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arxiv: 2605.20509 · v1 · pith:IGIMRL7Mnew · submitted 2026-05-19 · ✦ hep-lat · hep-ph· hep-th

The Causal Bootstrap: Bounding Smeared Spectral Functions from Non-Perturbative Euclidean Data

Ryan Abbott , Sarah Fields , William I. Jay , Patrick Oare , Matteo Saccardi This is my paper

Pith reviewed 2026-05-21 05:51 UTC · model grok-4.3

classification ✦ hep-lat hep-phhep-th
keywords causal bootstrapsmeared spectral functionsEuclidean dataspectral boundssemidefinite programmingpositive measureslattice field theoryconvex optimization
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The pith

The causal bootstrap computes rigorous bounds on smeared spectral functions from finite Euclidean data by optimizing over compatible positive spectral densities.

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

This paper introduces the causal bootstrap, a method that finds the tightest possible upper and lower bounds on a smeared spectral observable consistent with given non-perturbative Euclidean data. It works by searching over all positive spectral densities that match the data within the statistical uncertainties encoded in the covariance matrix. The approach uses Lagrange duality to turn the problem into finding the best bounding function for the smearing kernel, which for polynomial or rational kernels reduces to a semidefinite program. A reader would care because this gives certified, model-independent error estimates on physical quantities extracted from lattice simulations. If the method holds, it strengthens the reliability of non-perturbative calculations in quantum field theory.

Core claim

The central claim is that rigorous upper and lower bounds on any smeared spectral observable can be obtained by solving a convex optimization problem over the set of positive spectral densities that are compatible with the Euclidean data set, with statistical errors incorporated through covariance-matrix compatibility regions; these bounds are equivalent via duality to certified bounds on the smearing kernel itself.

What carries the argument

The causal bootstrap optimization over the convex set of positive spectral densities compatible with the data within covariance tolerances, which via Lagrange duality provides certified bounds on the smearing kernel.

If this is right

  • The bounds are rigorous upper and lower limits that incorporate all statistical correlations from the covariance matrix.
  • For polynomial, rational, and piecewise rational kernels the dual problems reduce to finite semidefinite programs.
  • The formulation is equivalent to certified bounds on the target smearing kernel.
  • It relates the problem to classical moment problems and Nevanlinna-Pick interpolation.
  • Numerical examples illustrate the production of non-trivial bounds.

Where Pith is reading between the lines

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

  • The method could be extended to incorporate additional physical constraints such as dispersion relations into the optimization.
  • Testing on synthetic data generated from known spectral functions would validate the enclosure of true values.
  • Infeasible optimizations would flag inconsistencies in the Euclidean data with positivity.
  • Applications to other areas like condensed matter physics where spectral functions are extracted from imaginary-time data could be explored.

Load-bearing premise

The assumption that a positive spectral density exists that is compatible with the finite Euclidean data set within the stated covariance; if no such density exists the bounds are undefined.

What would settle it

Generate synthetic Euclidean correlator data from a known exact positive spectral density with realistic noise; verify that the true smeared observable always lies within the computed bounds.

Figures

Figures reproduced from arXiv: 2605.20509 by Matteo Saccardi, Patrick Oare, Ryan Abbott, Sarah Fields, William I. Jay.

Figure 1
Figure 1. Figure 1: FIG. 1. Bounds on the smeared spectral function with kernel [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the resulting bounds on RL from solv￾ing the matrix (r = 2) versions of Eq. (52) and Eq. (53) using moment-problem and NP-based inputs, respec￾tively. In all cases, the bounds contain the exact value computed directly from the toy spectral density. The bounds shrink as the input uncertainty is reduced, as ex￾pected from the decreasing size of the feasible set. For this particular observable and finit… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Bounds on the toy HVP-like observable defined in [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical demonstration of the equivalent formulation of bounds on [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
read the original abstract

This work introduces the causal bootstrap, a framework for bounding smeared spectral observables from finite non-perturbative Euclidean data. The method optimizes over the convex set of positive spectral densities compatible with the data to compute rigorous upper and lower bounds on the target observable. Statistical uncertainties, including correlations, are incorporated through compatibility regions using the covariance matrix. The bounds are equivalent, via Lagrange duality, to certified bounds on the target smearing kernel. For polynomial, rational, and piecewise rational kernels, the resulting dual problems can be reduced to finite-dimensional semidefinite programs using techniques familiar, e.g., in the numerical conformal bootstrap. The present formulation clarifies the relation to moment problems, Nevanlinna--Pick interpolation, and linear kernel-reconstruction methods. Numerical examples demonstrate the method.

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 the causal bootstrap, a convex optimization method that derives rigorous upper and lower bounds on smeared spectral observables by optimizing over the set of positive spectral densities compatible with finite non-perturbative Euclidean data. Statistical uncertainties and correlations are incorporated via covariance-matrix-defined compatibility regions; Lagrange duality converts the problem to certified bounds on the target smearing kernel. For polynomial, rational, and piecewise-rational kernels the dual reduces to finite-dimensional semidefinite programs, with explicit connections drawn to moment problems, Nevanlinna–Pick interpolation, and linear kernel reconstruction. Numerical examples illustrate the approach.

Significance. If the central claims hold, the framework supplies a parameter-free, duality-certified route to bounding smeared spectral functions directly from lattice data, avoiding model-dependent fits. The reduction to SDPs and the clarification of relations to classical interpolation theory constitute genuine technical strengths that could be adopted in lattice QCD studies of hadronic spectral functions.

major comments (2)
  1. [§3] §3 (Primal formulation): the optimization is stated over the convex set of positive spectral densities whose smeared integrals lie inside the covariance-defined compatibility region. No mechanism is provided for detecting or handling the empty-feasible-set case that arises when noisy Euclidean data are inconsistent with positivity or causality; the bounds are then formally undefined. This is load-bearing for the central claim because lattice data routinely produce small violations.
  2. [§5] §5 (Numerical examples): all presented cases assume a non-empty feasible set. An explicit demonstration of an infeasible instance, together with the authors’ proposed regularization or diagnostic, is needed to establish practical robustness.
minor comments (2)
  1. [§2] Notation for the smearing kernel K(ω) and the target observable should be introduced once in §2 and used consistently thereafter.
  2. [Abstract] The abstract states that the dual yields ‘certified kernel bounds’; a one-sentence reminder of the precise certification (i.e., the dual objective equals the primal bound) would aid readers unfamiliar with convex duality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and for the constructive major comments. We address each point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§3] §3 (Primal formulation): the optimization is stated over the convex set of positive spectral densities whose smeared integrals lie inside the covariance-defined compatibility region. No mechanism is provided for detecting or handling the empty-feasible-set case that arises when noisy Euclidean data are inconsistent with positivity or causality; the bounds are then formally undefined. This is load-bearing for the central claim because lattice data routinely produce small violations.

    Authors: We agree that the primal formulation as written assumes a non-empty feasible set and does not explicitly treat the empty-set case. When finite noisy Euclidean data produce small violations of positivity or causality, the feasible set can indeed be empty and the bounds formally undefined. This is a practical issue for lattice applications. In the revised manuscript we will augment §3 with a short subsection on infeasibility detection: the dual problem is unbounded below if and only if the primal is infeasible, providing a rigorous certificate that can be read directly from any SDP solver. We will also introduce a simple regularization that relaxes the covariance-defined compatibility constraints by a small tolerance ε (either as a soft constraint or by minimizing the maximum violation), turning the problem into a always-feasible SDP while recovering the original bounds in the limit ε→0. These additions will be placed immediately after the statement of the primal problem. revision: yes

  2. Referee: [§5] §5 (Numerical examples): all presented cases assume a non-empty feasible set. An explicit demonstration of an infeasible instance, together with the authors’ proposed regularization or diagnostic, is needed to establish practical robustness.

    Authors: We accept the observation that every numerical example in the current §5 uses data for which the feasible set is non-empty. To address this, the revised §5 will contain an additional subsection that deliberately constructs an infeasible instance by adding a controlled perturbation to the Euclidean data that produces a small positivity violation. We will report the SDP solver output that certifies infeasibility via an unbounded dual, then apply the regularization procedure described in the revised §3 and display the resulting regularized bounds together with the size of the introduced violation. This will illustrate both the diagnostic and the practical regularization in a concrete setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the causal bootstrap derivation

full rationale

The paper frames the causal bootstrap as a convex optimization over the set of positive spectral densities rho(omega) >= 0 whose smeared integrals are compatible with Euclidean data inside a covariance-defined region. Bounds on the target smeared observable follow from Lagrange duality to certified kernel bounds, which is a standard equivalence in convex optimization and does not reduce the output to any fitted parameter or self-defined quantity. The reduction of dual problems to semidefinite programs for polynomial/rational kernels invokes general techniques from the numerical conformal bootstrap literature without load-bearing self-citation. The formulation explicitly relates to classical moment problems and Nevanlinna-Pick interpolation as independent context. No self-definitional steps, fitted-input predictions, or ansatz smuggling appear in the derivation chain; the method is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the domain assumption that spectral densities are positive measures and that the Euclidean data define a non-empty compatibility region; no free parameters or invented entities are explicitly introduced in the summary.

axioms (1)
  • domain assumption Spectral densities are positive measures
    Invoked when the method optimizes over the convex set of positive spectral densities compatible with the data.

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