Finite Free Information Theory

• Finite Free Information Theory is a framework that replaces asymptotic free-information constructs with exact finite structures such as polynomial zeros, matrix microstates, and rectangular transforms.
• It derives precise formulations of entropy, Fisher information, and additive inequalities by transitioning from limiting laws to finite-dimensional models.
• The theory unifies diverse approaches—from real-rooted polynomial analysis to microstate geometry and singular-value techniques—addressing challenges in noncommutative stochastic and quantum systems.

to=exa_search code 天天爱彩票怎么json {"^^^^^^^^9query9 \9^^^^^^^^"Finite Free Information Theory\"^^^^^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^^^^^", "^^^^^^^^9numResults9 ^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^}:-------------</analysis to=exa_search code 天天爱彩票提现json {"^^^^^^^^9query9 arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^^^^^", "^^^^^^^^9numResults9 ^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^}quotelevanalysis to=exa_search code 彩神争霸输钱json {"^^^^^^^^9query9 arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^^^^^"Finite Free Information Inequalities\"", "^^^^^^^^9numResults9 ^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^} Finite Free Information Theory is not a single standardized formalism in the current literature. The term is used for several finite-dimensional or finite-horizon analogues of free-probabilistic, information-geometric, and correlation-theoretic constructions. In one major line, it studies entropy, Fisher information, and convolution directly on zeros of real-rooted polynomials; in another, it develops a microstate-based multivariate free information geometry for noncommutative stochastic processes; in a third, it constructs a rectangular calculus on singular values of matrix polynomials. Related uses of the phrase also occur in finite-temperature free scalar quantum field theory and in finite-horizon Maximum-Caliber formulations of information. This suggests that “Finite Free Information Theory” presently functions as an umbrella designation rather than a uniquely fixed doctrine (&&&^^^^9query9^^^^&&&, &&&^^^^9site:arxiv.org \9^^^^&&&, &&&^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&, Katsinis et al., 2019, &&&^^^^9numResults9^^^^&&&).

^^^^9site:arxiv.org \9^^^^. Scope, meanings, and recurring structures

Across its current uses, the term consistently denotes a passage from asymptotic or continuum information notions to explicitly finite objects: finite degree PRESERVED_PLACEHOLDER_^^^^9query9^^^^, finite matrix size, finite time horizon, finite temperature, or finite state space. The common pattern is the replacement of limiting laws by exact finite structures together with transforms, entropy-like functionals, Fisher-information–like quantities, or transport metrics that survive at finite scale.

Formulation	Finite object	Representative result
Real-rooted polynomial theory	zeros of monic real-rooted degree-PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9^^^^ polynomials	finite free Stam inequality and entropy power inequality
Chronological microstate geometry	matrix microstates tested by chronological formulas	geodesic concavity of PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^ and EVIPRESERVED_PLACEHOLDER_^^^^9numResults9^^^^ for heat flow
Rectangular finite free probability	singular-value polynomials for PRESERVED_PLACEHOLDER_^^^^9query9^^^^ matrices	finite rectangular PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^-transform linearizing additive convolution
Free Fisher regularity	polynomial evaluations under finite PRESERVED_PLACEHOLDER_^^^^9numResults9^^^^	Hölder CDFs, finite logarithmic energy, finite PRESERVED_PLACEHOLDER_^^^^9query9^^^^
Finite-temperature free fields	mutual information across a spatial bipartition	area law with finite classical high-PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ remnant
MaxCal finite-horizon models	path ensembles over finite horizons	information as KL deviation from the constrained MaxCal ensemble

The literature also indicates that the adjective “finite” is context dependent. In the polynomial and singular-value programs it refers to finite algebraic degree; in chronological entropy it refers to finite-$n$ matrix microstates and ultralimit constructions; in free scalar field theory it refers to finite temperature; and in the MaxCal framework it refers to a finite horizon PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9^^^^ and finite state spaces (&&&^^^^9query9^^^^&&&, &&&^^^^9site:arxiv.org \9^^^^&&&, &&&^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&, Katsinis et al., 2019, &&&^^^^9numResults9^^^^&&&).

^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^. Real-rooted polynomials, zeros, and finite free information inequalities

A central formulation of Finite Free Information Theory works on monic real-rooted polynomials

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9^^^^

viewed through their root vector PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^. For distinct roots, the score vector is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9^^^^

the finite free Fisher information is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9^^^^

and the finite free entropy is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^

Equivalently, PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9^^^^ is the normalized logarithmic energy of the zeros and PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9^^^^, so the formalism has an explicit Coulomb-gas interpretation (&&&^^^^9query9^^^^&&&).

The basic finite free additive operation is Walsh’s finite free convolution. If PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ and PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \99^^^^, then

PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9query9^^^^

It preserves real-rootedness and has the symmetric-group average representation

PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org \9^^^^

Differentiation and finite free convolution are the two primary real-rootedness–preserving operators in the theory. The reverse heat flow

PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^

connects them to a Hermite benchmark, with PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9numResults9^^^^ the variance-PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9query9^^^^ monic Hermite polynomial (&&&^^^^9query9^^^^&&&).

The central information inequalities are exact finite analogues of classical and free inequalities. The finite free Stam inequality states

PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^

The finite free entropy power inequality states

PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9numResults9^^^^

There is also monotonicity of finite free Fisher information under variance-normalized differentiation and monotonicity of finite free entropy under variance-normalized differentiation, with Hermite polynomials as the sharp equality cases. In the large-degree limit, these results recover corresponding inequalities in free probability (&&&^^^^9query9^^^^&&&).

The proofs use a new link between score vectors and Jacobians of root maps. If PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9query9^^^^ maps PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ to the roots of PRESERVED_PLACEHOLDER_^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)99^^^^, then

PRESERVED_PLACEHOLDER_^^^^9numResults9query9^^^^

A similar identity holds for the derivative root map PRESERVED_PLACEHOLDER_^^^^9numResults9site:arxiv.org \9^^^^. Together with double stochasticity of the Jacobian blocks and convexity results for hyperbolic polynomials, these identities play the role of a finite Blachman-type mechanism and yield the contraction estimates behind Stam and Fisher monotonicity (&&&^^^^9query9^^^^&&&).

A later extension studies PRESERVED_PLACEHOLDER_^^^^9numResults9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^-generalizations under finite free additive convolution. For a root vector PRESERVED_PLACEHOLDER_^^^^9numResults9numResults9^^^^ with distinct entries,

PRESERVED_PLACEHOLDER_^^^^9numResults9query9^^^^

and the PRESERVED_PLACEHOLDER_^^^^9numResults9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^-Stam deficit is

PRESERVED_PLACEHOLDER_^^^^9numResults9numResults9^^^^

At PRESERVED_PLACEHOLDER_^^^^9numResults9query9^^^^, FlowBoost numerically recovers the Hermite pair as the equality case and reveals a spectral structure for the linearized convolution map at the Hermite diagonal. Conditional on the conjecture that the singular values of the doubly stochastic coupling matrix PRESERVED_PLACEHOLDER_^^^^9numResults9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ on the mean-zero subspace are PRESERVED_PLACEHOLDER_^^^^9numResults99^^^^, independent of PRESERVED_PLACEHOLDER_^^^^9query9query9^^^^, the work derives a sharp local stability constant and an PRESERVED_PLACEHOLDER_^^^^9query9site:arxiv.org \9^^^^-uniform finite free CLT convergence rate. For PRESERVED_PLACEHOLDER_^^^^9query9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^, the Hermite pair itself violates the proposed inequality; for PRESERVED_PLACEHOLDER_^^^^9query9numResults9^^^^, the numerically extremal configurations bifurcate into non-matching pairs with bimodal root structure, converging back to the Hermite diagonal as PRESERVED_PLACEHOLDER_^^^^9query9query9^^^^ (&&&^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^&&&).

^^^^9numResults9^^^^. Chronological entropy, matrix microstates, and free information geometry

A second major program develops a finite-PRESERVED_PLACEHOLDER_^^^^9query9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^, microstate-based “Finite Free Information Theory” for noncommutative stochastic processes. Its core object is a new multivariate free entropy PRESERVED_PLACEHOLDER_^^^^9query9numResults9^^^^, defined from matrix microstates tested by chronological formulas. These formulas belong to a filtered metric language PRESERVED_PLACEHOLDER_^^^^9query9query9^^^^ with domains PRESERVED_PLACEHOLDER_^^^^9query9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^, algebra operations, a metric, trace components, and constants PRESERVED_PLACEHOLDER_^^^^9query99^^^^ coding increments of a non-selfadjoint free Brownian motion compatible with the filtration. Chronological formulas are built from quantifier-free trace-polynomial expressions in resolvents and Brownian increments, then closed under continuous connectives, partial suprema and infima in chronological order, and the heat-shift PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9query9^^^^ (&&&^^^^9site:arxiv.org \9^^^^&&&).

For PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9site:arxiv.org \9^^^^, PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^, a finite set PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9numResults9^^^^ of restricted chronological formulas, and PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9query9^^^^, the finite-PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^ microstate space is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9numResults9^^^^

The Gaussian chronological entropy is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9query9^^^^

and the Lebesgue version is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^

The construction is based on finite-PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes99^^^^ pointwise functionals PRESERVED_PLACEHOLDER_^^^^9numResults9query9^^^^ and an ultrafiber quotient PRESERVED_PLACEHOLDER_^^^^9numResults9site:arxiv.org \9^^^^ that turns a random matrix ultraproduct into a PRESERVED_PLACEHOLDER_^^^^9numResults9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^ factor with filtration and free Brownian motion (&&&^^^^9site:arxiv.org \9^^^^&&&).

The information-geometric content is expressed on the space of conditional chronological types

PRESERVED_PLACEHOLDER_^^^^9numResults9numResults9^^^^

equipped with the free Wasserstein distance

PRESERVED_PLACEHOLDER_^^^^9numResults9query9^^^^

Optimal couplings exist, and there is a chronological Monge–Kantorovich duality with convex chronologically definable predicates of the form PRESERVED_PLACEHOLDER_^^^^9numResults9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^, where PRESERVED_PLACEHOLDER_^^^^9numResults9numResults9^^^^. If PRESERVED_PLACEHOLDER_^^^^9numResults9query9^^^^ is an optimal coupling of PRESERVED_PLACEHOLDER_^^^^9numResults9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ and PRESERVED_PLACEHOLDER_^^^^9numResults99^^^^, then

PRESERVED_PLACEHOLDER_^^^^9query9query9^^^^

Thus the new entropy is displacement-concave along free Wasserstein geodesics (&&&^^^^9site:arxiv.org \9^^^^&&&).

The heat semigroup is defined on chronologically definable predicates by

PRESERVED_PLACEHOLDER_^^^^9query9site:arxiv.org \9^^^^

Its evolution satisfies the metric EVIPRESERVED_PLACEHOLDER_^^^^9query9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^ inequality

PRESERVED_PLACEHOLDER_^^^^9query9numResults9^^^^

so heat evolution is the Wasserstein gradient flow of PRESERVED_PLACEHOLDER_^^^^9query9query9^^^^ in the metric sense. The corresponding minimizing-movement scheme is the free JKO iteration

PRESERVED_PLACEHOLDER_^^^^9query9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^

This places the framework squarely inside an information-geometric and optimal-transport paradigm (&&&^^^^9site:arxiv.org \9^^^^&&&).

The same theory also proves a true chain rule under iterated conditioning,

PRESERVED_PLACEHOLDER_^^^^9query9numResults9^^^^

invariance under definable closure of the conditioning variable, and a stochastic-control representation. For PRESERVED_PLACEHOLDER_^^^^9query9query9^^^^, the ultralimit pressure is

PRESERVED_PLACEHOLDER_^^^^9query9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^

with adapted bounded controls PRESERVED_PLACEHOLDER_^^^^9query99^^^^. The Gaussian chronological entropy admits the variational formula

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9query9^^^^

This gives the theory a finite-PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9site:arxiv.org \9^^^^/control-theoretic bridge analogous, in the paper’s terms, to Borell/Boué–Dupuis and Schrödinger-bridge/Benamou–Brenier structures (&&&^^^^9site:arxiv.org \9^^^^&&&).

^^^^9query9^^^^. Free Fisher information, distributional regularity, and operator-algebraic rigidity

A related strand studies the consequences of finite free Fisher information itself. In the tracial, non-microstates setting, if PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^ has finite PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9numResults9^^^^ and PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9query9^^^^ is a selfadjoint noncommutative polynomial of degree PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^, then the cumulative distribution function PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9numResults9^^^^ is Hölder continuous with explicit exponent

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9query9^^^^

If PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ admits Lipschitz conjugate variables, the exponent improves to

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \99^^^^

For linear polynomials, the exponent $n$^^^^9query9^^^^ under finite $n$^^^^9site:arxiv.org \9^^^^ is optimal, while Lipschitz conjugate variables yield Lipschitz continuity and hence absolute continuity with bounded density. These regularity estimates imply finite logarithmic energy and therefore finite non-microstates free entropy $n$^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^ for every selfadjoint nonconstant polynomial $n$^^^^9numResults9^^^^, partially resolving a conjecture of Charlesworth–Shlyakhtenko under the stronger assumption $n$^^^^9query9^^^^ (&&&^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9numResults9^^^^&&&).

The same work supplies an explicit route from weak convergence to rates in Kolmogorov distance. If the limiting law has Hölder CDF with exponent $n$^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^ and the Cauchy transforms satisfy suitable strip estimates, then

$n$^^^^9numResults9^^^^

with a compact-support refinement

$n$^^^^9query9^^^^

Applications include convergence in Kolmogorov distance for polynomial eigenvalue laws of Gibbs ensembles and explicit GUE rates such as

$n$^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^

in the block-GUE semi-flat case, and

$n$9

for selfadjoint polynomial GUE models (&&&^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9numResults9^^^^&&&).

In the non-tracial setting, finite free Fisher information for eigenvectors of a modular operator has much stronger structural consequences. If PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^ is generated by a finite selfadjoint set PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org \9^^^^ of eigenoperators of PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^ with finite free Fisher information, then

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9numResults9^^^^

In particular, PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^ is a PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^ factor, and if PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9numResults9^^^^ is the closed subgroup generated by the eigenvalues of PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^, then PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ is a factor of type PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query99^^^^, PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9query9^^^^ (PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9site:arxiv.org \9^^^^), or PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^ according as PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9numResults9^^^^, PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9query9^^^^, or PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^. The same hypotheses imply that PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9numResults9^^^^ does not have property PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9query9^^^^, and if PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ is type PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org \99^^^^ with PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9query9^^^^, then PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org \9^^^^ is full (&&&^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&).

These results rely on PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^-modular derivations, conjugate variables entire for the modular group, Dirichlet forms on the centralizer, contraction resolvents, and non-tracial PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9numResults9^^^^-homology estimates. In this line of work, finite free Fisher information is not merely a regularity parameter; it becomes a rigidity hypothesis controlling diffuseness, factoriality, and type classification (&&&^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&).

^^^^9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^. Rectangular finite free probability and singular-value information

Another formulation replaces eigenvalues by singular values of rectangular matrices. For PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9query9^^^^ with PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^, the basic polynomial is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9numResults9^^^^

Equivalently, if PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9query9^^^^ is a univariate polynomial with nonnegative roots, its rectangular polynomial extension of order PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ is PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)99^^^^. This encodes the squared singular values of PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9query9^^^^ directly at finite dimension (&&&^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&).

The finite rectangular additive convolution is defined by averaging over bi-orthogonal rotations:

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9site:arxiv.org \9^^^^

where PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^, PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9numResults9^^^^ are Haar and PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9query9^^^^. Setting PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^ yields a univariate real-rooted polynomial with nonnegative roots. The operation is bilinear, associative, and preserves real-rootedness. The theory also gives an explicit coefficient formula and a differential-operator expression for the convolution (&&&^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&).

This program introduces finite analogues of classical rectangular free-probability transforms. A PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9numResults9^^^^-point random variable PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9query9^^^^ is associated to a polynomial PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ so that its power sums are fixed linear functionals of the coefficients, and the finite rectangular PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults99^^^^-transform is defined by

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^

It linearizes finite rectangular additive convolution:

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org \9^^^^

A modified finite PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^-transform converges, as PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9numResults9^^^^, to the classical rectangular free PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^-transform, so the finite theory converges to asymptotic rectangular free probability (&&&^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&).

The same framework produces explicit finite-dimensional LLN and CLT analogues for polynomials. After PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^ rescaling, iterated finite rectangular convolution converges to the zero polynomial, while after PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9numResults9^^^^ rescaling it converges, up to scaling, to a generalized Laguerre polynomial. In transform terms,

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^

This identifies generalized Laguerre polynomials as the Gaussian analogues of the theory (&&&^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&).

Because the roots of the convolved polynomial approximate squared singular values, the construction interfaces directly with spectral information functionals. For a nonnegative spectral law PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^, the Shannon transform is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query99^^^^

and for a PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9query9^^^^ positive matrix it is PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9site:arxiv.org \9^^^^. In the finite rectangular framework, one computes PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^, extracts its nonnegative roots PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9numResults9^^^^, and evaluates

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9query9^^^^

The paper explicitly notes that a finite multiplicative convolution or finite PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^-transform is not introduced there, so multiplicative problems remain asymptotic in the current rectangular theory (&&&^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&).

^^^^9numResults9^^^^. Physical and finite-horizon extensions

In free scalar quantum field theory at finite temperature, Finite Free Information Theory refers to the mutual information across a spatial bipartition. For a free real scalar in PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9numResults9^^^^ dimensions, the mutual information between a region PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9query9^^^^ and its complement PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ satisfies an area law

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes99^^^^

with PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9query9^^^^ computable in an inverse-mass expansion. The high-temperature limit is finite and purely classical in origin:

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9site:arxiv.org \9^^^^

For general coupled harmonic systems, the high-PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^ expansion has no PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9numResults9^^^^ term, and in the field-theory setting this cancellation is recovered once a uniform angular cutoff is imposed. The low-temperature correction is exponentially small, proportional to PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9query9^^^^, while the PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^ limit matches the classical thermal calculation (Katsinis et al., 2019). A complementary numerical study of free scalar theory on concentric spherical shells established the same area-law behavior, reporting in the massless PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9numResults9^^^^-dimensional spherical setup

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9query9^^^^

thereby exhibiting the finite classical remnant directly in the lattice computation (&&&^^^^9numResults9numResults9^^^^&&&).

A conceptually different finite-horizon usage defines information as deviation from a constrained Maximum-Caliber ensemble. For a one-step transition network, if PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^ is the MaxCal input marginal

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9numResults99^^^^

then information is

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^

For longer horizons, large-deviation theory yields

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org \9^^^^

Within this framework, Integrated Information Theory repertoires are re-derived from constrained MaxEnt/MaxCal posteriors, and partition-based integration is defined by

PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^

The same paper states a duality to active-inference free-energy functionals and gives CLT- and LDP-based predictive-coding reductions in which PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9numResults9^^^^ becomes a Bayesian-surprise or transition-accuracy term (&&&^^^^9numResults9^^^^&&&).

The open-problem landscape is correspondingly plural. In the chronological microstate program, open directions include ultrafilter independence of PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^, a chronological free Fisher functional, and EVIPRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org arXiv (Jekel, 14 Apr 2026) free information geometry model theory noncommutative stochastic processes9^^^^ for free Ornstein–Uhlenbeck semigroups (&&&^^^^9site:arxiv.org \9^^^^&&&). In the polynomial program, the main unresolved issues are proof of the dyadic singular-value conjecture for PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9numResults9^^^^, proof of the PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9query9^^^^-Stam inequality for all PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^, uniqueness of the Hermite equality case at PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9query99^^^^, and characterization of the PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9query9^^^^ bifurcating extremizers (&&&^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^&&&). In the rectangular program, the missing finite multiplicative analogue remains explicit (&&&^^^^9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&). In the non-tracial Fisher-information line, an important question is how much of the factoriality and fullness theory survives without an eigenoperator generating set (&&&^^^^9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9^^^^&&&). In the MaxCal/IIT/FEP program, open questions include explicit non-equilibrium current formulations, non-Gaussian regimes beyond the CLT, and empirical validation relating PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9site:arxiv.org \9^^^^, PRESERVED_PLACEHOLDER_^^^^9site:arxiv.org \9site:arxiv.org arXiv (Garza-Vargas et al., 17 Feb 2026) \9 arXiv 2026 (Garza-Vargas et al., 17 Feb 2026, Jekel, 14 Apr 2026, Hashemi, 13 Apr 2026, Kearney, 3 May 2026)9^^^^, and PCI (&&&^^^^9numResults9^^^^&&&).

Taken together, these programs show that Finite Free Information Theory currently names a family of finite analogues rather than a single invariant package. What unifies them is a shared strategy: replace asymptotic free-information objects by exact finite structures—zeros, microstate sets, singular-value polynomials, path ensembles, or finite-temperature Gaussian subsystems—and recover entropy, Fisher information, transport, or mutual-information phenomena before passing to a limit.