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Algorithmically Presented Numbers and Canonical Representations in Cryptographic Protocols

T0 review · 3 major / 0 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read Mathematical uniqueness of a value is not the same as a unique protocol byte sequence; arbitrary approximation programs of computable reals cannot be uniformly turned into canonical codes.

desk verdict Solid conceptual transfer of classical computable-analysis limits into crypto protocol language; useful taxonomy and worked examples, modest novelty, high soundness. read the letter →

arxiv 2607.05016 v1 pith:N27LWX64 submitted 2026-07-06 cs.CR

Arslan Brömme This is my paper · ORCID
classification cs.CR
keywords computablerealnumberscanonicalserializationrepresentationtheorycryptographicprotocolscanonicalizationbarrieranalysisprotocolinteroperability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that cryptographic protocols do not run on abstract mathematical values; they run on finite byte sequences, so the way a value is presented matters as much as which value it is. It separates three ideas: values that can be algorithmically approximated (the computable reals), values that admit a finite exact description inside a fixed representation system, and systems that are canonically normalizable so that every value has one computable normal form. The central negative claim is a canonicalization barrier: there is no computable procedure that takes arbitrary approximation programs for computable reals and outputs unique finite codes that coincide exactly when the reals are equal. The practical upshot is that protocols wanting byte-input invariance must restrict their inputs to a previously fixed, canonically normalizable presentation, of which the rational system with a normative fraction encoding is the core example. The same requirement is transferred to everyday protocol objects such as files, hashes, transaction IDs, and serialized payloads, and is illustrated with toy encryption and hashing examples plus a blockchain hash-anchoring protocol.

What carries the argument

The canonicalization barrier (Theorem 1): a classical undecidability-of-equality argument (via reduction from the halting problem) re-read as the non-existence of a universal extensional map from approximation-program indices to unique finite value codes. It is paired with the operational core class (Q, Sigma_Q) of rationals under a fixed canonical fraction encoding.

What would settle it

Exhibit a single partially computable function that, given any two machine indices of fast rational Cauchy names, returns identical finite codes exactly when the named reals are equal (or prove that some concrete, unrestricted protocol already does this without restricting the name language).

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Extended reading notes

Core claim

There is no partially computable extensional canonicalizer that is defined on all machine indices of fast rational Cauchy names of computable reals and produces the same finite code exactly when two names denote the same real. Protocols that need semantically equal values to produce identical protocol bytes must therefore restrict their input interface to a fixed, canonically normalizable presentation (such as the rational core class with specification Sigma_Q) rather than accepting arbitrary approximation programs.

Load-bearing premise

The barrier applies only when inputs are arbitrary approximation programs from an undecidable name space of all total Cauchy names; if the protocol already restricts to a decidable subclass of names that carries a computable equality test, the obstruction does not apply.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 0 minor

Summary. The paper develops a representation-theoretic framing for cryptographic protocol objects. It separates algorithmically approximable reals (A_app), finite exact describability in a system (A_fin(S)), and canonical normalizability of a system, and states a canonicalization barrier (Theorem 1): there is no partially computable extensional map that sends arbitrary machine indices of fast rational Cauchy names of computable reals to unique finite codes with c(e)=c(f) iff the denoted values are equal. As an operational core it proposes the rational system with a fixed encoding specification Sigma_Q, so A_ex = Q, and generalizes the idea to canonically serializable object classes (files, hashes, transaction IDs, payloads). Worked toy examples (XOR stream cipher, toy RSA, toy hash) and the author's snaproot hash-anchoring protocol illustrate that mathematical determinacy of a value does not yield operational byte uniqueness without a normative presentation, after which byte-level correctness and well-definedness arguments become available without further serialization or rounding decisions.

Significance. If accepted as a framing paper, the contribution is a clear transfer of classical computable-analysis limits (undecidability of equality of computable reals) into the language of cryptographic protocol specification, together with an explicit design trilemma and a taxonomy of operational representation levels (exact byte identity, fixed serialization, canonical value presentation). The paper is explicit that Theorem 1 is not a new undecidability result, that the toy schemes carry no security claims, and that canonicity is always relative to a fixed specification. Strengths include the self-contained reduction sketch for Lemma 1, the carefully scoped injectivity lemma (Lemma 2), the worked examples that make representation dependence concrete, and the honest limitations section. The novelty is primarily didactic and organizational rather than technical; for a cs.CR or interdisciplinary venue that values conceptual clarification of protocol interfaces, that can still be useful.

major comments (3)
  1. The central negative result (Theorem 1 / Corollary 1) is correctly stated as an immediate consequence of classical undecidability of equality on A_app (Lemma 1). The manuscript already acknowledges this in the introduction and Section 4. For journal publication the contribution claim should be tightened further so that readers cannot mistake the paper for a new undecidability theorem: the original content is the protocol-theoretic reading, the design trilemma, and the transfer to canonically serializable object classes, not the barrier itself.
  2. Section 7.5 and the well-definedness argument in Section 8.3 rest heavily on snaproot, which the paper correctly labels as the author's own case study and not independent external evidence (Limitations). The illustration is useful, but the manuscript should either (a) add at least one independent, widely deployed standard (e.g., a concrete DER/CBOR/JCS interoperability failure or success) as a second real-world anchor, or (b) clearly demote snaproot to a purely didactic example so that the empirical weight of the real-world claim matches the evidence.
  3. Definition 4 (canonically serializable object class) and the informal shorthand 'A_ex-compatible' transfer the rational-core idea to files, hashes, and payloads. The transfer is conceptually natural, but the formal link between the number-theoretic notions of Section 3 and the protocol-object notion of Section 3.5 remains mostly analogical. A short proposition or remark stating precisely which properties of (Q, Sigma_Q) are inherited by an arbitrary class K under a fixed (C, enc, dec) would make the taxonomy load-bearing rather than terminological.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1 is a direct classical reformulation of undecidability of equality on computable reals; self-citation of snaproot is only illustrative.

full rationale

The load-bearing negative result (Theorem 1 / Corollary 1) reduces immediately to the classical, externally established undecidability of equality of computable reals (Lemma 1, with an explicit halting-problem reduction sketch). The paper is explicit that the totality of approximation programs is an external non-decidable name space, not a representation system with decidable L, so the barrier applies only there and does not obstruct the recommended design of restricting to a fixed canonically normalizable presentation such as (Q, Σ_Q). The rational core class and its encoding rules Σ_Q are introduced by explicit normative stipulations, not fitted to data or derived from a self-referential uniqueness claim. The sole self-citation (snaproot [9]) is the author’s own prior protocol and is used strictly as a worked illustration of already-fixed byte identity and prescribed serialization; the paper itself labels it “the author’s own case study, not independent external evidence” and lists it under Limitations. No parameter is fitted and then re-presented as a prediction, no uniqueness theorem is imported from overlapping authors to force the main claim, and no known empirical pattern is merely renamed. The derivation chain is therefore self-contained against classical computability results and free of circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The central negative claim rests entirely on standard recursion-theoretic and computable-analysis facts already in the literature the paper cites. No free parameters are fitted. The only invented entities are definitional packaging (A_ex, Σ_Q, canonically serializable object class) introduced for the transfer to protocols; they do not carry independent empirical content. Domain assumptions are the usual model of deterministic Turing machines and the decidability of the description language L of a representation system.

assumptions (4)
  • standard math Equality of computable real numbers given by arbitrary fast rational Cauchy names is undecidable (classical, reduced from the halting problem).
    Invoked as Lemma 1 and used to prove Theorem 1; cited to Weihrauch and standard recursion theory.
  • domain assumption A representation system has a decidable language L of finite descriptions and a total evaluation algorithm Eval that produces rational approximations converging at rate 2^{-n}.
    Definition 1; deliberately narrower than general Type-2 representations, as the paper notes in Limitations.
  • standard math The set of all total approximation programs satisfying the Cauchy condition is not itself a decidable language (Rice-style index-set undecidability).
    Used in Section 3.3 to justify treating the totality of approximation programs as an external name space rather than a representation system.
  • domain assumption Once a normative encoding specification (C, enc, dec) is fixed, encoder and decoder are total computable relative to the fixed representations of the object class.
    Definition 3; standard relative-computability assumption for concrete encodings.
invented entities (2)
  • Operational core class A_ex = Q together with specification Σ_Q
    purpose: Provide a concrete, smallest field-structured class of values that admits a computable canonical normal form usable as protocol parameters.
    Definitional packaging of the familiar rationals under fixed fraction syntax; no new mathematical object is postulated.
  • Canonically serializable object class independent evidence
    purpose: Transfer the uniqueness property of (Q, Σ_Q) to practical protocol objects (files, hashes, txids, payloads).
    Purely definitional (Definition 4); independent evidence is the existence of already-standardized encodings (DER, JCS, etc.).

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Cite this review

Pith. "Pith review of Algorithmically Presented Numbers and Canonical Representations in Cryptographic Protocols." pith.science (2026). https://pith.science/paper/N27LWX64

@misc{pith2026260705016,
  author       = {Pith},
  title        = {Pith review of: Algorithmically Presented Numbers and Canonical Representations in Cryptographic Protocols},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N27LWX64}},
  note         = {Machine review of arXiv:2607.05016}
}
read the original abstract

This paper develops a representation-theoretic perspective on cryptographic protocols. The focus is not solely on the computability of the abstract value as an extensional property, but on the algorithmic structure of its presentation in a representation system: for operational use in protocols, algorithmic accessibility of the value does not suffice; its fixed presentation is also decisive. We distinguish three representation-theoretic notions -- algorithmically approximable (A_app, the computable real numbers), finitely exactly describable in a system (A_fin(S)), and canonical normalizability of a system -- and show that there is no computable extensional canonicalizer that uniformly transforms arbitrary approximation programs of computable real numbers into unique finite value encodings. As the operational rational core presentation we use the rational system with its canonical encoding specification Sigma_Q (fixed rules for valid fraction descriptions, canonical codes, and normalization); the associated value set is A_ex = Q. The notion of a canonically serializable object class transfers this core idea to practical protocol objects (files as byte sequences, hash values, transaction IDs, and normatively serialized payloads). We illustrate the consequences for interoperability, well-definedness, and verification with fully worked toy examples from symmetric and asymmetric encryption and hashing, and with a real-world example, the snaproot hash-anchoring protocol for blockchain-based file integrity verification. The paper thereby shows that the mathematical determinacy of a value and its operational uniqueness as a protocol object are two different requirements. Once a normative representation specification has been fixed, byte-level correctness and well-definedness arguments can be carried out without further implementation-dependent serialization or rounding decisions.

Figures

Figures reproduced from arXiv: 2607.05016 by the authors.

Figure 1
Figure 1. From the abstract value via the operational representation to cryptographic processing. The [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Canonicalization Failures as a Recurring Vulnerability Class: Representation Divergence in Cryptographic Systems and Its Avoidance

    cs.CR 2026-08 conditional novelty 4.0 of 10

    A cross-ecosystem taxonomy unifies canonicalization failures into one violated uniqueness condition with two break directions, plus a review checklist.

Reference graph

Works this paper leans on

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