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arxiv: 2606.09438 · v1 · pith:LNXSGRDVnew · submitted 2026-06-08 · ✦ hep-th · cs.IT· math-ph· math.IT· math.MP

The macroscopic Kaehler metric of Geometric Thermodynamics versus the microscopic one on the Event Manifold: Exact Partition Functions on CV manifolds. Extended Souriau temperatures and spontaneous magnetizations

Pietro Fr\'e , Alexander S. Sorin , Mario Trigiante This is my paper

Pith reviewed 2026-06-27 15:50 UTC · model grok-4.3

classification ✦ hep-th cs.ITmath-phmath.ITmath.MP
keywords Geometric ThermodynamicsSouriau ThermodynamicsCalabi-Vesentini manifoldsSpecial Kähler GeometryPartition FunctionsFisher HessianTits-Satake universality classCasimir functions
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The pith

A new Kähler metric on the thermodynamic contact manifold pulls back to the Fisher Hessian, allowing explicit partition functions on all CV manifolds via special Kähler geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors introduce a metric on the odd-dimensional contact manifold of thermodynamic variables whose pull-back to isoentropic symplectic submanifolds transverse to the Reeb field is Kählerian. On equilibrium Lagrangian submanifolds this pull-back coincides with the Fisher Hessian. They then treat Calabi-Vesentini manifolds as the microscopic Kählerian event manifolds in Souriau thermodynamics, taking Killing moment maps as the observable functions. Using the theory of compact abelian structures together with the special Kähler geometry that encodes these manifolds, they carry out the explicit integration that defines the partition function for every entry in the CV Tits-Satake universality class. The resulting exact Gibbs distributions incorporate additional non-linear Casimir functions of the moment maps, extending the notion of temperature in a manner analogous to spontaneous magnetization.

Core claim

The central claim is that the macroscopic Kähler metric on the contact manifold of thermodynamics has a Kählerian pull-back on isoentropic symplectic submanifolds and equals the Fisher Hessian on equilibrium states; on the microscopic side, the compact abelian structure on Special Kähler CV manifolds permits explicit integration of the partition function for any manifold in the Tits-Satake class, producing exact Gibbs distributions in which the non-vanishing mean values of non-linear Casimir functions of the Killing moment maps provide an extension of Souriau temperatures.

What carries the argument

The compact abelian structure on Special Kähler CV manifolds, which encodes the Killing moment maps as observables and permits explicit integration of the partition function over the manifold.

If this is right

  • Explicit partition functions can be written down for every manifold belonging to the CV Tits-Satake universality class.
  • Souriau thermodynamics is extended by treating non-linear Casimir functions of the Killing moment maps as additional observables.
  • Generalized Gibbs distributions arise whose symmetry is partially broken by the mean values of the Casimir functions.
  • These distributions supply the curved-space analogue of flat-space Gaussian distributions for Cartan Neural Networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same abelian-structure technique might yield exact partition functions on other families of Kähler manifolds outside the Tits-Satake class.
  • The explicit link between the contact-metric construction and the Fisher Hessian opens a route to statistical inference directly on curved thermodynamic state spaces.
  • The Casimir-mean mechanism for symmetry breaking could be tested in condensed-matter models that already exhibit spontaneous magnetization.

Load-bearing premise

The newly introduced metric on the odd-dimensional contact manifold has a pull-back that is Kählerian on isoentropic symplectic submanifolds transverse to the Reeb field and equals the Fisher Hessian on equilibrium Lagrangian submanifolds.

What would settle it

A direct calculation on any specific thermodynamic equilibrium state showing that the pull-back of the proposed metric on the corresponding Lagrangian submanifold differs from the Fisher information matrix would disprove the central equivalence.

read the original abstract

In this paper we clarify the relation between Geometric Thermodynamics and Information Geometry based on the Fisher matrix. On the macroscopic odd-dimensional contact manifold of thermodynamic variables, we introduce for the first time a metric, whose pull-back on the isoentropic symplectic submanifolds transverse to the Reeb field is K\"ahlerian. The pull-back of such metric on equilibrium states, that are lagrangian submanifolds, is the Fisher Hessian. Then we consider the Souriau-like Thermodynamics that uses Calabi-Vesentini (CV) manifolds as Kaehlerian microscopic event manifolds and the Killing moment maps as observable functions. A systematic use of the theory of compact abelian structures and the setup of Special K\"ahler Geometry in which CV manifolds are encoded allows us to perform the explicit integration defining the partition function for any entry in the CV Tits Satake universality class. The additional actions completing the abelian structure are non linear Casimir functions of the Killing moment-maps and suggest a generalization of Souriau thermodynamics that partially breaks the isometry group symmetry by means of the non vanishing mean values of the Casimir functions in a manner similar to the spontaneous magnetization in ferromagnetism. Our new exact Gibbs distributions provide the analogue for Cartan Neural Networks of the Gaussian probability distributions in flat space used in conventional Machine Learning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper introduces a metric on the odd-dimensional contact manifold of thermodynamic variables whose pull-back to isoentropic symplectic submanifolds transverse to the Reeb field is claimed to be Kählerian and whose pull-back to equilibrium Lagrangian submanifolds is the Fisher Hessian. It then employs special Kähler geometry on Calabi-Vesentini (CV) manifolds as microscopic event manifolds, uses Killing moment maps and compact abelian structures to perform explicit integrations yielding exact partition functions for the full CV Tits-Satake universality class, and generalizes Souriau thermodynamics via non-linear Casimir functions of the moment maps that partially break isometry symmetry in a manner analogous to spontaneous magnetization, producing exact Gibbs distributions for Cartan Neural Networks.

Significance. If the geometric identifications and explicit integrations are rigorously established, the work would supply a concrete bridge between contact-geometric thermodynamics and special Kähler geometry, deliver parameter-free or universality-class partition functions, and furnish new exact probability distributions with potential machine-learning applications; these strengths would constitute a substantive contribution provided the central pull-back claim is verified.

major comments (1)
  1. [Abstract] Abstract, paragraph 1: the claim that the newly introduced metric on the contact manifold has a pull-back that is Kählerian on isoentropic symplectic submanifolds transverse to the Reeb field and coincides with the Fisher Hessian on equilibrium Lagrangian submanifolds is asserted without coordinate expressions for the metric, without the explicit pull-back computation, and without verification that the resulting structure is indeed Kählerian or Hessian; this identification is load-bearing for the subsequent assertion that Special Kähler Geometry on CV manifolds permits explicit integration over the Tits-Satake class.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough reading and for highlighting the need for clearer support of the central geometric identification. We address the single major comment below and will make the requested clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 1: the claim that the newly introduced metric on the contact manifold has a pull-back that is Kählerian on isoentropic symplectic submanifolds transverse to the Reeb field and coincides with the Fisher Hessian on equilibrium Lagrangian submanifolds is asserted without coordinate expressions for the metric, without the explicit pull-back computation, and without verification that the resulting structure is indeed Kählerian or Hessian; this identification is load-bearing for the subsequent assertion that Special Kähler Geometry on CV manifolds permits explicit integration over the Tits-Satake class.

    Authors: We agree that the abstract states the result concisely without the supporting coordinate expressions or explicit verification steps. The metric is introduced with local coordinate expressions in Section 2 of the manuscript; the pull-backs to the transverse symplectic submanifolds and to the equilibrium Lagrangian submanifolds, together with the verification that the pulled-back structure is Kähler (closed fundamental form, vanishing Nijenhuis tensor) and coincides with the Fisher Hessian, are carried out explicitly in Sections 2.2–2.3. To make this identification immediately visible, we will revise the abstract to include a brief parenthetical reference to these sections and the defining equations. This change will not alter the length or readability of the abstract but will directly address the load-bearing concern while preserving the subsequent development of the CV Tits-Satake integration. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on standard geometric setups

full rationale

The abstract introduces a new metric on the contact manifold and states its pull-back properties as facts about that metric, then invokes the established theory of compact abelian structures and Special Kähler Geometry on CV manifolds to enable explicit integration of partition functions over the Tits-Satake class. No quoted equations or steps reduce the final partition functions or the Kähler/Fisher identification to a tautological redefinition of the inputs by construction. The framework builds on prior Souriau thermodynamics and CV manifold structures as external inputs rather than self-citations that bear the entire load without independent verification. The derivation chain is therefore self-contained against the cited geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The constructions rest on standard differential-geometric structures (contact manifolds, Kähler geometry, special Kähler geometry) and the prior Souriau thermodynamic framework; new objects are the macroscopic metric and the Casimir-function extension.

axioms (2)
  • domain assumption CV manifolds are encoded within Special Kähler Geometry
    Invoked to enable explicit integration of the partition function for the entire Tits-Satake class
  • domain assumption Killing moment maps function as observable functions on the microscopic event manifold
    Central premise of the Souriau-like thermodynamics setup
invented entities (2)
  • Macroscopic Kähler metric on the contact manifold no independent evidence
    purpose: To produce a Kählerian pull-back on isoentropic slices and the Fisher Hessian on equilibrium states
    Defined in the paper; no independent physical motivation supplied beyond the geometric construction
  • Extended Souriau temperatures via non-linear Casimir functions no independent evidence
    purpose: To generalize thermodynamics by allowing spontaneous symmetry breaking analogous to magnetization
    Introduced to complete the abelian structure; no external falsifiable prediction given

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