Landauer, Relative Entropy, and Fisher Length in Capacity Geometry

Abstract

Landauer, Relative Entropy, and Fisher Length in Capacity Geometry. This paper is a standalone companion note adjacent to the broader SGOC programme. It gives a thermodynamic translation of the capacity-geometry slow scalar Sc=−log⁡(1−R2)S_c = -\log(1 - R^2)Sc=−log(1−R2), working in a lane that is deliberately narrower than the programme’s quantum, inverse-probe, or many-body notes. Its central theorem-lane points are: the thermodynamically meaningful object is the normalized surprisal Sp=−log⁡p,p=e−Sc/Z[Sc]S_p = -\log p,\quad p = e^{-S_c}/Z[S_c]Sp=−logp,p=e−Sc/Z[Sc], not bare ScS_cSc, and this normalized lift is gauge invariant under additive shifts of ScS_cSc; Shannon entropy and relative entropy become endpoint ledgers of mean normalized capacity; the exact state-space Fisher audit is the weighted companion Ix[p]=Z[Sc]−1F0[Sc;Ω]I_x[p] = Z[S_c]^{-1}\mathcal{F}_0[S_c;\Omega]Ix[p]=Z[Sc]−1F0[Sc;Ω], so the precise probability-lane Fisher object is weighted and should not be conflated with the unweighted geometric audits REG0\mathrm{REG}_0REG0 or REG\mathrm{REG}REG; the control-space Fisher metric Iij(λ)=Cov⁡pλ(∂iSc,∂jSc)\mathcal{I}_{ij}(\lambda) = \operatorname{Cov}_{p_\lambda}(\partial_i S_c,\partial_j S_c)Iij(λ)=Covpλ(∂iSc,∂jSc) induces the natural protocol Fisher length and protocol action along a control path; no universal static single-state identity collapses entropy or Landauer cost into REG0\mathrm{REG}_0REG0 or F0\mathcal{F}_0F0. The paper is explicit about scope. It does not claim a new derivation of Landauer’s principle from the SGOC ledger alone, a completed read/write asymmetry theorem, a bulk-boundary theorem, or a many-body completion. Its claim is narrower and sharper: the thermodynamic lane of the capacity scalar can be written exactly in normalized capacity variables, with a clean separation between endpoint entropy, state-space Fisher localization, and protocol-space Fisher geometry. KeywordsLandauer principle; relative entropy; Fisher information; thermodynamic length; capacity geometry; normalized surprisal; protocol Fisher metric; entropy dissipation; gauge invariance; control-space geometry DOI10.5281/zenodo.19186309 Related SGOC programme records for interested readers:The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge, The Spectral Geometry of Coherence (SGOC), Volume Two, Part 1A, v2.0. DOI: 10.5281/zenodo.19156316 Localisation Rather than Saturation in Capacity Geometry: REG Dissipation, Forced Balance Laws, and a Restricted Ricci Bridge, The Spectral Geometry of Coherence (SGOC), Volume Two, Part 6A, v2.0. DOI: 10.5281/zenodo.19174434