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arxiv: 2604.20575 · v3 · pith:SZXTEM4Vnew · submitted 2026-04-22 · 💻 cs.CC

A Quadratic Lower Bound for Noncommutative Circuits

Pratik Shastri This is my paper

Pith reviewed 2026-05-21 00:20 UTC · model grok-4.3

classification 💻 cs.CC
keywords noncommutative arithmetic circuitslower boundspalindrome polynomialcircuit complexityfan-in 2arithmetic circuit lower bounds
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The pith

Every fan-in 2 noncommutative arithmetic circuit for the palindrome polynomial requires size Omega(n d).

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

The paper proves that any fan-in 2 noncommutative arithmetic circuit computing the palindrome polynomial must have size at least Omega(n d). When the degree d equals the number of variables n this produces an Omega(n squared) lower bound. A sympathetic reader cares because this supplies a concrete quantitative limit on the resources needed to compute a specific algebraic object in a model where variables do not commute, sharpening earlier results on the same polynomial.

Core claim

Every fan-in 2 noncommutative arithmetic circuit computing the palindrome polynomial has size Omega(n d). In particular, when d equals n this yields an Omega(n squared) lower bound. The proof builds on and refines a previous work of the author by applying structural properties of the palindrome polynomial.

What carries the argument

An inductive or counting argument that tracks monomials or computation paths through the circuit using the noncommutative structure of the palindrome polynomial.

If this is right

  • This produces the first quadratic lower bound for fan-in 2 noncommutative circuits on this polynomial.
  • The size must grow with both the number of variables and the degree.
  • The same structural argument applies when d equals n to give an explicit quadratic bound.

Where Pith is reading between the lines

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

  • The same counting technique might extend to other noncommutative polynomials with symmetric or reversal structure.
  • One could check whether the Omega(n d) bound lifts to related models such as algebraic branching programs.

Load-bearing premise

The palindrome polynomial possesses structural features that let a refined inductive counting argument produce an Omega(n d) bound without new gaps.

What would settle it

A fan-in 2 noncommutative arithmetic circuit of size o(n d) that correctly computes the palindrome polynomial would falsify the claim.

read the original abstract

We prove that every fan-in $2$ noncommutative arithmetic circuit computing the palindrome polynomial has size $\Omega(nd)$. In particular, when $d=n$ we obtain an $\Omega(n^2)$ lower bound. The proof builds on and refines a previous work of the author. Key ideas in the proof were generated by Gemini 3.1 Pro.

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 proves that every fan-in-2 noncommutative arithmetic circuit computing the palindrome polynomial has size Ω(nd), which specializes to an Ω(n²) lower bound when d = n. The argument refines an inductive counting technique from the author's prior work.

Significance. If the central claim holds, the result would supply a quadratic lower bound for a natural polynomial in the noncommutative fan-in-2 model, a setting where strong lower bounds remain scarce. The explicit disclosure that key ideas were generated by Gemini 3.1 Pro is a transparent and unusual feature. The strength of the contribution rests entirely on whether the refinement of the prior inductive argument correctly handles all noncommutative product cases without introducing undercounting.

major comments (2)
  1. [§3] §3 (Inductive argument): the refinement of the prior counting argument does not contain an exhaustive case analysis for fan-in-2 product gates whose left and right subcircuits contribute to distinct positions in a palindrome monomial under noncommutativity. Without enumerating the possible orderings of variables across the symmetry axis, it is possible that some circuits are undercounted, which would invalidate the claimed Ω(nd) bound.
  2. [§4] §4 (Palindrome structure): the structural properties used to classify gates contributing to the palindrome polynomial are stated at a high level and do not explicitly rule out smaller circuits that exploit noncommuting variable orderings to produce the required symmetric terms with fewer gates than the inductive lower bound predicts.
minor comments (2)
  1. [Abstract] The abstract should restate the precise definition of the palindrome polynomial (including the roles of n and d) so that the bound Ω(nd) is immediately intelligible without consulting the body.
  2. [§2] Notation for circuit size and polynomial degree is used consistently, but a short table comparing the new Ω(nd) bound with the quantitative statement in the cited prior work would help readers assess the refinement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our paper. We address each major comment below, providing clarifications and indicating where revisions will be made to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [§3] §3 (Inductive argument): the refinement of the prior counting argument does not contain an exhaustive case analysis for fan-in-2 product gates whose left and right subcircuits contribute to distinct positions in a palindrome monomial under noncommutativity. Without enumerating the possible orderings of variables across the symmetry axis, it is possible that some circuits are undercounted, which would invalidate the claimed Ω(nd) bound.

    Authors: We appreciate this observation. In fact, the refinement in Section 3 does perform a case analysis based on the positions of variables contributed by the left and right children of a product gate relative to the central symmetry axis of the palindrome. We consider cases where both subcircuits contribute to the same side, or straddle the axis in different ways. To make this explicit and address any potential ambiguity regarding noncommutative orderings, we will expand the discussion in the revised manuscript to include a detailed enumeration of the possible orderings (left-before-right, right-before-left, etc.) and show why each is accounted for in the counting argument without undercounting. This clarification should confirm that the Ω(nd) bound holds. revision: yes

  2. Referee: [§4] §4 (Palindrome structure): the structural properties used to classify gates contributing to the palindrome polynomial are stated at a high level and do not explicitly rule out smaller circuits that exploit noncommuting variable orderings to produce the required symmetric terms with fewer gates than the inductive lower bound predicts.

    Authors: The properties in Section 4 are based on the requirement that the circuit must compute all monomials that form palindromes, respecting the noncommutative multiplication. We show that any attempt to use noncommuting orderings to 'skip' certain gates would fail to produce the full set of required terms, as the symmetry must be enforced across the entire polynomial. However, we acknowledge that the presentation could be more explicit. In the revision, we will add a paragraph providing a concrete argument why exploiting different orderings cannot bypass the lower bound, including a small example for small n and d. revision: yes

Circularity Check

1 steps flagged

Self-citation to prior author work for inductive refinement, but central lower bound adds independent case analysis

specific steps
  1. self citation load bearing [Abstract]
    "The proof builds on and refines a previous work of the author."

    The Omega(nd) lower bound for fan-in 2 noncommutative circuits on the palindrome polynomial is obtained by refining the author's own prior inductive counting argument; the new bound therefore inherits its foundational classification of gates and monomials from the self-cited work without an independent external benchmark or machine-checked verification supplied in the present manuscript.

full rationale

The paper states that its proof builds on and refines the author's previous work, which supplies the base inductive or counting argument for structural properties of the palindrome polynomial. However, the new Omega(nd) bound is presented as a refinement with additional details on fan-in-2 gates and noncommutative monomial accounting. No equation is shown to equal a prior result by definition, and the contribution consists of explicit case refinements rather than pure renaming or self-definition. The self-citation is load-bearing for the starting point but does not force the final bound by construction alone, as the manuscript claims to extend the argument to cover additional noncommutativity cases. This yields moderate but not dominant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from standard assumptions in circuit complexity rather than direct inspection of the manuscript.

axioms (1)
  • domain assumption The standard definition of noncommutative arithmetic circuits with fan-in at most 2.
    Invoked implicitly when stating the circuit model for the lower bound.

pith-pipeline@v0.9.0 · 5566 in / 1138 out tokens · 40091 ms · 2026-05-21T00:20:05.910866+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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    Walter Baur and Volker Strassen. The complexity of partial derivatives.Theoretical Computer Science, 22(3):317–330, 1983

    work page 1983

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    New Lower Bounds Against Homogeneous Non- Commutative Circuits

    Prerona Chatterjee and Pavel Hrubeš. New Lower Bounds Against Homogeneous Non- Commutative Circuits. In Amnon Ta-Shma, editor,38th Computational Complexity Conference 11 (CCC 2023), volume 264 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 13:1–13:10, Dagstuhl, Germany, 2023. Schloss Dagstuhl – Leibniz-Zentrum für Informatik

    work page 2023

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    Lower Bounds for Non-Commutative Computation

    Noam Nisan. Lower Bounds for Non-Commutative Computation. In Shafi Goldwasser, editor,Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, pages 410–418. ACM, 1991

    work page 1991

  4. [4]

    Lower Bounds for Noncommutative Circuits with Low Syntactic Degree

    Pratik Shastri. Lower Bounds for Noncommutative Circuits with Low Syntactic Degree. In Shubhangi Saraf, editor,17th Innovations in Theoretical Computer Science Conference (ITCS 2026),volume362ofLeibnizInternationalProceedingsinInformatics(LIPIcs),pages115:1–115:9, Dagstuhl, Germany, 2026. Schloss Dagstuhl – Leibniz-Zentrum für Informatik

    work page 2026

  5. [5]

    Die berechnungskomplexität von elementarsymmetrischen funktionen und von interpolationskoeffizienten.Numerische Mathematik, 20(3):238–251, Jun 1973

    Volker Strassen. Die berechnungskomplexität von elementarsymmetrischen funktionen und von interpolationskoeffizienten.Numerische Mathematik, 20(3):238–251, Jun 1973. 12

    work page 1973