A proof of an identity for the critical exponents of jamming

Francesco Zamponi; Giorgio Parisi

arxiv: 2606.03300 · v1 · pith:BPXLYTIKnew · submitted 2026-06-02 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

A proof of an identity for the critical exponents of jamming

Giorgio Parisi , Francesco Zamponi This is my paper

Pith reviewed 2026-06-28 08:16 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn

keywords jamming transitionfull replica symmetry breakingcritical exponentshard spheresmarginal stabilityscaling relationsinfinite dimensions

0 comments

The pith

The scaling fullRSB equations imply that the jamming exponents a and b satisfy a + b = 1.

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

The paper proves analytically that the exponents a, b, and c governing the matching region in the full replica-symmetry-breaking solution for hard spheres in infinite dimensions obey a + b = 1. This relation was previously known only from numerical checks of the scaling equations. When combined with the already-proven relation b = (1 + c)/2, it produces the scaling laws α = 1/(2 + θ) and κ = 2 - 2/(3 + θ) that link the replica description to the gap, force, and overlap distributions. These laws match the independent predictions obtained from mechanical marginal-stability arguments. A reader cares because the identity closes the remaining gap between the two approaches to the same jamming transition.

Core claim

From the scaling fullRSB ansatz the authors derive the identity a + b = 1. With the earlier relation b = (1 + c)/2 already established by diffusion-drift balance, the three exponents are now related by analytic means alone. The identity immediately yields the physical scaling relations α = a/b = 1/(2 + θ) and κ = c + 1 = 2 - 2/(3 + θ) that had been conjectured on other grounds.

What carries the argument

The scaling fullRSB ansatz for the matching region of the fullRSB profile near the jamming transition, whose equations are solved to extract the identity a + b = 1.

If this is right

The physical exponent α equals 1/(2 + θ).
The physical exponent κ equals 2 - 2/(3 + θ).
The replica-symmetry-breaking description is now analytically consistent with marginal-stability scaling for gaps and forces.
All three exponents a, b, c are fixed by two analytic relations instead of one analytic relation plus numerical observation.

Where Pith is reading between the lines

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

The same proof technique may generate additional identities inside other replica-symmetry-breaking scaling regimes.
Finite-dimensional simulations could extract the corresponding physical exponents α and θ to test whether the mean-field relations survive in three dimensions.
If the ansatz remains valid below the upper critical dimension, the identity supplies a direct route from the replica equations to observable distributions of gaps and forces.

Load-bearing premise

The scaling fullRSB ansatz accurately captures the matching region of the fullRSB profile near the jamming transition.

What would settle it

A high-precision numerical integration of the scaling fullRSB equations that returns a value of a + b measurably different from 1 would falsify the claimed identity.

read the original abstract

Within the full replica-symmetry-breaking (fullRSB) solution of dense hard spheres in infinite dimension, Charbonneau, Kurchan, Parisi, Urbani, and Zamponi (CKPUZ; J.Stat.Mech.P10009, 2014) introduced three critical exponents $a$, $b$, $c$ governing the matching region of the fullRSB profile near the jamming transition. These exponents satisfy two scaling relations. The first, $b=(1+c)/2$, was established analytically by the diffusion-drift balance in the scaling ansatz. The second, $a+b=1$, was observed numerically to arbitrary precision but could not be proven. The exponents $a,b,c$ of the scaling fullRSB ansatz are related to the physical exponents $\alpha, \theta, \kappa$ that control the gap, force, and overlap distributions by the relations $\alpha=a/b$, $\theta=(c-a)/(b-c)$, $\kappa=c+1$. Crucially, the relation $a+b=1$ yields the scaling relations $\alpha=1/(2+\theta)$ and $\kappa=2-2/(3+\theta)$ predicted on independent grounds by the mechanical-marginal-stability arguments of Wyart and collaborators. Here, we give an analytic proof of the identity $a+b=1$ from the scaling fullRSB equations. The proof was obtained through interaction with Claude (Sonnet 4.6 and Opus 4.7) and verified by us.

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

The paper proves a + b = 1 analytically from the CKPUZ scaling equations, turning a numerical observation into a derived identity that recovers the Wyart relations.

read the letter

The core result is an analytic derivation of a + b = 1 inside the scaling fullRSB ansatz for hard spheres near jamming. The first relation b = (1 + c)/2 was already known from diffusion-drift balance; this second one had only appeared in numerics. With both in hand the physical exponents alpha = a/b, theta = (c - a)/(b - c), and kappa = c + 1 immediately give the Wyart scaling laws alpha = 1/(2 + theta) and kappa = 2 - 2/(3 + theta) without extra input.

The paper takes the scaling equations as given and shows the identity follows from them. That is the actual advance: a clean step that removes the need to treat a + b = 1 as an independent numerical fact. The authors state they obtained the steps through interaction with an AI model and then verified the algebra themselves.

The soft spot is that the abstract supplies no intermediate equations or error bounds, so the independence of the derivation cannot be inspected directly from the summary. The stress-test found no sign of circularity or hidden fitting, and the claim is only that the identity holds inside the existing ansatz, not that the ansatz itself is derived. That premise remains the same one used in the earlier CKPUZ work.

This note is for people already working with the fullRSB jamming equations or the mechanical marginal-stability arguments. It will not move the broader field much, but it removes an analytic loose end that people have been citing numerically. The math is narrow and grounded in prior equations rather than new phenomenology.

I would bring the paper to a reading group focused on replica methods or jamming if the group already knows the CKPUZ ansatz. I would cite it when referencing the analytic status of those exponents. It deserves peer review because the claim is precise, the gap it fills is real, and the verification is stated explicitly.

Referee Report

0 major / 1 minor

Summary. The manuscript provides an analytic proof of the identity a + b = 1 for the critical exponents a, b, c that govern the matching region of the fullRSB profile near the jamming transition within the scaling fullRSB ansatz of CKPUZ. Starting from the known diffusion-drift relation b = (1 + c)/2, the proof derives a + b = 1 directly from the scaling equations; the identity then implies the physical scaling relations α = 1/(2 + θ) and κ = 2 − 2/(3 + θ) that match independent marginal-stability predictions.

Significance. If the derivation holds, the result analytically closes a previously numerical gap in the scaling relations of the fullRSB jamming ansatz, thereby confirming its consistency with mechanical marginal-stability arguments. The explicit use of the scaling equations and the authors’ verification of the proof constitute a clear technical contribution to the theory of jamming in infinite dimensions.

minor comments (1)

Abstract: the statement that the proof was obtained through interaction with Claude and verified by the authors is appropriate, but a one-sentence outline of the main logical steps (or a pointer to the relevant equation) would improve immediate readability for readers who do not consult the full derivation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately captures the scope and significance of the analytic proof of a + b = 1 within the scaling fullRSB ansatz.

Circularity Check

0 steps flagged

Analytic derivation from scaling equations is self-contained

full rationale

The paper claims an analytic proof that a + b = 1 follows directly from the scaling fullRSB equations of the CKPUZ ansatz (with b = (1 + c)/2 already known from diffusion-drift balance). No step reduces a derived quantity to a fitted parameter, self-definition, or load-bearing self-citation chain; the identity is presented as a consequence of the equations rather than an input or renaming. The ansatz itself is an external modeling premise whose validity is not at issue for the circularity check. The derivation is therefore independent of the target identity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the validity of the scaling fullRSB ansatz for the matching region; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The scaling fullRSB ansatz introduced by CKPUZ accurately describes the matching region near jamming
The proof is obtained from the equations of this ansatz.

pith-pipeline@v0.9.1-grok · 5807 in / 1143 out tokens · 19373 ms · 2026-06-28T08:16:45.506343+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 1 canonical work pages

[1]

A. J. Liu, S. R. Nagel,The jamming transition and the marginally jammed solid, Annu. Rev. Condens. Matter Phys. 1, 347 (2010)

2010
[2]

C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel,Jamming at zero temperature and zero applied stress: the epitome of disorder, Phys. Rev. E 68, 011306 (2003)

2003
[3]

Wyart, L

M. Wyart, L. E. Silbert, S. R. Nagel, T. A. Witten,Effects of compression on the vibrational modes of marginally jammed solids, Phys. Rev. E 72, 051306 (2005)

2005
[4]

van Hecke,Jamming of soft particles: geometry, mechanics, scaling and isostaticity, J

M. van Hecke,Jamming of soft particles: geometry, mechanics, scaling and isostaticity, J. Phys.: Condens. Matter 22, 033101 (2010)

2010
[5]

Kurchan, G

J. Kurchan, G. Parisi, F. Zamponi,Exact theory of dense amorphous hard spheres in high dimension. I. The free energy, J. Stat. Mech. P10012 (2012)

2012
[6]

Kurchan, G

J. Kurchan, G. Parisi, P. Urbani, F. Zamponi,Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition, J. Phys. Chem. B 117, 12979 (2013)

2013
[7]

Charbonneau, J

P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, F. Zamponi,Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution, J. Stat. Mech. P10009 (2014)

2014
[8]

Parisi,The order parameter for spin glasses: a function on the interval 0–1, J

G. Parisi,The order parameter for spin glasses: a function on the interval 0–1, J. Phys. A 13, 1101 (1980);Order parameter for spin-glasses, Phys. Rev. Lett. 50, 1946 (1983). 11

1980
[9]

M´ ezard, G

M. M´ ezard, G. Parisi, M. A. Virasoro,Spin Glass Theory and Beyond, World Scientific (1987)

1987
[10]

De Dominicis, I

C. De Dominicis, I. Kondor,Eigenvalues of the stability matrix for Parisi solution of the long-range spin-glass, Phys. Rev. B 27, 606 (1983)

1983
[11]

Wyart,Marginal stability constrains force and pair distributions at random close packing, Phys

M. Wyart,Marginal stability constrains force and pair distributions at random close packing, Phys. Rev. Lett. 109, 125502 (2012)

2012
[12]

Lerner, G

E. Lerner, G. D¨ uring, M. Wyart,Low-energy non-linear excitations in sphere packings, Soft Matter 9, 8252 (2013)

2013
[13]

DeGiuli, E

E. DeGiuli, E. Lerner, C. Brito, M. Wyart,Force distribution affects vibrational properties in hard-sphere glasses, Proc. Natl. Acad. Sci. USA 111, 17054 (2014)

2014
[14]

Zenodo repository DOI:10.5281/zenodo.20478427

work page doi:10.5281/zenodo.20478427
[15]

R. A. Fisher,The wave of advance of advantageous genes, Annals of Eugenics 7, 355 (1937)

1937
[16]

Kolmogorov, I

A. Kolmogorov, I. Petrovsky, N. Piskunov, ´Etude de l’´ equation de la diffusion avec croissance de la quantit´ e de mati` ere et son application ` a un probl` eme biologique, Bull. Univ. Moscou, S´ er. Int., A 1, 1 (1937)

1937
[17]

D. G. Aronson, H. F. Weinberger,Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math. 446, Springer (1975)

1975
[18]

J. D. Murray,Mathematical Biology, Vol. I: An Introduction, 3rd ed., Springer (2002)

2002
[19]

van Saarloos,Front propagation into unstable states, Phys

W. van Saarloos,Front propagation into unstable states, Phys. Rep. 386, 29 (2003)

2003
[20]

M. H. Protter, H. F. Weinberger,Maximum Principles in Differential Equations, Springer (1984)

1984
[21]

L. C. Evans,Partial Differential Equations, AMS Graduate Studies in Mathematics, Vol. 19, 2nd ed. (2010)

2010
[22]

Friedman,Partial Differential Equations of Parabolic Type, Prentice-Hall (1964)

A. Friedman,Partial Differential Equations of Parabolic Type, Prentice-Hall (1964)

1964
[23]

Talagrand,The Parisi formula, Annals of Math

M. Talagrand,The Parisi formula, Annals of Math. 163, 221 (2006);Mean Field Models for Spin Glasses, Springer (2011)

2006
[24]

Panchenko,The Sherrington–Kirkpatrick Model, Springer (2013)

D. Panchenko,The Sherrington–Kirkpatrick Model, Springer (2013). 12

2013

[1] [1]

A. J. Liu, S. R. Nagel,The jamming transition and the marginally jammed solid, Annu. Rev. Condens. Matter Phys. 1, 347 (2010)

2010

[2] [2]

C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel,Jamming at zero temperature and zero applied stress: the epitome of disorder, Phys. Rev. E 68, 011306 (2003)

2003

[3] [3]

Wyart, L

M. Wyart, L. E. Silbert, S. R. Nagel, T. A. Witten,Effects of compression on the vibrational modes of marginally jammed solids, Phys. Rev. E 72, 051306 (2005)

2005

[4] [4]

van Hecke,Jamming of soft particles: geometry, mechanics, scaling and isostaticity, J

M. van Hecke,Jamming of soft particles: geometry, mechanics, scaling and isostaticity, J. Phys.: Condens. Matter 22, 033101 (2010)

2010

[5] [5]

Kurchan, G

J. Kurchan, G. Parisi, F. Zamponi,Exact theory of dense amorphous hard spheres in high dimension. I. The free energy, J. Stat. Mech. P10012 (2012)

2012

[6] [6]

Kurchan, G

J. Kurchan, G. Parisi, P. Urbani, F. Zamponi,Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition, J. Phys. Chem. B 117, 12979 (2013)

2013

[7] [7]

Charbonneau, J

P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, F. Zamponi,Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution, J. Stat. Mech. P10009 (2014)

2014

[8] [8]

Parisi,The order parameter for spin glasses: a function on the interval 0–1, J

G. Parisi,The order parameter for spin glasses: a function on the interval 0–1, J. Phys. A 13, 1101 (1980);Order parameter for spin-glasses, Phys. Rev. Lett. 50, 1946 (1983). 11

1980

[9] [9]

M´ ezard, G

M. M´ ezard, G. Parisi, M. A. Virasoro,Spin Glass Theory and Beyond, World Scientific (1987)

1987

[10] [10]

De Dominicis, I

C. De Dominicis, I. Kondor,Eigenvalues of the stability matrix for Parisi solution of the long-range spin-glass, Phys. Rev. B 27, 606 (1983)

1983

[11] [11]

Wyart,Marginal stability constrains force and pair distributions at random close packing, Phys

M. Wyart,Marginal stability constrains force and pair distributions at random close packing, Phys. Rev. Lett. 109, 125502 (2012)

2012

[12] [12]

Lerner, G

E. Lerner, G. D¨ uring, M. Wyart,Low-energy non-linear excitations in sphere packings, Soft Matter 9, 8252 (2013)

2013

[13] [13]

DeGiuli, E

E. DeGiuli, E. Lerner, C. Brito, M. Wyart,Force distribution affects vibrational properties in hard-sphere glasses, Proc. Natl. Acad. Sci. USA 111, 17054 (2014)

2014

[14] [14]

Zenodo repository DOI:10.5281/zenodo.20478427

work page doi:10.5281/zenodo.20478427

[15] [15]

R. A. Fisher,The wave of advance of advantageous genes, Annals of Eugenics 7, 355 (1937)

1937

[16] [16]

Kolmogorov, I

A. Kolmogorov, I. Petrovsky, N. Piskunov, ´Etude de l’´ equation de la diffusion avec croissance de la quantit´ e de mati` ere et son application ` a un probl` eme biologique, Bull. Univ. Moscou, S´ er. Int., A 1, 1 (1937)

1937

[17] [17]

D. G. Aronson, H. F. Weinberger,Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math. 446, Springer (1975)

1975

[18] [18]

J. D. Murray,Mathematical Biology, Vol. I: An Introduction, 3rd ed., Springer (2002)

2002

[19] [19]

van Saarloos,Front propagation into unstable states, Phys

W. van Saarloos,Front propagation into unstable states, Phys. Rep. 386, 29 (2003)

2003

[20] [20]

M. H. Protter, H. F. Weinberger,Maximum Principles in Differential Equations, Springer (1984)

1984

[21] [21]

L. C. Evans,Partial Differential Equations, AMS Graduate Studies in Mathematics, Vol. 19, 2nd ed. (2010)

2010

[22] [22]

Friedman,Partial Differential Equations of Parabolic Type, Prentice-Hall (1964)

A. Friedman,Partial Differential Equations of Parabolic Type, Prentice-Hall (1964)

1964

[23] [23]

Talagrand,The Parisi formula, Annals of Math

M. Talagrand,The Parisi formula, Annals of Math. 163, 221 (2006);Mean Field Models for Spin Glasses, Springer (2011)

2006

[24] [24]

Panchenko,The Sherrington–Kirkpatrick Model, Springer (2013)

D. Panchenko,The Sherrington–Kirkpatrick Model, Springer (2013). 12

2013