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arxiv: 2605.22477 · v1 · pith:KXFAFNRBnew · submitted 2026-05-21 · 💻 cs.CR · math.CO

Exact Hidden Paths in Noisy High Dimensional Path Spaces

Victor Duarte Melo This is my paper

Pith reviewed 2026-05-22 05:09 UTC · model grok-4.3

classification 💻 cs.CR math.CO
keywords hidden path recoverynoisy discrete pathscryptographic frameworkexact recoveryhigh-dimensional path spacesobservable vectorspost-quantum cryptography
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The pith

Exact recovery of hidden trajectories is possible from incomplete noisy projections in high dimensional discrete path spaces when using large observable vectors.

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

The paper establishes a mathematical and cryptographic framework for recovering one exact hidden trajectory from noisy, projected, and aggregated observables in high dimensional discrete path spaces. It shifts from the usual path integral approach of summing over trajectories to the inverse problem of isolating the single planted path that includes macro steps, microscopic perturbations, and discrete noise. Public information is supplied as large observable vectors rather than short hash digests to avoid bounding recovery by digest size. The work distinguishes approximate reconstruction of coarse geometry from exact recovery of the precise microscopic sequence and formalizes several recovery notions including planted exact recovery and quotient recovery. It also surveys attack surfaces such as lattice methods and quantum search as preparation for possible future cryptographic use.

Core claim

The central claim is that exact hidden path recovery can be formalized in noisy high dimensional discrete path spaces where the hidden object is a planted discrete path whose transitions may include macro steps, microscopic perturbations, and discrete noise, with public information represented by large observable vectors that allow recovery of the precise sequence without being bounded by the amount of public data. Approximate reconstruction and exact recovery are fundamentally different tasks, and multiple recovery notions can be defined including planted exact recovery, arbitrary witness recovery, canonical recovery, quotient recovery, and recovery of derived encodings.

What carries the argument

The planted discrete path whose transitions include macro steps, microscopic perturbations, and discrete noise, serving as the hidden object to be recovered exactly from the large observable vectors.

If this is right

  • Approximate reconstruction of dominant regions or coarse geometry does not recover the precise microscopic sequence that defines the hidden path.
  • Multiple distinct recovery notions exist, such as planted exact recovery, quotient recovery, and recovery of derived encodings.
  • Attack surfaces for any future construction include linearization attacks, lattice-style recovery, dynamic programming, meet-in-the-middle methods, SAT and SMT solvers, and generic quantum search.
  • Using short hash digests would inherently limit the recovery problem to the digest size, whereas large observable vectors avoid this bound.

Where Pith is reading between the lines

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

  • The framework could serve as a starting point for designing cryptographic primitives whose security rests on the hardness of exact path recovery rather than on traditional number-theoretic problems.
  • Small-scale computational tests with controlled noise levels could directly measure the transition between approximate and exact recovery regimes.
  • The approach may connect to inverse problems in statistical physics where path-based models are inverted rather than summed.

Load-bearing premise

Large observable vectors can provide enough information to allow exact recovery of the microscopic path sequence without the problem being fundamentally bounded by the amount of public data.

What would settle it

A concrete calculation or small-instance experiment in which exact recovery of the full microscopic sequence fails for all observable vector sizes once noise or dimension exceeds a specific threshold.

Figures

Figures reproduced from arXiv: 2605.22477 by Victor Duarte Melo.

Figure 1
Figure 1. Figure 1: The path sum intuition. Many trajectories contribute to a global quantity, but the inverse [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A low dimensional observable can preserve a visible trend while losing a large number of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A transition decomposed into macro increment, micro perturbation, and noise. Approxi [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Observable generation pipeline. The observables are structured mathematical data, not [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A dominant region may contain many microscopic path objects. Locating the region does [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two roles of noise. The present framework mainly uses internal noise to increase [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A hierarchy of recovery notions. Approximate reconstruction and exact microscopic [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Cryptographic one way interpretation. The public object is a large observable vector, [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
read the original abstract

We introduce a mathematical and cryptographic framework for exact recovery of noisy hidden paths in high dimensional discrete path spaces. The work is inspired by the path integral viewpoint, where global quantities arise from contributions over many possible trajectories. Instead of approximating a global path sum, we study the inverse problem of recovering one exact hidden trajectory from incomplete, noisy, projected, and aggregated observables. The hidden object is a planted discrete path whose transitions may include macro steps, microscopic perturbations, and discrete noise. Public information is represented by large observable vectors rather than short hash digests, since excessive compression would bound the effective recovery problem by the digest size. We formalize several recovery notions, including planted exact recovery, arbitrary witness recovery, canonical recovery, quotient recovery, and recovery of derived encodings. The main distinction is that approximate reconstruction and exact recovery are fundamentally different tasks. A method may reveal coarse geometry or dominant regions without recovering the precise microscopic sequence defining the hidden path. We also discuss attack surfaces relevant to future cryptographic use, including linearization, lattice style recovery, dynamic programming, meet in the middle attacks, SAT and SMT formulations, approximation followed by rounding, witness collisions, and generic quantum search. This work does not claim a complete post quantum cryptosystem. It provides a formal framework for studying exact hidden path recovery as a possible foundation for future cryptographic constructions

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

1 major / 2 minor

Summary. The paper introduces a mathematical and cryptographic framework for exact recovery of noisy hidden paths in high dimensional discrete path spaces. It is inspired by the path integral viewpoint but focuses on the inverse problem of recovering one exact hidden trajectory from incomplete, noisy, projected, and aggregated observables represented by large observable vectors. The work formalizes recovery notions including planted exact recovery, arbitrary witness recovery, canonical recovery, quotient recovery, and recovery of derived encodings, while distinguishing approximate reconstruction from exact recovery. It also outlines attack surfaces such as linearization, lattice-style recovery, dynamic programming, meet-in-the-middle, SAT/SMT, approximation-plus-rounding, witness collisions, and generic quantum search. The manuscript explicitly states that it does not claim a complete post-quantum cryptosystem but provides a formal framework for studying exact hidden path recovery as a foundation for future constructions.

Significance. If the distinctions between approximate and exact recovery can be made rigorous and the attack surfaces operationalized, the framework could help guide the design of cryptographic primitives whose security relies on the hardness of microscopic path recovery in high-dimensional noisy spaces. The deliberate use of large observable vectors (rather than short digests) to avoid information-theoretic bounds is a coherent design choice consistent with the stated scope. The paper's primary contribution is conceptual and definitional; its long-term significance will depend on whether follow-up work supplies the missing formalizations, examples, or hardness results.

major comments (1)
  1. Abstract and overall manuscript: the central claim is that a 'formal framework' is provided for recovery notions and attack surfaces, yet the text contains no explicit mathematical definitions, axioms, equations, or theorems that operationalize planted exact recovery, quotient recovery, or the listed attack surfaces. Without these, it is not possible to verify internal consistency or assess whether the framework is load-bearing for future cryptographic constructions.
minor comments (2)
  1. The distinction between 'approximate reconstruction' and 'exact recovery' is stated clearly in the abstract but would benefit from a short illustrative example (even a low-dimensional toy case) to make the difference concrete for readers.
  2. Terms such as 'macro steps, microscopic perturbations, and discrete noise' are introduced without immediate definitions or references; adding a dedicated preliminary section with notation would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below and will strengthen the formal aspects of the framework in revision.

read point-by-point responses

  1. Referee: Abstract and overall manuscript: the central claim is that a 'formal framework' is provided for recovery notions and attack surfaces, yet the text contains no explicit mathematical definitions, axioms, equations, or theorems that operationalize planted exact recovery, quotient recovery, or the listed attack surfaces. Without these, it is not possible to verify internal consistency or assess whether the framework is load-bearing for future cryptographic constructions.

    Authors: We acknowledge the validity of this observation. The current manuscript introduces the framework at a conceptual and definitional level, outlining recovery notions and attack surfaces in descriptive terms without supplying explicit mathematical definitions, axioms, or theorems. This approach was chosen to establish the high-level distinctions (e.g., between approximate reconstruction and exact recovery) and to identify relevant attack surfaces for future work. However, to make the notions operational, internally consistent, and suitable as a foundation for cryptographic constructions, we agree that more rigorous formalization is required. In the revised version we will add: (i) precise mathematical definitions for planted exact recovery, arbitrary witness recovery, canonical recovery, quotient recovery, and recovery of derived encodings; (ii) basic propositions or characterizations that relate these notions; and (iii) structured descriptions or pseudocode outlines for the listed attack surfaces (linearization, lattice-style recovery, dynamic programming, meet-in-the-middle, SAT/SMT, approximation-plus-rounding, witness collisions, and quantum search) to render them more concrete. These additions will allow verification of consistency and better evaluation of the framework's load-bearing capacity. We continue to view the work as a starting point rather than a complete theory. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a conceptual framework for formalizing recovery notions and attack surfaces in noisy high-dimensional path spaces but advances no mathematical derivations, predictions, or first-principles results. It explicitly states that it does not claim a complete post-quantum cryptosystem and instead provides definitions and discussion points as a possible foundation for future constructions. The distinction between approximate and exact recovery, along with the choice of large observable vectors, is presented as a deliberate scoping decision rather than a derived claim. No equations, fitted parameters, or self-citation chains appear in the provided text that would reduce any asserted result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the framework rests on the domain assumption that exact recovery remains possible from aggregated observables; no free parameters, invented entities, or additional axioms are specified.

axioms (1)
  • domain assumption Large observable vectors suffice for exact microscopic path recovery without being limited by digest size.
    Stated explicitly when contrasting with short hash digests.

pith-pipeline@v0.9.0 · 5760 in / 1222 out tokens · 47903 ms · 2026-05-22T05:09:07.488636+00:00 · methodology

discussion (0)

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Reference graph

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