A Constraint-Modulated Rate Law Outperforming VFT and Its Modern Alternatives Across Canonical Glass-Forming Liquids

Christian E. Precker; Debra S. Gavant

arxiv: 2511.16791 · v3 · submitted 2025-11-20 · ❄️ cond-mat.mtrl-sci · cond-mat.stat-mech

A Constraint-Modulated Rate Law Outperforming VFT and Its Modern Alternatives Across Canonical Glass-Forming Liquids

Debra S. Gavant , Christian E. Precker This is my paper

Pith reviewed 2026-05-17 20:03 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.stat-mech

keywords glass-forming liquidsviscosity rate lawVFT equationconstraint loadglass transitionmodel comparisonAIC penalization

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The pith

A constraint-modulated rate law for viscosity in glass-forming liquids outperforms VFT and its alternatives after parameter penalization.

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

The paper presents Continuous Present Actualization with Constraints (CPA+C), a new rate law for viscosity in glass-forming liquids. It is based on the idea that configurational states are resolved based on current structural constraints that tighten with decreasing temperature. When tested on datasets for ortho-terphenyl, salol, and boron trioxide, CPA+C provides better fits than the Vogel-Fulcher-Tammann (VFT) equation and modern alternatives like MYEGA and Avramov-Milchev on four of five cases, even after using AIC to penalize for its two extra parameters. The improvements are significant, reaching Delta-AIC values of 141, and leave-one-out cross-validation confirms better predictive performance.

Core claim

The central discovery is that introducing a temperature-dependent constraint load C(T) to modulate the rate law allows the model to capture the narrowing of configurational access near the glass transition, leading to superior agreement with experimental viscosity data compared to divergence-based or other free-energy surface models.

What carries the argument

The temperature-dependent constraint load C(T) that quantifies the narrowing of configurational access in the liquid as temperature decreases toward the glass transition.

Load-bearing premise

Each configurational state is resolved independently under its current structural constraints rather than as a point on a predetermined free-energy surface.

What would settle it

A direct test would be to apply CPA+C to a new high-precision viscosity dataset spanning a wide temperature range and check whether the AIC-penalized fit remains superior and whether the constraint load C(T) exhibits a clear sigmoid-like transition.

Figures

Figures reproduced from arXiv: 2511.16791 by Christian E. Precker, Debra S. Gavant.

**Figure 2.** Figure 2: Validation on independent OTP dataset (Plazek et al., 1994). The CPA + C model demonstrates robustness across independent experimental trials. Notably, in this dataset, the CPA + C model slightly outperforms the VFT reference (AIC = −33.81 vs −33.42) with R2 = 0.9932, indicating that the constraint-based mechanism effectively captures the viscosity evolution without requiring the VFT singularity [PITH_FU… view at source ↗

**Figure 3.** Figure 3: Extension to hydrogen-bonded networks (Glycerol-Water mixtures, Kumar et al., 1994). The CPA + C formulation maintains statistical parity (R2 ≈ 0.9981) across three distinct mixture concentrations. This suggests the constraint-driven actualization mechanism is generalizable beyond van der Waals liquids to systems with different bonding topologies. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

read the original abstract

A constraint-modulated rate law for viscosity in glass-forming liquids is reported. The key assumption is that each configurational state is resolved independently under its current structural constraints, rather than as a point on a predetermined free-energy surface. This approach, termed Continuous Present Actualization (CPA), requires a rate law that tracks resolution cost as it changes with temperature. The formulation, CPA + Constraint (CPA+C), introduces a temperature-dependent constraint load C(T) that quantifies how configurational access narrows as a liquid approaches the glass transition. Tested against VFT and its modern divergence-free successors MYEGA and Avramov-Milchev on canonical datasets for ortho-terphenyl, salol, and boron trioxide, CPA+C outperforms all three on four of five datasets after full AIC penalization for its two additional parameters, with margins reaching Delta-AIC = 141. On two datasets the baseline kinetic parameter vanishes, reducing the effective model to four free parameters. BIC confirms the same ranking. A smooth sigmoid variant fits equally well or better. The single exception occurs on the narrowest-range dataset, where the temperature range is too narrow for the constraint transition to separate the model from simpler alternatives. Leave-one-out cross-validation on salol (n=95) confirms that CPA+C generalizes to held-out data with mean prediction error 3x lower than the next-best model.

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.

Desk Editor's Note private letter to a colleague

CPA+C delivers stronger empirical fits than VFT and its successors on most canonical datasets after penalizing for extra parameters, backed by cross-validation.

read the letter

The main thing to know is that this CPA+C rate law fits viscosity data for ortho-terphenyl, salol, and boron trioxide better than VFT, MYEGA, and Avramov-Milchev on four of five sets, with AIC gains up to 141 even after adding two parameters, and leave-one-out checks on the salol set show roughly three times lower prediction error on held-out points. The single loss comes on the narrowest temperature window, which lines up with the model's own logic about needing enough range to resolve the constraint shift. What is new is the explicit temperature-dependent constraint load C(T) built around the Continuous Present Actualization premise that each configurational state is handled independently under its instantaneous structural limits rather than on a fixed free-energy landscape. The paper does a clean job of running full AIC/BIC rankings plus cross-validation instead of just claiming better curves, and it notes that a smooth sigmoid version of C(T) works at least as well. That adds a bit of practical flexibility. The soft spots are the two added parameters and the fact that the baseline kinetic term drops out on two datasets, which suggests the functional shape is doing some data-driven work on top of the physical story. The core assumption about independent resolution is stated clearly but the fits themselves do not test it directly; they mainly show that the resulting form captures the data shape. Nothing in the reported numbers looks internally inconsistent or circular beyond the usual extra-parameter cost. This paper is aimed at people who fit viscosity or relaxation data in glass-forming liquids and want a divergence-free alternative that still tracks the slowing near the transition. Anyone who already works with these specific canonical liquids or needs a practical upgrade to existing rate laws will find it directly useful. The statistical comparisons and physical motivation are solid enough to justify sending it to a serious referee for closer scrutiny of the derivation and data handling.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a constraint-modulated viscosity rate law called Continuous Present Actualization plus Constraint (CPA+C) for glass-forming liquids. It rests on the premise that configurational states are resolved independently under their instantaneous structural constraints rather than along a fixed free-energy landscape. The model adds a temperature-dependent constraint load C(T) and two extra parameters relative to VFT; after AIC/BIC penalization it is reported to outperform VFT, MYEGA and Avramov-Milchev on four of five canonical datasets (ortho-terphenyl, salol, B2O3), with Delta-AIC reaching 141, and to generalize better under leave-one-out cross-validation on the salol set (n=95). The single exception occurs on the narrowest temperature-range dataset.

Significance. If the reported AIC/BIC rankings and LOO-CV results hold after explicit verification of the functional form of C(T), the work supplies a statistically competitive, divergence-free description of viscosity that could improve empirical modeling of canonical glass-formers. The explicit use of AIC/BIC penalization for the two additional parameters together with leave-one-out cross-validation on a large dataset (n=95) constitutes a strength in model-selection rigor that is not always present in the glass-transition literature.

major comments (2)

[Results section, paragraph on parameter reduction] Results section, paragraph on parameter reduction: on two of the five datasets the baseline kinetic parameter vanishes, reducing the effective free-parameter count from six to four. Because the central claim relies on 'full AIC penalization for its two additional parameters,' the AIC/BIC values must be recomputed with the actual effective parameter count for those datasets; otherwise the reported Delta-AIC margins (up to 141) are not directly comparable to the three-parameter reference models.
[Model derivation (section introducing CPA+C)] Model derivation (section introducing CPA+C): the explicit functional form of the temperature-dependent constraint load C(T) and the precise manner in which it encodes the 'independent resolution under current structural constraints' assumption are load-bearing for both the physical interpretation and the statistical claim. Without the closed-form expression and its derivation, it is impossible to judge whether C(T) is independently motivated or partly tuned to the viscosity data.

minor comments (2)

[Abstract and main-text model section] The abstract states that a smooth sigmoid variant 'fits equally well or better'; the explicit equation, parameter values, and AIC comparison for this variant should be moved from any supplementary material into the main text or a dedicated table.
[Data and methods] Table or figure listing the five datasets should include the exact temperature ranges and number of data points for each liquid so that the statement 'narrowest-range dataset' can be verified quantitatively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive overall assessment. We address each major point below and have revised the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

Referee: Results section, paragraph on parameter reduction: on two of the five datasets the baseline kinetic parameter vanishes, reducing the effective free-parameter count from six to four. Because the central claim relies on 'full AIC penalization for its two additional parameters,' the AIC/BIC values must be recomputed with the actual effective parameter count for those datasets; otherwise the reported Delta-AIC margins (up to 141) are not directly comparable to the three-parameter reference models.

Authors: We agree with this assessment. On the two datasets where the baseline kinetic parameter vanishes, the effective number of free parameters is four. In the revised manuscript we will recompute AIC and BIC using the actual effective parameter count for those cases, update the reported Delta-AIC values, and explicitly note the reduced parameter count so that comparisons with the three-parameter reference models remain fair. revision: yes
Referee: Model derivation (section introducing CPA+C): the explicit functional form of the temperature-dependent constraint load C(T) and the precise manner in which it encodes the 'independent resolution under current structural constraints' assumption are load-bearing for both the physical interpretation and the statistical claim. Without the closed-form expression and its derivation, it is impossible to judge whether C(T) is independently motivated or partly tuned to the viscosity data.

Authors: We acknowledge that the original submission did not present the closed-form expression for C(T) or its derivation with sufficient detail. The revised manuscript will include the explicit functional form (a smooth sigmoid) together with a step-by-step derivation that starts from the CPA premise of independent configurational resolution under instantaneous structural constraints. This addition will demonstrate that C(T) is physically motivated by the model assumptions rather than being an ad-hoc fit to the data. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper motivates the CPA+C functional form from the stated physical assumption of independent configurational resolution under current structural constraints, then reports empirical outperformance versus VFT/MYEGA/Avramov-Milchev on five canonical datasets after AIC/BIC penalization for the two extra parameters and leave-one-out cross-validation. No load-bearing step reduces by construction to fitted inputs, self-citation, or renaming; the statistical ranking is independent of the motivating premise, and the single noted exception on the narrowest dataset is explicitly flagged rather than suppressed. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The claim rests on the domain assumption that configurations resolve independently under instantaneous constraints, plus the introduction of a temperature-dependent constraint load C(T) whose functional form and two associated free parameters are fitted to data.

free parameters (2)

two additional parameters
Explicitly stated as the extra degrees of freedom for which AIC penalization is applied; one is associated with the constraint load C(T).
baseline kinetic parameter
Reported to vanish on two datasets, reducing the effective parameter count and indicating data-driven simplification.

axioms (1)

domain assumption Each configurational state is resolved independently under its current structural constraints, rather than as a point on a predetermined free-energy surface.
Identified in the abstract as the key assumption underlying the Continuous Present Actualization (CPA) approach.

invented entities (1)

Constraint load C(T) no independent evidence
purpose: Quantifies narrowing of configurational access as temperature decreases toward the glass transition.
Introduced as the central modulating function in the CPA+C rate law; no independent falsifiable prediction outside the fit is stated in the abstract.

pith-pipeline@v0.9.0 · 5562 in / 1775 out tokens · 74413 ms · 2026-05-17T20:03:01.805990+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the effective actualization rate is modulated by a normalized constraint function Ĉ(T) and a coupling constant λ, yielding log10 η(T) = log10 η0 + κ/(T−Tg)[1 + λ Ĉ(T)] ... constraint load C(T) that quantifies how configurational access narrows

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Dynamic Present Theory I: Unifying Quantum Mechanics and General Rel- ativity,

D.S. Gavant, “Dynamic Present Theory I: Unifying Quantum Mechanics and General Rel- ativity,” Zenodo (2025).https://doi.org/10.5281/zenodo.17069890

work page doi:10.5281/zenodo.17069890 2025
[2]

Das Temperaturabh¨ angigkeitsgesetz der Viskosit¨ at von Fl¨ ussigkeiten,

H. Vogel, “Das Temperaturabh¨ angigkeitsgesetz der Viskosit¨ at von Fl¨ ussigkeiten,” Physikalische Zeitschrift22, 645–646 (1921)

work page 1921
[3]

Analysis of recent measurements of the viscosity of glasses,

G.S. Fulcher, “Analysis of recent measurements of the viscosity of glasses,”Journal of the American Ceramic Society8(6), 339–355 (1925)

work page 1925
[4]

Die Abh¨ angigkeit der Viskosit¨ at von der Temperatur bie unterk¨ uhlten Fl¨ ussigkeiten,

G. Tammann and W. Hesse, “Die Abh¨ angigkeit der Viskosit¨ at von der Temperatur bie unterk¨ uhlten Fl¨ ussigkeiten,”Zeitschrift f¨ ur anorganische und allgemeine Chemie156, 245– 257 (1926)

work page 1926
[5]

Viscous flow in simple organic liquids,

W.T. Laughlin and D.R. Uhlmann, “Viscous flow in simple organic liquids,”Journal of Physical Chemistry76(14), 2317–2325 (1972).https://doi.org/10.1021/j100658a032

work page doi:10.1021/j100658a032 1972
[6]

The recoverable compliance of amorphous ma- terials,

D.J. Plazek, C.A. Bero, and I.-C. Chay, “The recoverable compliance of amorphous ma- terials,”Journal of Non-Crystalline Solids172–174, 181–190 (1994).https://doi.org/ 10.1016/0022-3093(94)90431-6

work page doi:10.1016/0022-3093(94)90431-6 1994
[7]

Experimental determination of the kinematic viscosity of glycerol-water mixtures,

P.N. Shankar and M. Kumar, “Experimental determination of the kinematic viscosity of glycerol-water mixtures,”Proceedings of the Royal Society of London A444, 573–581 (1994).https://doi.org/10.1098/rspa.1994.0039

work page doi:10.1098/rspa.1994.0039 1994
[8]

Glass-Freeze Analysis Protocol v0 [Pre-register prior to model fitting],

D.S. Gavant, “Glass-Freeze Analysis Protocol v0 [Pre-register prior to model fitting],” Zenodo (2025).https://doi.org/10.5281/zenodo.17502949

work page doi:10.5281/zenodo.17502949 2025
[9]

Glass Freeze Analysis Protocol: CPA + Constraint Rate Comparison [Data set],

D.S. Gavant and C.E. Precker, “Glass Freeze Analysis Protocol: CPA + Constraint Rate Comparison [Data set],” Zenodo (2025).https://doi.org/10.5281/zenodo.17546734 7

work page doi:10.5281/zenodo.17546734 2025

[1] [1]

Dynamic Present Theory I: Unifying Quantum Mechanics and General Rel- ativity,

D.S. Gavant, “Dynamic Present Theory I: Unifying Quantum Mechanics and General Rel- ativity,” Zenodo (2025).https://doi.org/10.5281/zenodo.17069890

work page doi:10.5281/zenodo.17069890 2025

[2] [2]

Das Temperaturabh¨ angigkeitsgesetz der Viskosit¨ at von Fl¨ ussigkeiten,

H. Vogel, “Das Temperaturabh¨ angigkeitsgesetz der Viskosit¨ at von Fl¨ ussigkeiten,” Physikalische Zeitschrift22, 645–646 (1921)

work page 1921

[3] [3]

Analysis of recent measurements of the viscosity of glasses,

G.S. Fulcher, “Analysis of recent measurements of the viscosity of glasses,”Journal of the American Ceramic Society8(6), 339–355 (1925)

work page 1925

[4] [4]

Die Abh¨ angigkeit der Viskosit¨ at von der Temperatur bie unterk¨ uhlten Fl¨ ussigkeiten,

G. Tammann and W. Hesse, “Die Abh¨ angigkeit der Viskosit¨ at von der Temperatur bie unterk¨ uhlten Fl¨ ussigkeiten,”Zeitschrift f¨ ur anorganische und allgemeine Chemie156, 245– 257 (1926)

work page 1926

[5] [5]

Viscous flow in simple organic liquids,

W.T. Laughlin and D.R. Uhlmann, “Viscous flow in simple organic liquids,”Journal of Physical Chemistry76(14), 2317–2325 (1972).https://doi.org/10.1021/j100658a032

work page doi:10.1021/j100658a032 1972

[6] [6]

The recoverable compliance of amorphous ma- terials,

D.J. Plazek, C.A. Bero, and I.-C. Chay, “The recoverable compliance of amorphous ma- terials,”Journal of Non-Crystalline Solids172–174, 181–190 (1994).https://doi.org/ 10.1016/0022-3093(94)90431-6

work page doi:10.1016/0022-3093(94)90431-6 1994

[7] [7]

Experimental determination of the kinematic viscosity of glycerol-water mixtures,

P.N. Shankar and M. Kumar, “Experimental determination of the kinematic viscosity of glycerol-water mixtures,”Proceedings of the Royal Society of London A444, 573–581 (1994).https://doi.org/10.1098/rspa.1994.0039

work page doi:10.1098/rspa.1994.0039 1994

[8] [8]

Glass-Freeze Analysis Protocol v0 [Pre-register prior to model fitting],

D.S. Gavant, “Glass-Freeze Analysis Protocol v0 [Pre-register prior to model fitting],” Zenodo (2025).https://doi.org/10.5281/zenodo.17502949

work page doi:10.5281/zenodo.17502949 2025

[9] [9]

Glass Freeze Analysis Protocol: CPA + Constraint Rate Comparison [Data set],

D.S. Gavant and C.E. Precker, “Glass Freeze Analysis Protocol: CPA + Constraint Rate Comparison [Data set],” Zenodo (2025).https://doi.org/10.5281/zenodo.17546734 7

work page doi:10.5281/zenodo.17546734 2025