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arxiv: 2604.11464 · v1 · submitted 2026-04-13 · 🧮 math-ph · math.MP· nlin.AO

Passive two-plateau relaxation from Tricomi confluent hypergeometric kernels

Marc Tudela-Pi , Ivano Colombaro This is my paper

Pith reviewed 2026-05-10 14:58 UTC · model grok-4.3

classification 🧮 math-ph math.MPnlin.AO
keywords anomalous relaxationTricomi confluent hypergeometric functionStieltjes representationpassivityCole-Cole responsedielectric spectroscopyelectrochemical impedanceFoster approximation
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The pith

Tricomi confluent hypergeometric kernels with Moebius normalisation produce passive two-plateau relaxation responses that admit nonnegative Stieltjes spectral measures.

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

The paper develops a non-fractional model for anomalous relaxation observed in disordered solids, soft matter, and electrochemical systems. It uses the Tricomi confluent hypergeometric function together with a bounded Moebius transform to enforce independent low-frequency and high-frequency plateaux while keeping a broad transition region. This construction recovers the Debye and Cole-Cole responses exactly as special cases and extends them to asymmetric dispersive laws. For an admissible range of parameters the resulting function possesses a Stieltjes integral representation whose density is nonnegative, which directly implies complete monotonicity, passivity, and causality. The same representation supplies a systematic route to finite-dimensional passive approximations suitable for circuit simulation and state-space modelling.

Core claim

For an admissible parameter range, the bounded Tricomi-Moebius relaxation block admits a Stieltjes representation whose spectral density is nonnegative; this representation guarantees complete monotonicity, passivity, causality, and direct compatibility with standard passive circuit and state-space realisations.

What carries the argument

The bounded Tricomi-Moebius kernel, a normalised confluent hypergeometric function that satisfies prescribed low- and high-frequency plateaux and carries a nonnegative Stieltjes spectral measure.

If this is right

  • The model supplies first-order state-space realisations with positive poles and residues via Gauss-Stieltjes discretisation.
  • Rational Foster-type approximations converge for both moderate-memory and long-tail regimes.
  • Multi-block mixtures improve complex-domain fitting accuracy on broadband dielectric spectra and battery impedance data relative to classical Cole-Cole baselines.
  • The framework remains fully compatible with standard passive network synthesis and finite-dimensional simulation tools.

Where Pith is reading between the lines

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

  • The same kernel structure may be reusable for other memory kernels that require two distinct asymptotic exponents.
  • Because the realisations are passive by construction, they can be embedded directly into larger circuit models without introducing artificial energy sources.
  • The explicit Stieltjes measure opens a route to stochastic interpretations of the underlying relaxation processes.

Load-bearing premise

There exists a nonempty set of parameters for which the Tricomi kernel plus Moebius normalisation simultaneously meets the two-plateau boundary conditions and produces a nonnegative spectral density.

What would settle it

A numerical or analytic counter-example inside the claimed admissible range in which the Stieltjes density takes a negative value at some frequency.

read the original abstract

Anomalous relaxation with memory spectra arises in disordered solids, soft matter, biological tissues and electrochemical interfaces. Fractional-order models capture broad power-law behaviour efficiently, but they can obscure spectral structure and are not always convenient for passive realisation or finite-dimensional simulation. We introduce a non-fractional passive framework based on the Tricomi confluent hypergeometric function, combined with a bounded Moebius normalisation that enforces prescribed low-frequency and high-frequency plateaux while preserving a broad dispersive transition. The resulting family contains the Debye and Cole-Cole responses as exact subcases, while extending them to asymmetric two-plateau dispersive laws with independently tunable low- and high-frequency exponents. For an admissible parameter range, we prove that the bounded block admits a Stieltjes representation with nonnegative spectral density, implying complete monotonicity, passivity, causality and compatibility with standard circuit and state-space descriptions. Building on this structure, we derive a passive Gauss-Stieltjes discretisation leading to Foster-type rational approximations and first-order state-space realisations with positive poles and residues. Numerical experiments show convergence of these finite-dimensional approximations across moderate-memory and long-tail regimes, enabling passive reduced-order representations of broad-memory responses. The framework is then validated on broadband dielectric data and battery electrochemical impedance spectra. In tissues, multi-block Tricomi mixtures improve complex-domain fitting accuracy relative to classical Cole-Cole baselines while preserving interpretable modal structure.

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

2 major / 2 minor

Summary. The manuscript introduces a family of relaxation kernels based on the Tricomi confluent hypergeometric function subject to a bounded Möbius normalisation chosen to enforce independent low- and high-frequency plateaux. It asserts that, inside an admissible parameter range, the resulting response admits a Stieltjes representation with nonnegative spectral density (hence complete monotonicity and passivity), contains the Debye and Cole-Cole models as exact subcases, and yields passive rational approximations via a Gauss-Stieltjes discretisation. The claims are supported by an analytical argument, numerical convergence studies, and fits to dielectric and battery impedance data.

Significance. If the central nonnegativity result holds for a nonempty admissible set, the construction supplies a tunable, non-fractional, passive alternative to fractional-order models that is directly compatible with circuit synthesis and state-space simulation. The explicit recovery of classical models and the provision of first-order realisations with positive poles/residues are concrete strengths that could facilitate reduced-order modelling in soft-matter and electrochemical applications.

major comments (2)
  1. [main theorem / admissible-range statement] The main theorem (the statement that begins 'For an admissible parameter range, we prove...') invokes an admissible set of parameters (α,β,γ,…) that must simultaneously satisfy the two-plateau boundary conditions and guarantee a nonnegative Stieltjes measure. No explicit system of inequalities or interval is supplied, nor is a proof given that this set is nonempty. This is load-bearing for the strongest claim.
  2. [Stieltjes-representation derivation] The Möbius normalisation is constructed to enforce the prescribed plateaux; the subsequent claim that the Stieltjes density remains nonnegative therefore depends on showing that the normalisation does not violate the sign condition inside the same parameter set. The manuscript does not isolate this compatibility step or provide an independent verification (e.g., via the inverse Stieltjes transform or an integral representation) that survives the plateau constraints.
minor comments (2)
  1. [§2 / parameter definitions] Notation for the low- and high-frequency exponents is introduced without a compact table relating them to the Tricomi parameters; a small summary table would improve readability.
  2. [numerical-experiments section] Several figures display fitted spectra but omit the numerical values of the admissible-range parameters used; adding these values (or the interval bounds) would allow readers to reproduce the examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised concern the explicitness of the admissible parameter range and the compatibility verification for the Stieltjes representation. We address each below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [main theorem / admissible-range statement] The main theorem (the statement that begins 'For an admissible parameter range, we prove...') invokes an admissible set of parameters (α,β,γ,…) that must simultaneously satisfy the two-plateau boundary conditions and guarantee a nonnegative Stieltjes measure. No explicit system of inequalities or interval is supplied, nor is a proof given that this set is nonempty. This is load-bearing for the strongest claim.

    Authors: We agree that the admissible parameter range requires an explicit definition to make the main theorem fully rigorous. In the revised manuscript we will supply the complete system of inequalities on the parameters (α, β, γ, …) that simultaneously enforce the two-plateau boundary conditions and guarantee nonnegativity of the Stieltjes measure. We will also add an explicit demonstration that the set is nonempty, either by exhibiting concrete admissible parameter values or by deriving the relevant bounds from the existing analytical arguments. revision: yes

  2. Referee: [Stieltjes-representation derivation] The Möbius normalisation is constructed to enforce the prescribed plateaux; the subsequent claim that the Stieltjes density remains nonnegative therefore depends on showing that the normalisation does not violate the sign condition inside the same parameter set. The manuscript does not isolate this compatibility step or provide an independent verification (e.g., via the inverse Stieltjes transform or an integral representation) that survives the plateau constraints.

    Authors: We acknowledge that the compatibility between the Möbius normalisation and the nonnegativity of the spectral density must be isolated and verified explicitly. The revised manuscript will contain a dedicated step that demonstrates this compatibility inside the admissible parameter range. The verification will use an independent method, such as the inverse Stieltjes transform or a suitable integral representation, that explicitly accounts for and survives the plateau constraints imposed by the normalisation. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from known special function via explicit normalisation and asserted analytic proof

full rationale

The central claim rests on starting with the Tricomi confluent hypergeometric kernel, applying a Möbius transformation chosen to enforce the two prescribed plateaux, and then proving (rather than assuming or fitting) that the resulting kernel admits a Stieltjes representation with nonnegative measure inside a nonempty admissible parameter set. No equation reduces by construction to its own inputs, no parameter is fitted to data and then relabeled as a prediction, and no load-bearing step is justified solely by self-citation. The construction is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; free parameters and axioms are inferred from the stated construction rather than explicit listing in the provided text.

free parameters (2)
  • low- and high-frequency exponents
    Independently tunable parameters controlling the power-law slopes on either side of the dispersive transition.
  • Tricomi kernel parameters
    Parameters of the confluent hypergeometric function that shape the memory spectrum.
axioms (2)
  • standard math The Tricomi confluent hypergeometric function satisfies the required analytic continuation and asymptotic properties for the chosen parameter range.
    Invoked implicitly when defining the kernel and proving the Stieltjes representation.
  • domain assumption A bounded Moebius transformation can be chosen that simultaneously enforces the low- and high-frequency plateaux without destroying nonnegativity of the spectral measure.
    Central to the two-plateau construction and the passivity proof.

pith-pipeline@v0.9.0 · 5558 in / 1516 out tokens · 56008 ms · 2026-05-10T14:58:14.667916+00:00 · methodology

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Reference graph

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