pith. sign in

arxiv: 2605.20062 · v1 · pith:RQGCLB7Nnew · submitted 2026-05-19 · 🧮 math.AC · cs.IT· math.GR· math.IT

Cyclotomic finite-field Fourier spectra: Galois descent, native subfields, and residual coding

David Kumallagov , Daniil Sizikov , Anton Zarubin This is my paper

Pith reviewed 2026-05-20 03:13 UTC · model grok-4.3

classification 🧮 math.AC cs.ITmath.GRmath.IT
keywords Galois descentfinite fieldsFourier spectracyclotomic classesresidual codingfinite abelian groupssubfield products
0
0 comments X

The pith

Fourier spectra of base-field vectors over finite fields obey a Frobenius consistency relation that forces them into products of subfields indexed by cyclotomic classes.

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

The paper develops a Galois descent approach showing that when a Fourier transform is applied to a vector valued in a finite base field K, the resulting spectrum cannot be arbitrary but must satisfy a consistency condition under the Frobenius automorphism. This condition allows the one-dimensional spectra to be characterized exactly as products of subfields corresponding to the orbits under multiplication by q modulo the group order. The authors further introduce a two-stage decomposition for general vectors that separates a class-consistent component from a residual term, with the residual optimization factoring independently over the cyclotomic classes. These structures lead to concrete coding results including exact support minimization and recovery guarantees.

Core claim

We prove a general Galois-descent theorem for Fourier transforms on finite abelian groups. When the input vector takes values in the base field K, its spectrum satisfies the relation V_s^q equals V_{q s mod n}. We characterize the one-dimensional spectra as products of subfields indexed by q-cyclotomic classes and show that the orbit-seed representation is optimal in base-field coordinates. For arbitrary vectors we study the decomposition g equals f plus h where f is class-consistent and h is residual, and we derive exact support minimization, a symbol weight enumerator, recovery guarantees, global covering-radius formulas, random residual tail bounds, and entropy-type lower bounds.

What carries the argument

The Frobenius consistency relation V_s^q = V_{qs mod n} together with the Galois-descent theorem that enforces it on spectra of finite abelian group Fourier transforms.

If this is right

  • One-dimensional spectra of base-field vectors are exactly products of subfields indexed by the q-cyclotomic classes.
  • The orbit-seed representation uses the minimal number of base-field coordinates.
  • Residual optimization and support minimization separate independently over each cyclotomic class.
  • The class-consistent component admits an exact symbol weight enumerator and explicit recovery guarantees.
  • Global covering radius and random residual tail bounds are determined by the decomposition.

Where Pith is reading between the lines

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

  • The cyclotomic decomposition may allow trace maps and normal bases to be aligned with the subfield factors for simpler arithmetic.
  • Sparse-polynomial techniques for residuals could be implemented directly over the canonical subfield embeddings.
  • The separation of consistent and residual parts might extend to other linear transforms on finite groups.

Load-bearing premise

Applying a Fourier transform to a vector valued in the base field produces a spectrum obeying the relation that each coefficient raised to the q power equals the coefficient at the index multiplied by q.

What would settle it

Compute the Fourier transform of any vector with entries in the base field K and test whether there exists an index s such that the s-th spectrum coefficient raised to the q power differs from the coefficient at position q s mod n.

read the original abstract

We develop a Galois descent approach to finite-field Fourier spectra over an arbitrary finite base field. Let $\mathbb K=\mathbb F_q$ and $\mathbb L=\mathbb F_{q^m}$. If a Fourier transform is applied to a $\mathbb K$-valued vector, then its spectrum is not an arbitrary element of $\mathbb L^n$: it satisfies the Frobenius consistency relation \[ V_s^q=V_{qs \bmod n}. \] We prove a general Galois-descent theorem for Fourier transforms on finite abelian groups, characterize the one-dimensional spectra as products of subfields indexed by $q$-cyclotomic classes, and show that the orbit-seed representation is optimal in base-field coordinates. For arbitrary vectors in $\mathbb L^n$, we study a two-stage representation $g=f+h$, where $f$ is class-consistent and $h$ is a residual. The residual optimization separates over cyclotomic classes. We give exact support minimization, a symbol weight enumerator for the class-consistent code, recovery guarantees, global covering-radius formulas, random residual tail bounds, and entropy-type lower bounds. We also discuss implementation consequences for trace decompositions, normal bases, canonical subfield embeddings, and sparse-polynomial residual backends.

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

0 major / 3 minor

Summary. The manuscript develops a Galois descent framework for Fourier transforms on finite abelian groups over finite fields K = F_q and L = F_{q^m}. It establishes the Frobenius consistency relation V_s^q = V_{qs mod n} for spectra of K-valued vectors, proves a general descent theorem, characterizes one-dimensional spectra as products of subfields indexed by q-cyclotomic orbits, and shows optimality of the orbit-seed representation in base-field coordinates. For arbitrary vectors it introduces a two-stage decomposition g = f + h separating a class-consistent component f from a residual h, with optimization separating over cyclotomic classes, and derives exact support minimization, a symbol weight enumerator, recovery guarantees, global covering-radius formulas, random residual tail bounds, and entropy-type lower bounds, together with implementation remarks on trace decompositions, normal bases, and sparse-polynomial backends.

Significance. If the stated theorems hold, the work supplies a parameter-free algebraic structure for finite-field Fourier spectra that directly exploits the Galois action, yielding dimension counts equal to the number of orbits and separable residual problems. The explicit covering-radius and entropy bounds, together with the optimality claim for orbit-seed coordinates, constitute concrete, falsifiable contributions to algebraic coding and finite-field signal processing. The absence of invented entities or fitted parameters, and the grounding in standard Galois theory, are strengths.

minor comments (3)
  1. [Abstract] The abstract and opening paragraph state the consistency relation V_s^q = V_{qs mod n} but do not indicate the precise section where the general Galois-descent theorem for arbitrary finite abelian groups is proved; an explicit forward reference would help readers.
  2. [Section 5] In the discussion of residual optimization, the precise objective function (e.g., support size or Hamming weight) minimized over each cyclotomic class should be written as an equation rather than described in prose.
  3. [Section 6] The symbol weight enumerator for the class-consistent code is announced but its generating function is not displayed; inserting the explicit polynomial would make the result immediately usable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed and accurate summary of our work, the positive significance assessment, and the recommendation of minor revision. We appreciate the recognition of the Galois-descent framework, the orbit-seed optimality, and the residual-coding results as concrete contributions grounded in standard Galois theory. Since the report contains no specific major comments or requested changes, we interpret the minor-revision recommendation as an invitation to improve exposition, add clarifying remarks, and polish implementation notes. We outline our planned revisions below and confirm that all stated theorems remain unchanged.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained in standard Galois theory

full rationale

The paper's central consistency relation V_s^q = V_{qs mod n} is obtained directly by applying the Frobenius automorphism to the defining sum of the Fourier transform on a K-valued vector, without any fitted parameters or prior results from the same authors. The characterization of one-dimensional spectra as products of subfields indexed by q-cyclotomic classes, the separation of residual optimization over classes, and the optimality of the orbit-seed representation all follow immediately from partitioning the index set into orbits under s ↦ qs mod n and equating the K-dimension to the number of orbits. No self-citation is load-bearing, no ansatz is smuggled, and no prediction reduces to a fitted input by construction; the entire chain rests on elementary properties of finite fields and group actions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper rests on standard properties of finite fields and the Frobenius automorphism; it introduces the class-consistent/residual split as a new modeling device without independent external evidence.

axioms (2)
  • standard math Finite fields admit a Frobenius automorphism x |-> x^q of order equal to the extension degree.
    Invoked to define the consistency relation V_s^q = V_{qs mod n} that drives the entire descent.
  • domain assumption Fourier transforms exist and are well-defined on finite abelian groups over finite fields.
    Background assumption required for the Galois-descent theorem to apply.
invented entities (2)
  • class-consistent component f no independent evidence
    purpose: The part of the vector representation that obeys the Galois/Frobenius consistency relations.
    Central to the two-stage g = f + h decomposition introduced for residual optimization.
  • residual h no independent evidence
    purpose: The correction term that allows separate optimization and support minimization over cyclotomic classes.
    New modeling device whose properties (exact support minimization, tail bounds) are claimed in the abstract.

pith-pipeline@v0.9.0 · 5768 in / 1599 out tokens · 57623 ms · 2026-05-20T03:13:33.593384+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    Babai.The Fourier Transform and Equations over Finite Abelian Groups

    L. Babai.The Fourier Transform and Equations over Finite Abelian Groups. Lecture notes, University of Chicago, 2002. Available:https://people.cs.uchicago.edu/ ~laci/reu02/fourier.pdf

    work page 2002

  2. [2]

    Conrad.Characters of Finite Abelian Groups

    K. Conrad.Characters of Finite Abelian Groups. Expository notes. Available: https://kconrad.math.uconn.edu/blurbs/grouptheory/charthy.pdf

    work page

  3. [3]

    Ben-Or and P

    M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation. InProceedings of STOC 1988, pages 301–309, 1988

    work page 1988

  4. [4]

    R. E. Blahut. Transform techniques for error control codes.IBM Journal of Research and Development, 23(3):299–315, 1979. 19

    work page 1979

  5. [5]

    R. E. Blahut.Theory and Practice of Error Control Codes. Addison-Wesley, Reading, MA, 1983

    work page 1983

  6. [6]

    Bosma, J

    W. Bosma, J. Cannon, and A. Steel. Lattices of compatibly embedded finite fields. Journal of Symbolic Computation, 24(3–4):351–369, 1997

    work page 1997

  7. [7]

    W. C. Huffman and V. Pless.Fundamentals of Error-Correcting Codes. Cambridge University Press, 2003

    work page 2003

  8. [8]

    Delsarte

    P. Delsarte. On subfield subcodes of modified Reed-Solomon codes.IEEE Transactions on Information Theory, 21(5):575–576, 1975

    work page 1975

  9. [9]

    S. V. Fedorenko. The discrete Fourier transform over the binary finite field.IEEE Access, 11:62771–62779, 2023

    work page 2023

  10. [10]

    S. Gao, T. Mateer. Additive fast Fourier transforms over finite fields.IEEE Transac- tions on Information Theory, 56(12):6265–6272, 2010

    work page 2010

  11. [11]

    D. Y. Grigoriev, M. Karpinski, and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields.SIAM Journal on Computing, 19(6):1059–1063, 1990

    work page 1990

  12. [12]

    M. D. Huang and A. J. Rao. Interpolation of sparse multivariate polynomials over large finite fields with applications.Journal of Algorithms, 33(2):204–228, 1999

    work page 1999

  13. [13]

    Kaltofen and W

    E. Kaltofen and W. S. Lee. Early termination in sparse interpolation algorithms. Journal of Symbolic Computation, 36(3–4):365–400, 2003

    work page 2003

  14. [14]

    H. W. Lenstra and R. J. Schoof. Primitive normal bases for finite fields.Mathematics of Computation, 48(177):217–231, 1987

    work page 1987

  15. [15]

    Lidl and H

    R. Lidl and H. Niederreiter.Finite Fields. 2nd ed., Cambridge University Press, Cambridge, 1997

    work page 1997

  16. [16]

    F. Lübeck. Standard generators of finite fields and their cyclic subgroups.Journal of Symbolic Computation, 117:51–67, 2023

    work page 2023

  17. [17]

    F. J. MacWilliams. A theorem on the distribution of weights in a systematic code. Bell System Technical Journal, 42(1):79–94, 1963

    work page 1963

  18. [18]

    G. L. Mullen, D. Panario.Handbook of Finite Fields. Chapman and Hall/CRC, 2013

    work page 2013

  19. [19]

    R. C. Mullin, I. M. Onyszchuk, S. A. Vanstone, R. M. Wilson. Optimal normal bases inF pn.Discrete Applied Mathematics, 22(2):149–161, 1988

    work page 1988

  20. [20]

    J. M. Pollard. The fast Fourier transform in a finite field.Mathematics of Computation, 25(114):365–374, 1971

    work page 1971

  21. [21]

    D. S. Roche. What can and cannot we do with sparse polynomials? InProceedings of ISSAC 2018, pages 25–30, 2018

    work page 2018

  22. [22]

    Scheerhorn

    A. Scheerhorn. Trace- and norm-compatible extensions of finite fields.Applicable Algebra in Engineering, Communication and Computing, 3:199–209, 1992. 20

    work page 1992