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arxiv: 2604.26400 · v2 · pith:SLV6QXZQ · submitted 2026-04-29 · cs.SC · cs.LO

Pseudo-Complex Quantifier Elimination

Nicolas Faro{\ss} , Thomas Sturm This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-01 08:54 UTCgrok-4.3pith:SLV6QXZQrecord.jsonopen to challenge →

classification cs.SC cs.LO
keywords quantifier eliminationcomplex numbersreal closed fieldssymbolic computationdecision proceduresordered rings
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The pith

A quantifier elimination procedure for the complex numbers reduces problems to real quantifier elimination and reinterprets the output.

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

The paper develops a method for eliminating quantifiers from logical formulas whose variables range over the complex numbers. The language includes the imaginary unit along with operations that extract real and imaginary parts and form conjugates. The central technique translates each such problem into an equivalent problem over the real numbers, solves it with existing real quantifier elimination software, and then maps the resulting formula back into the complex language by a direct reinterpretation step. If the reinterpretation step is faithful, the procedure yields a quantifier-free formula that is logically equivalent over the complexes to the original input.

Core claim

The framework performs quantifier elimination for the complex numbers by first mapping each input formula into the language of ordered rings over the reals, applying a real quantifier elimination algorithm, and then applying a heuristic reinterpretation that replaces the real variables and operations with their complex counterparts including the imaginary unit, real-part, imaginary-part, and conjugate symbols.

What carries the argument

Reduction of complex formulas to real quantifier elimination followed by heuristic reinterpretation of the resulting real formula inside the complex language.

If this is right

  • Existing real quantifier elimination implementations become directly usable for complex formulas that mention the imaginary unit and conjugates.
  • Decision procedures for statements in the language of ordered rings extended by imaginary-unit symbols become available without building a separate complex solver from scratch.
  • Computational examples can be run immediately in the prototype implementation inside the Python system Logic1.

Where Pith is reading between the lines

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

  • The same reduction-plus-reinterpretation pattern might be tested on other field extensions or on ordered fields with additional algebraic structure.
  • If the reinterpretation step can be proved correct rather than merely heuristic, the method would supply a fully rigorous decision procedure for the extended complex language.

Load-bearing premise

The heuristic reinterpretation of the real quantifier elimination result always produces a formula that is logically equivalent over the complexes to the original input.

What would settle it

A concrete quantified formula over the complexes for which the reinterpreted output formula evaluates to a different truth value than the original formula under the standard interpretation of the complex numbers.

Figures

Figures reproduced from arXiv: 2604.26400 by Nicolas Faro{\ss}, Thomas Sturm.

Figure 1
Figure 1. Figure 1: Circuit diagram of a passive RC high-pass filter Example 10 (Gain of passive RC high-pass filter) view at source ↗
Figure 2
Figure 2. Figure 2: Circuit diagram of an active RC filter; see [18, Example 1] normal form used time (s) 3 Cartesian coordinates conjugate 0.04 4 Roots of unity, φ1 conjugate 0.03 4 Roots of unity, φ2 conjugate 1.23 5 Counterexample for geometry provers conjugate 0.07 6 Orthogonality (n = 3) conjugate 0.2 7 Cauchy-Schwarz inequality (n = 4) conjugate 24.57 8 Self-adjoint matrices, φ1 conjugate 1.28 8 Self-adjoint matrices, φ… view at source ↗
read the original abstract

We describe the design of a quantifier elimination framework for the complex numbers in the language of ordered rings supplemented with symbols for the imaginary unit, real parts, imaginary parts, and conjugates. Technically, we use a reduction to real quantifier elimination followed by a heuristic reinterpretation of the results within our complex framework. We present computational examples using a prototypical implementation of our approach in our Python-based open-source system Logic1.

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 / 0 minor

Summary. The paper describes the design of a quantifier elimination framework for the complex numbers in the language of ordered rings augmented with symbols for the imaginary unit i, real parts Re, imaginary parts Im, and conjugates. The approach reduces complex QE to real quantifier elimination followed by a heuristic reinterpretation of the results back into the complex language, and demonstrates the method via a prototypical open-source implementation in the Python-based Logic1 system together with computational examples.

Significance. If the heuristic reinterpretation step can be shown to preserve logical equivalence, the framework would provide a practical reduction-based method for complex QE that leverages existing real QE tools without requiring a fully independent decision procedure. The open-source prototypical implementation supports reproducibility and allows direct testing of the examples. However, the heuristic nature of the core step limits the result's theoretical weight absent a soundness argument.

major comments (1)
  1. [Abstract] Abstract (technical approach paragraph): The central claim depends on reducing to real QE and then applying a heuristic reinterpretation to recover results in the language with i, Re, Im, and conj; no details, algorithm, or argument establishing that this reinterpretation preserves logical equivalence are supplied, leaving the soundness of the entire framework unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and the recommendation for major revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (technical approach paragraph): The central claim depends on reducing to real QE and then applying a heuristic reinterpretation to recover results in the language with i, Re, Im, and conj; no details, algorithm, or argument establishing that this reinterpretation preserves logical equivalence are supplied, leaving the soundness of the entire framework unverified.

    Authors: The manuscript describes the core step explicitly as a 'heuristic reinterpretation' (abstract and introduction), without claiming or providing a general argument that it preserves logical equivalence. The contribution is positioned as a practical reduction to existing real QE tools, supported by a prototypical open-source implementation and computational examples rather than a complete decision procedure. We agree that this leaves the framework without a verified soundness guarantee in the theoretical sense noted by the referee. We will revise the abstract to state the heuristic limitation more explicitly and to avoid any implication of guaranteed equivalence. revision: yes

Circularity Check

0 steps flagged

No significant circularity: reduction to external real QE plus explicit heuristic

full rationale

The paper's central technical step is a reduction to existing real quantifier elimination followed by a heuristic reinterpretation step that is explicitly labeled as such in the abstract. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. The derivation chain relies on standard real QE (an independent external method) and does not reduce any claimed result to its own inputs by construction. This is the normal case of a design paper presenting a new framework without circular reasoning.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach assumes standard real quantifier elimination exists and can be applied after reduction; the complex language extension is treated as a domain assumption. No free parameters or invented entities are introduced.

axioms (2)
  • standard math Quantifier elimination procedures exist for the theory of real ordered rings
    The method explicitly reduces complex problems to real quantifier elimination.
  • domain assumption The language of ordered rings extended with symbols for imaginary unit, real/imaginary parts, and conjugates is semantically well-defined
    The framework is defined in this supplemented language.

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