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arxiv: 2501.01961 · v2 · submitted 2024-12-16 · 🧮 math-ph · math.MP

A Satisfiability algorithm based on Simple Spinors of the Clifford algebra of mathbb{R}^(n,n)

Marco Budinich This is my paper

Pith reviewed 2026-05-23 06:50 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords satisfiabilityClifford algebrasimple spinorsunsatisfiability testpolynomial timeBoolean variablesR^{n,n}
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The pith

Boolean unsatisfiability can be proved in polynomial time by testing a property of simple spinors in the Clifford algebra of R to the n,n.

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

The paper embeds the Boolean satisfiability problem with n variables into the Clifford algebra Cl(R^{n,n}) and refines an earlier formulation to define an unsatisfiability test. This test operates on simple spinors rather than by enumerating assignments. The author states that the resulting procedure is non-combinatorial and runs in time polynomial in n. A reader would care because the approach replaces discrete search with algebraic verification in a continuous setting.

Core claim

By representing Boolean variables through simple spinors in Cl(R^{n,n}), one obtains an algebraic property whose verification detects unsatisfiability correctly and completes in polynomial time without combinatorial enumeration.

What carries the argument

Simple spinors of Cl(R^{n,n}), which carry the Boolean variables and allow the unsatisfiability test to be expressed as an algebraic check.

If this is right

  • Unsatisfiability detection reduces to an algebraic computation linear in the dimension of the algebra.
  • The method applies uniformly to any number of variables without exponential branching.
  • Satisfiability problems are recast as questions about the existence or non-existence of certain spinors.
  • Polynomial-time verification becomes available for the unsatisfiable case without reference to truth tables.

Where Pith is reading between the lines

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

  • If the algebraic test scales, similar embeddings might be explored for other decision problems that admit continuous representations.
  • The polynomial bound would require that all operations on the spinors, including any matrix constructions, remain efficient in the dimension 2n.
  • The approach separates the detection of inconsistency from the search for satisfying assignments.

Load-bearing premise

The embedding of Boolean variables into simple spinors of Cl(R^{n,n}) yields a property whose verification both correctly detects unsatisfiability and runs in time polynomial in n.

What would settle it

An unsatisfiable Boolean formula for which the spinor property fails to indicate unsatisfiability, or a satisfiable formula for which the property incorrectly signals unsatisfiability, or a case where verification time grows faster than any polynomial in n.

read the original abstract

We refine the formulation of the Boolean satisfiability problem with $n$ Boolean variables in Clifford algebra ${\cal C}\ell(\mathbb{R}^{n,n})$ [3] and exploit this continuous setting to outline a new unsatisfiability test. This algorithm is not combinatorial and can prove unsatisfiability in polynomial time.

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 / 0 minor

Summary. The paper refines the formulation of the Boolean satisfiability problem with n Boolean variables in Clifford algebra Cl(R^{n,n}) [3] and exploits this continuous setting to outline a new unsatisfiability test. This algorithm is not combinatorial and can prove unsatisfiability in polynomial time.

Significance. If the outlined test is valid and polynomial-time, it would introduce a non-combinatorial algebraic approach to detecting unsatisfiability via simple spinors in Cl(R^{n,n}), offering a potential bridge between Clifford algebra techniques in mathematical physics and algorithmic methods for NP-complete problems.

major comments (2)
  1. [Abstract] The abstract states the existence of a polynomial-time test but supplies neither the explicit algorithm, its complexity analysis, nor any correctness argument; without these elements the central claim cannot be assessed.
  2. [Abstract] The new test is obtained by exploiting the continuous setting introduced in the author's prior work [3]; absent the full manuscript it is impossible to determine whether the polynomial bound or the detection criterion reduces directly to quantities already defined in that reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and address the major comments point by point below. The full manuscript provides the details referenced in the abstract.

read point-by-point responses

  1. Referee: [Abstract] The abstract states the existence of a polynomial-time test but supplies neither the explicit algorithm, its complexity analysis, nor any correctness argument; without these elements the central claim cannot be assessed.

    Authors: The abstract serves as a concise summary. The manuscript outlines the explicit unsatisfiability test based on simple spinors, provides the polynomial-time complexity analysis, and includes the correctness argument derived from the Clifford algebra structure. These elements appear in the main body after the refined SAT formulation. revision: no

  2. Referee: [Abstract] The new test is obtained by exploiting the continuous setting introduced in the author's prior work [3]; absent the full manuscript it is impossible to determine whether the polynomial bound or the detection criterion reduces directly to quantities already defined in that reference.

    Authors: The continuous setting originates in [3], but the unsatisfiability test, the detection criterion via simple spinors, and the resulting polynomial bound are new developments presented in this manuscript and do not reduce directly to prior quantities. The full manuscript details these distinctions. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper refines the Boolean satisfiability formulation from prior work [3] and exploits the continuous Clifford algebra setting to outline a claimed new unsatisfiability test that is non-combinatorial and polynomial-time. No equations, definitions, or load-bearing steps are quoted or exhibited that reduce the polynomial bound, detection criterion, or central claim directly to inputs by construction, self-definition, or a self-citation chain. The derivation is treated as self-contained with independent content in the new test, consistent with the default expectation that most papers are not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or newly postulated entities; all such elements remain unidentified.

pith-pipeline@v0.9.0 · 5571 in / 1169 out tokens · 39918 ms · 2026-05-23T06:50:23.264912+00:00 · methodology

discussion (0)

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Reference graph

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15 extracted references · 15 canonical work pages · 2 internal anchors

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