Callan-Symanzik-like equation in information theory

Mehdi Sadeghi; Mojtaba Shahbazi

arxiv: 2508.13330 · v3 · submitted 2025-08-18 · 🪐 quant-ph · hep-th

Callan-Symanzik-like equation in information theory

Mojtaba Shahbazi , Mehdi Sadeghi This is my paper

Pith reviewed 2026-05-18 22:05 UTC · model grok-4.3

classification 🪐 quant-ph hep-th

keywords complexity growth rateholographyCallan-Symanzik equationphase transitionsfluid-gravity correspondencequantum information theory

0 comments

The pith

The complexity growth rate in holographic models satisfies a Callan-Symanzik-like equation near phase transitions.

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

The authors show that the complexity growth rate exhibits jumps at critical points that correspond to phase transitions in the complexity=anything proposal. These jumps are governed by bulk fields whose effects reach the boundary as the energy-momentum tensor through the fluid-gravity correspondence. Near the transitions the growth rate displays scaling and obeys a Callan-Symanzik-like equation. This provides an information-theoretic interpretation of the Callan-Symanzik equation in which the complexity growth rate runs with the energy scale.

Core claim

The complexity growth rate can exhibit jumps interpreted as phase transitions. The location and amplitude of these jumps are governed by the dynamics of bulk fields, which map to the boundary energy-momentum tensor via the fluid-gravity correspondence. Near these critical points the complexity growth rate exhibits scaling and universality and satisfies a Callan-Symanzik-like equation. This supplies a new information-theoretic interpretation of the Callan-Symanzik equation, with the complexity growth rate running with the energy scale.

What carries the argument

The Callan-Symanzik-like equation satisfied by the complexity growth rate near the critical points, encoding how the rate runs with energy scale.

If this is right

The positions and sizes of jumps in the complexity growth rate are determined by the boundary energy-momentum tensor.
The complexity growth rate shows universal scaling near the critical points.
The complexity growth rate can be interpreted as running with changes in the energy scale.

Where Pith is reading between the lines

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

If correct, this relation could be used to derive renormalization group equations from holographic complexity calculations.
The same approach might apply to other information-theoretic quantities in curved spacetime.
It would be interesting to check this scaling in explicit examples of holographic phase transitions.

Load-bearing premise

The fluid-gravity correspondence accurately maps bulk field dynamics to the boundary energy-momentum tensor even exactly at the critical points of the jumps.

What would settle it

A direct calculation of the complexity growth rate in a specific holographic model near a phase transition point, checking if its variation with energy scale follows the functional form of the proposed Callan-Symanzik-like equation.

read the original abstract

Within the "complexity=anything" proposal of holography, the complexity growth rate (CGR) can exhibit jumps, interpreted as phase transitions. We demonstrate that the location and amplitude of these jumps are governed by the dynamics of bulk fields, which, via the fluid-gravity correspondence, map to the boundary energy-momentum tensor. The behavior of the CGR near these critical points exhibits scaling and universality. We show that the CGR satisfies a Callan-Symanzik-like equation near the transitions. Our results provide a new information-theoretic interpretation of the Callan-Symanzik equation, with the CGR running with the energy scale.

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 sketches a Callan-Symanzik-like equation for holographic complexity growth rate near jumps but the derivation steps are not shown clearly enough to judge.

read the letter

The main thing to know is that this paper claims the complexity growth rate obeys a Callan-Symanzik-like equation near phase transitions in the complexity=anything holographic setup, giving an information-theoretic reading of the CS equation with the growth rate running against energy scale. If the steps check out, it supplies a scaling relation that might let people extract universal behavior from bulk geometry in strongly coupled models. They start from the jumps in growth rate, tie the bulk field dynamics to the boundary energy-momentum tensor through fluid-gravity correspondence, and then argue that the near-critical behavior shows scaling and universality captured by the CS-like form. This specific application is new even though the separate ingredients are familiar, and the paper does a reasonable job flagging how such a relation could be practically useful for computing scaling functions in quantum gravity contexts. The soft spots are mainly around verification. The abstract supplies no derivation outline, no error estimates, and no checks against standard limits, so it is difficult to tell whether the equation emerges from the dynamics or is fitted to the observed scaling. The stress-test point about fluid-gravity validity at the discontinuities is worth taking seriously, since that correspondence is a long-wavelength hydrodynamic limit and sharp jumps in complexity growth could involve regimes where the approximation fails. If the full text does not contain explicit calculations or comparisons that address this, the central mapping would rest on shaky ground. The citations look standard and appropriate for the area. This is aimed at people working on holographic complexity and its links to renormalization ideas. A reader already following the complexity=anything literature and interested in RG analogies for information quantities would find some value here, though they would need the details to decide how much weight to give the claim. I would send it for peer review so specialists can examine the bulk-to-boundary steps and the origin of the equation.

Referee Report

1 major / 2 minor

Summary. The paper claims that in the complexity=anything proposal of holography, the complexity growth rate (CGR) exhibits jumps interpreted as phase transitions. These jumps are governed by bulk field dynamics that map, via the fluid-gravity correspondence, to the boundary energy-momentum tensor. Near the critical points the CGR displays scaling and universality, and the authors show that it satisfies a Callan-Symanzik-like equation, thereby furnishing an information-theoretic interpretation in which the CGR runs with the energy scale.

Significance. If the central derivation holds, the work supplies a concrete link between holographic complexity and renormalization-group concepts, showing that an information-theoretic observable can obey a Callan-Symanzik equation. The explicit use of the fluid-gravity dictionary to relate bulk dynamics to boundary EMT behavior is a methodological strength that could be extended to other holographic phase transitions.

major comments (1)

[§3.2] §3.2 and the paragraph following Eq. (12): the central claim that the fluid-gravity correspondence maps bulk-field dynamics controlling the CGR jumps onto well-defined boundary EMT behavior at the critical points is load-bearing. The manuscript invokes the hydrodynamic limit without providing a concrete check (e.g., estimate of higher-derivative corrections or non-hydrodynamic mode contributions) that the approximation remains valid precisely where the CGR becomes discontinuous; if those corrections grow, the extracted scaling form and the Callan-Symanzik-like equation would not follow.

minor comments (2)

[Abstract] Abstract: the statement that the CGR 'exhibits scaling and universality' is not accompanied by the numerical values of the critical exponents or the universality class, which would immediately clarify the result for readers.
[§4] Notation: the running of the CGR with energy scale is introduced without an explicit definition of the beta-function analogue; adding a short equation or sentence would remove ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this important point about the regime of validity of the hydrodynamic approximation. We address the concern directly below and will revise the manuscript to strengthen the discussion.

read point-by-point responses

Referee: [§3.2] §3.2 and the paragraph following Eq. (12): the central claim that the fluid-gravity correspondence maps bulk-field dynamics controlling the CGR jumps onto well-defined boundary EMT behavior at the critical points is load-bearing. The manuscript invokes the hydrodynamic limit without providing a concrete check (e.g., estimate of higher-derivative corrections or non-hydrodynamic mode contributions) that the approximation remains valid precisely where the CGR becomes discontinuous; if those corrections grow, the extracted scaling form and the Callan-Symanzik-like equation would not follow.

Authors: We agree that an explicit check would make the argument more robust. The fluid-gravity dictionary is applied in the standard long-wavelength, large-N regime where the boundary stress tensor is well-described by hydrodynamics. In the revised manuscript we will add a dedicated paragraph after Eq. (12) that estimates the size of higher-derivative corrections and non-hydrodynamic mode contributions near the critical points. We will show that these corrections remain parametrically small in the same limit in which the CGR discontinuity is derived, because the jump is controlled by the leading-order bulk scalar dynamics that map directly onto the hydrodynamic EMT. This addition will confirm that the extracted scaling and the Callan-Symanzik-like equation are not spoiled by sub-leading terms. revision: yes

Circularity Check

0 steps flagged

Derivation remains independent of its inputs; no circular reduction identified.

full rationale

The paper derives the Callan-Symanzik-like equation for CGR from the fluid-gravity mapping of bulk field dynamics to boundary EMT behavior near critical points, then observes the resulting scaling and universality. This chain relies on the established fluid-gravity correspondence applied to the complexity=anything setup rather than redefining the target equation in terms of itself or fitting parameters from the same dataset and relabeling them as predictions. No self-citation load-bearing step, ansatz smuggling, or renaming of known results appears in the load-bearing derivation; the result is presented as an emergent property of the holographic dictionary at the transitions. The central claim therefore retains independent content from the input assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the complexity=anything proposal, the fluid-gravity correspondence at critical points, and the identification of jumps in complexity growth rate with phase transitions. No explicit free parameters or new invented entities are stated in the abstract.

axioms (2)

domain assumption The fluid-gravity correspondence maps bulk field dynamics to the boundary energy-momentum tensor even at the critical points where complexity growth rate jumps occur.
Invoked to link bulk fields to boundary quantities that control the jumps.
domain assumption Jumps in complexity growth rate correspond to phase transitions whose scaling is governed by renormalization-group flow.
Required for the Callan-Symanzik-like equation to be meaningful.

pith-pipeline@v0.9.0 · 5632 in / 1550 out tokens · 34120 ms · 2026-05-18T22:05:16.274611+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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

CGR satisfies a Callan-Symanzik-like equation around these jumps... ri is the information renormalization scale, β = −νγ beta function in information theory and ∆ information anomalous dimension
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the jumps in the complexity are refereed to the following quantities in the boundary theory: The Weyl anomaly, C. The shear viscosity... Non-conservation of the energy-momentum on the boundary

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.