation on the discrete R-hat lattice. Forward steps increase Z-complexity while reverse steps traverse the same loop with opposite orientation; absolute Z remains non-decreasing, producing an intrinsic before-after ordering. The eight-tick phase definition supplies the periodic angles $kπ/4$ for $k=0..7$ that underlie the Berry phase. The Anchor.Z definition maps sectors and charges to the integer ladder used for complexity counting. entropyFromZ implements the entropy_from_berry result listed in the module documentation.
proof idea
The definition is a direct one-line expression that applies Real.log to the term 1 + z * density.
why it matters
This definition is invoked by the downstream theorem entropy_monotone, which proves strict increase under z1 < z2 and thereby encodes the second law. It realizes the entropy_from_berry item in the module's key-results list, closing the link from eight-tick Berry phase to thermodynamic entropy. The construction supports the overall claim that the arrow of time emerges topologically from Z-monotonicity without importing external thermodynamics.
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papers checked against this theorem (showing 3 of 3)
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String entanglement explains 2D and 3D black hole entropy
"the thermal entropy of 2d and 3d black holes is accounted for by the string entanglement entropy between folded strings arising in the dual sine-Liouville CFT"
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Spiral arms self-quench, sustaining disk evolution through successive transients
"non-resonant spiral excitation through azimuthal forcing... cavernae... GDI"
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Modular flows extend to celestial and Klein CFTs
"Modular Hamiltonian K_B = ∫_B β(r) T_{tt}(x) d^d x with β(r) = 2π · ½(1 − r²/R²) — standard Casini-Huerta-Myers parabolic weight"