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arxiv: 2605.25178 · v1 · pith:UUGPES54new · submitted 2026-05-24 · ✦ hep-th · quant-ph

Pseudorandom Dynamics in the SYK Model and Cryptographic Censorship in JT Gravity

Pouya Golmohammadi This is my paper

Pith reviewed 2026-06-29 23:41 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords SYK modelJT gravityunitary designscryptographic censorshipevent horizonspseudorandom unitariesblack hole interiorplanted-SYK conjecture
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The pith

Typical SYK states in the microcanonical window dual to JT gravity have event horizons, with the horizonless fraction doubly exponentially small, conditional on the planted-SYK hardness conjecture.

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

The paper proves that the SYK disorder ensemble forms an approximate unitary k-design for all k polynomial in N, with the error bounded by the spectral form factor, via Weingarten calculus and random matrix universality. It introduces the planted-SYK hardness conjecture, backed by spectral universality and low-degree polynomial evidence, to promote this design to a gravitationally pseudorandom unitary. Combined with efficient causal wedge reconstruction in JT gravity, the result is that typical states in the SYK microcanonical window correspond to bulk geometries containing event horizons. The fraction of horizonless states is doubly exponentially small. The regularized geodesic length of the maximal interior slice is the explicit distinguishing operator, whose prediction gap grows linearly with time from interior stretching.

Core claim

Under the planted-SYK hardness conjecture, the SYK model supplies a conditional realization of cryptographic censorship in JT gravity: typical microcanonical states must have event horizons in their bulk duals, and the horizonless fraction is doubly exponentially small.

What carries the argument

The planted-SYK hardness conjecture, which converts the approximate k-design into a gravitationally pseudorandom unitary that cannot be efficiently distinguished from Haar-random by low-complexity observers.

If this is right

  • Typical states in the SYK microcanonical window have event horizons in their JT gravity bulk duals.
  • The horizonless fraction of such states is doubly exponentially small.
  • The regularized geodesic length of the maximal interior slice distinguishes horizonful from horizonless geometries.
  • This length gap grows linearly with time due to stretching of the black hole interior.

Where Pith is reading between the lines

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

  • If the conjecture holds, the interior geodesic length provides a direct link between cryptographic censorship and the complexity-equals-volume conjecture.
  • Analogous hardness assumptions in other holographic models could similarly enforce the rarity of horizonless geometries.
  • A counterexample to the conjecture would permit efficient boundary reconstruction of horizonless bulk states.

Load-bearing premise

The planted-SYK hardness conjecture holds so that the approximate design can be treated as a gravitationally pseudorandom unitary.

What would settle it

An efficient algorithm that distinguishes a typical SYK unitary from a random one without knowledge of the disorder, or an explicit typical SYK state whose JT gravity dual lacks an event horizon.

Figures

Figures reproduced from arXiv: 2605.25178 by Pouya Golmohammadi.

Figure 1
Figure 1. Figure 1: Numerical verification of the frame potential analysis for SYK4 with N = 8, 10, 12 (L = 16, 32, 64), averaged over 100–300 disorder realizations. (a) The first frame potential F (1)(T) computed directly from cross-realization averages of |tr(U † 1U2)| 2 (solid), compared with the analytic formula (1.1) evaluated from the disorder-averaged SFF (dashed). (b) The deviation F (1)(T) − 1 as a function of T /TH,… view at source ↗
read the original abstract

We argue that the SYK model provides a conditional realization of Cryptographic Censorship in JT gravity. By using the Weingarten calculus and random matrix universality, we prove that the SYK disorder ensemble is an approximate unitary $k$-design for all $k=\poly(N)$, with deviation controlled by the spectral form factor. We then formulate the planted-SYK hardness conjecture and provide evidence from spectral universality and the low-degree polynomial framework. Under this conjecture, the approximate design becomes a gravitationally pseudorandom unitary. Together with the efficient causal wedge reconstruction in JT gravity, this leads to the conclusion that typical states in the SYK microcanonical window must have event horizons in their bulk duals, with the horizonless fraction doubly exponentially small. We further identify the regularized geodesic length of the maximal interior slice as the explicit distinguishing operator. Its prediction gap grows linearly with time due to the stretching of the black hole interior, linking Cryptographic Censorship to the complexity equals volume conjecture.

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 / 1 minor

Summary. The manuscript claims that the SYK model provides a conditional realization of Cryptographic Censorship in JT gravity. Using Weingarten calculus and random-matrix universality, it proves that the SYK disorder ensemble is an approximate unitary k-design for all k = poly(N), with deviation controlled by the spectral form factor. It formulates the planted-SYK hardness conjecture and supplies supporting evidence from spectral universality and the low-degree polynomial framework. Under this conjecture the approximate design is upgraded to a gravitationally pseudorandom unitary; combined with efficient causal-wedge reconstruction in JT gravity, this implies that typical states in the SYK microcanonical window have event horizons in their bulk duals, with the horizonless fraction doubly exponentially small. The regularized geodesic length of the maximal interior slice is identified as the explicit distinguishing operator whose prediction gap grows linearly with time, linking the result to the complexity-equals-volume conjecture.

Significance. If the planted-SYK hardness conjecture holds, the result would be significant in connecting quantum chaos, pseudorandomness, and holographic gravity by exhibiting a concrete microscopic model in which cryptographic hardness enforces horizon formation for typical states. The independent technical contribution of proving the approximate k-design property via Weingarten calculus and the spectral form factor stands on its own, as does the explicit identification of a bulk operator whose linear growth distinguishes horizonless configurations. The work supplies a falsifiable prediction in the form of doubly exponential suppression of horizonless states.

major comments (2)
  1. [Section formulating the planted-SYK hardness conjecture] The section formulating the planted-SYK hardness conjecture: the central claim that typical microcanonical states must have event horizons (with horizonless fraction doubly exponentially small) follows only after invoking this unproven conjecture to promote the approximate design to a gravitationally pseudorandom unitary. The supplied evidence from spectral universality and low-degree polynomials is suggestive but does not constitute a proof, so the censorship conclusion remains conditional on an assumption that is also used to define the pseudorandomness property.
  2. [Abstract and design-proof paragraphs] Abstract and the design-proof paragraphs: the claim that the spectral form factor controls the deviation for all k = poly(N) is presented as a proof via Weingarten calculus, yet only a sketch of the route is given; without the explicit derivation it is impossible to confirm whether the stated bounds are strong enough to support the subsequent upgrade step even under the conjecture.
minor comments (1)
  1. The definition of 'gravitationally pseudorandom unitary' is introduced only after the conjecture; a self-contained definition placed earlier would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment below. The manuscript already frames the censorship result as conditional on the planted-SYK hardness conjecture and presents supporting evidence rather than a proof of the conjecture. For the design property we clarify the location of the full derivation and agree to minor clarifications in the abstract.

read point-by-point responses

  1. Referee: [Section formulating the planted-SYK hardness conjecture] The section formulating the planted-SYK hardness conjecture: the central claim that typical microcanonical states must have event horizons (with horizonless fraction doubly exponentially small) follows only after invoking this unproven conjecture to promote the approximate design to a gravitationally pseudorandom unitary. The supplied evidence from spectral universality and low-degree polynomials is suggestive but does not constitute a proof, so the censorship conclusion remains conditional on an assumption that is also used to define the pseudorandomness property.

    Authors: We agree that the central censorship claim is conditional on the planted-SYK hardness conjecture. This conditional status is stated explicitly in the abstract, the introduction, and the dedicated section that formulates the conjecture. The spectral-universality and low-degree-polynomial arguments are supplied only as supporting evidence for the plausibility of the conjecture, not as a proof. Because the manuscript never claims an unconditional proof, no revision is required on this point. revision: no

  2. Referee: [Abstract and design-proof paragraphs] Abstract and the design-proof paragraphs: the claim that the spectral form factor controls the deviation for all k = poly(N) is presented as a proof via Weingarten calculus, yet only a sketch of the route is given; without the explicit derivation it is impossible to confirm whether the stated bounds are strong enough to support the subsequent upgrade step even under the conjecture.

    Authors: The explicit derivation of the approximate k-design property, including the Weingarten-calculus steps and the precise bounds controlled by the spectral form factor, appears in Sections 3 and 4 of the main text. The abstract summarizes the result. To make the strength of the bounds immediately visible to readers who consult only the abstract, we will add a short parenthetical reference to the key steps and the resulting exponential suppression. This is a minor clarification; the technical bounds themselves remain unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; result is explicitly conditional

full rationale

The paper first establishes that the SYK disorder ensemble forms an approximate unitary k-design (k=poly(N)) via Weingarten calculus and random-matrix universality, with the deviation bounded by the spectral form factor; this step is self-contained and uses standard external techniques. It then separately formulates the planted-SYK hardness conjecture, supplies supporting evidence from spectral universality and low-degree polynomials, and states that only under this conjecture does the design become a gravitationally pseudorandom unitary, which combined with causal-wedge reconstruction yields the horizon conclusion. Because the central claim is presented as conditional on an additional unproven assumption rather than derived unconditionally or by re-labeling the design property itself, no step reduces the output to the inputs by construction. This is a standard conditional argument, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the planted-SYK hardness conjecture (ad_hoc_to_paper) and on standard random-matrix universality assumptions; no free parameters or new invented entities with independent evidence are introduced in the abstract.

axioms (2)
  • ad hoc to paper planted-SYK hardness conjecture
    Invoked to convert the approximate k-design property into gravitational pseudorandomness
  • domain assumption random matrix universality of SYK spectral form factor
    Used to control the deviation from exact k-design
invented entities (1)
  • gravitationally pseudorandom unitary no independent evidence
    purpose: To realize cryptographic censorship in the JT dual
    Defined via the hardness conjecture; no independent falsifiable handle supplied in the abstract

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discussion (0)

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