The Incompatibility of Quantum and Gravitational Formalisms: Why Quantum Gravity is a Category Error

Abstract

For nearly a century, theoretical physics has pursued the unification of quantum mechanics and general relativity under the rubric of "quantum gravity." Despite immense mathematical sophistication, this endeavor has produced neither empirical verification nor conceptual consensus. In this paper, we demonstrate that the persistent impasse of quantum gravity is not an empirical deficit or technical limitation, but a foundational category error resulting from a syntactic collision. Building upon the two-level ontic/epistemic architecture (𝒢, 𝒟, Π) and probe-screening theorems established in [22]–[25], we prove that the quantum register 𝒟_QM (a linear, projective, Hilbertian screening syntax) and the gravitational register 𝒟_GR (a non-linear, pseudo-Riemannian, transmitting syntax) are mutually incompatible mathematical habitats formalizing structurally asymmetric descriptive projections of a single underlying non-linear substrate 𝒢. We analyze this structural incompatibility across eight rigorous mathematical dimensions and expose four false presuppositions anchoring the quantization programme: the fundamental status of the gravitational field, the universal validity of quantum syntax, the necessity of quantizing metric geometry, and the misidentification of the conflict as lying between "quantum mechanics" and "general relativity" rather than between the informationally-constrained epistemic syntax — whose mathematical structure (non-commutativity, linear screening, microcausality, spectral condition) emerges from the three probe axioms A1–A3 acting through the Hilbert-space channel — and the ontological geometry generated by A2+A3 acting through the metric channel. The key differentiator is A1: the energy cost of observation, whose coupling regime — ranging continuously from negligible collective imprint on the substrate geometry (where metric transmission remains commutative as a limiting case) to constitutive excitation of individual substrate modes (forcing non-commutative Hilbertian screening) — determines whether the constitutive embeddedness of the probe (A3) manifests as commutative geometric transmission (𝒟_GR) or non-commutative linear screening syntax (𝒟_QM). We provide a structural diagnostic of three major contemporary programmes: string theory (syntactic juxtaposition), loop quantum gravity (misapplication of linear quantization to an emergent metric interface), and Alain Connes' non-commutative geometry (a mathematically coherent spectral interface directly probing substrate structure). Furthermore, we establish that gravitation and gauge interactions constitute the same geometric syntax (𝒟_GR) operating across two distinct observational scales: macroscopic metric curvature and soliton-scale moduli holonomy (Aharonov-Bohm logic). The framework yields direct falsifiable predictions, notably the exact persistence of continuous Lorentz invariance without Planck-scale spacetime discretization (Prediction), in agreement with current astrophysical bounds.

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BibTeX RIS

BibTeX

@misc{dechery2026incompatibility,
  title = {{The Incompatibility of Quantum and Gravitational Formalisms: Why Quantum Gravity is a Category Error}},
  author = {Fabien Déchery},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05857958v1},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - The Incompatibility of Quantum and Gravitational Formalisms: Why Quantum Gravity is a Category Error
AU  - Fabien Déchery
PY  - 2026
DA  - 2026/10/29
PB  - HAL
UR  - https://hal.science/hal-05857958v1
AB  - For nearly a century, theoretical physics has pursued the unification of quantum mechanics and general relativity under the rubric of "quantum gravity." Despite immense mathematical sophistication, this endeavor has produced neither empirical verification nor conceptual consensus. In this paper, we demonstrate that the persistent impasse of quantum gravity is not an empirical deficit or technical limitation, but a foundational category error resulting from a syntactic collision. Building upon the two-level ontic/epistemic architecture (𝒢, 𝒟, Π) and probe-screening theorems established in [22]–[25], we prove that the quantum register 𝒟_QM (a linear, projective, Hilbertian screening syntax) and the gravitational register 𝒟_GR (a non-linear, pseudo-Riemannian, transmitting syntax) are mutually incompatible mathematical habitats formalizing structurally asymmetric descriptive projections of a single underlying non-linear substrate 𝒢. We analyze this structural incompatibility across eight rigorous mathematical dimensions and expose four false presuppositions anchoring the quantization programme: the fundamental status of the gravitational field, the universal validity of quantum syntax, the necessity of quantizing metric geometry, and the misidentification of the conflict as lying between "quantum mechanics" and "general relativity" rather than between the informationally-constrained epistemic syntax — whose mathematical structure (non-commutativity, linear screening, microcausality, spectral condition) emerges from the three probe axioms A1–A3 acting through the Hilbert-space channel — and the ontological geometry generated by A2+A3 acting through the metric channel. The key differentiator is A1: the energy cost of observation, whose coupling regime — ranging continuously from negligible collective imprint on the substrate geometry (where metric transmission remains commutative as a limiting case) to constitutive excitation of individual substrate modes (forcing non-commutative Hilbertian screening) — determines whether the constitutive embeddedness of the probe (A3) manifests as commutative geometric transmission (𝒟_GR) or non-commutative linear screening syntax (𝒟_QM). We provide a structural diagnostic of three major contemporary programmes: string theory (syntactic juxtaposition), loop quantum gravity (misapplication of linear quantization to an emergent metric interface), and Alain Connes' non-commutative geometry (a mathematically coherent spectral interface directly probing substrate structure). Furthermore, we establish that gravitation and gauge interactions constitute the same geometric syntax (𝒟_GR) operating across two distinct observational scales: macroscopic metric curvature and soliton-scale moduli holonomy (Aharonov-Bohm logic). The framework yields direct falsifiable predictions, notably the exact persistence of continuous Lorentz invariance without Planck-scale spacetime discretization (Prediction), in agreement with current astrophysical bounds.
ER  -

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