Papers on the quantum-geometric correspondence—the hypothesis that quantum mechanics and general relativity are complementary projections of a single underlying reality.
Quantum-Geometric Correspondence Series
Eleven papers, grouped into three themes: a foundations quartet that fixes the axioms, derives gravitational decoherence at the Wheeler-DeWitt rate, reviews programme status, extends geometric modular flow to the physical graviton, and bounds the obstruction to the modular identity; four decoherence papers turning that prediction into laboratory and cosmic tests; and three cosmology papers on holographic dark energy, emergent gravity, and the de Sitter horizon. Start with the canonical core; each satellite is self-contained and cites it.
Reading order & paper relationships
Foundations — the canonical core
The Quantum-Geometric Correspondence — the three axioms, the gravitational-decoherence mechanism, the Wheeler-DeWitt G¹ rate, the information-theoretic bound, and the second-quantized extension, all in one place. Start here.
Status and outlook — programme-level review: what is established, what is conjecture, falsifiability map, and roadmap
Graviton modular flow — geometric diamond modular flow for the physical spin-2 field via dimensional reduction; Type II crossed product extends to gravitons
Modular-identity obstruction — bounding the deviation from K = 2π H_phys in a finite causal diamond; the (d/h)⁴ quadrupole and the open constants that decide the rate
Decoherence — laboratory & cosmic tests
BMV prediction — mass-independent cube law (and magic-angle quintic) for the tabletop entanglement test
Decoherence floor & signatures — the finite-size saturated floor (Γ_sat ∝ M^5/3), platform reach, and the entanglement-decay signature unique to gravity
Cosmic decoherence — resolving the naive 121-order infinity on a de Sitter background
Modular spectroscopy — the full noise spectrum S(ω) and the C-independent knee that discriminates the modular mechanism from collapse
Cosmology — dark energy & emergent gravity
Holographic cosmology — one horizon entropy budget giving ρ_DE ≈ 0.082 c²H²/G and the Past Hypothesis / arrow of time
Horizon clock — the MOND scale and cosmic decoherence rate as one de Sitter frequency
Foundations
Canonical CoreJune 9, 2026
The Quantum-Geometric Correspondence: Three Axioms and Gravitational Decoherence at Order G
Quantum mechanics and general relativity disagree about what spacetime is; we develop the hypothesis that they are complementary projections of a single underlying reality. Three primitive axioms carry the framework: Generalized Entropy (the area–entanglement ledger of Type II gravitational algebras), Entanglement Equilibrium (stationarity of that entropy for every small causal diamond, from which the Einstein equations follow as a theorem), and Modular Time (the diamond's modular flow is physical time, K = 2π H_phys + O(G²)—established in expectation values and solvable models, conjectured as a full operator identity).
The falsifiable consequence is gravitational decoherence. The linearized Wheeler–DeWitt constraint entangles a matter superposition with its gravitational dressing; tracing out the geometry decoheres the matter at Γ_dec = C·GM²/(ℏd) with C ∈ [2/π, 1] and C = 1 for interferometric visibility, giving τ_dec = ℏd/(GM²) ≈ 1.6 ns for a 1 μg particle separated by 1 mm. Standard perturbative quantum field theory, whose product initial state violates the constraint, predicts a G² rate ~3×10³⁴ times slower—so even an order-of-magnitude measurement is decisive.
This canonical core absorbs the framework's axioms, the gravitational-decoherence mechanism, the Wheeler–DeWitt rate, the information-theoretic bound, and the second-quantized field extension. The holographic dark energy and emergent-gravity results are developed in companion papers. Experiment at the microgram scale decides the scaling.
@article{Sperzel2026QGC,
author = {Sperzel, Marc},
title = {The Quantum-Geometric Correspondence: Three Axioms and Gravitational Decoherence at Order G},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/canonical-theory.html}
}
FoundationsJune 16, 2026
The Quantum-Geometric Correspondence: Status and Outlook
The Quantum-Geometric Correspondence treats quantum mechanics and general relativity as complementary projections of a single quantum-informational reality. We review the programme's status. Three primitive axioms—generalized entropy, entanglement equilibrium, and modular time—recover the Bekenstein-Hawking entropy, the holographic bound, the Unruh temperature, the Einstein equations, and a thermodynamic dark-energy density ρ_DE = α c² H² / G.
The programme's one new dynamical prediction is gravitational decoherence of a mass superposition at the rate Γ_dec = C E_G / ℏ with E_G = G M² / d and C ∈ [2/π, 1], giving τ_dec ~ 1.6 ns for a microgram over a millimetre—some thirty-five orders of magnitude faster than the perturbative (~G²) result, and hence experimentally decisive. We distinguish what is established from what is not: the energy scale E_G is proven, while the operator identity K = 2π H_phys that converts it into a rate is a conjecture reduced, through a finite causal diamond (Type II crossed-product) construction, to a single open question—the clock-model normalization that fixes C.
We collect the falsifiability map, summarize the cosmological sector (the MOND scale a₀ = c H₀ / (2π), holographic dark energy, and a cosmic decoherence rate Γ_total = g(1) H / (4π) that share a single de Sitter frequency), position the programme against spontaneous-collapse, Diósi-Penrose, and stochastic-gravity approaches, and give the open problems and roadmap.
@article{Sperzel2026QGCStatus,
author = {Sperzel, Marc},
title = {The Quantum-Geometric Correspondence: Status and Outlook},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/qgc-status-outlook.html}
}
FoundationsJune 13, 2026
Geometric Modular Flow for Gravitons in a Causal Diamond: Dimensional Reduction and the Type II Crossed Product for a Non-Conformal Field
The modular Hamiltonian of a causal diamond in the Minkowski vacuum is geometric—local with the conformal-Killing weight f(r) = (R² − r²)/(2R)—for conformally invariant fields, underpinning recent Type II crossed-product constructions in which modular flow is an observer's proper time. The linearized graviton is not conformally invariant, and the conformal map behind these results does not act on it; this has been regarded as a structural obstruction to extending diamond modular theory to the physical spin-2 field.
We show the obstruction dissolves. The vacuum modular flow of the gauge-invariant graviton ball algebra is geometric. Three published results assemble: Benedetti–Casini decompose the graviton's sphere content into two towers of massless-scalar spherical modes with ℓ ≥ 2; rotational invariance makes the reduced quasi-free vacuum block-diagonal over the towers; and Huerta–van der Velde show each dimensionally reduced, non-conformal angular mode inherits a local modular Hamiltonian with the parent weight f(r) that generates its flow. At M → 0 the Regge–Wheeler and Zerilli master potentials degenerate to the scalar's ℓ(ℓ+1)/r², verified symbolically and on a radial lattice at machine precision. Maxwell validates the assembly independently.
Consequences: the diamond's Tolman temperature is helicity-blind; the graviton crossed product is Type II∞ with a canonical trace; and a centrally localized excitation of energy ΔE accumulates modular phase at ΔE/ℏ, independent of diamond radius—the same helicity-blind modular clock as conformal matter. Remaining open items (edge modes, trace normalization, intrinsic observer construction) are stated explicitly.
@article{Sperzel2026GravitonModularFlow,
author = {Sperzel, Marc},
title = {Geometric Modular Flow for Gravitons in a Causal Diamond: Dimensional Reduction and the Type II Crossed Product for a Non-Conformal Field},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/graviton-modular-flow.html}
}
FoundationsJuly 1, 2026
Bounding the Obstruction to the Modular-Physical Identity in a Finite Causal Diamond
The identity K = 2π H_phys—the modular Hamiltonian of an observer's algebra equals 2π times the physical Hamiltonian—is the Bisognano-Wichmann theorem for the free field, and for gravity it underlies the thermal-time hypothesis, the crossed-product construction of subregion algebras, and a first-order-in-G gravitational decoherence rate Γ_dec = E_G/ℏ with E_G = GM²/d. Its proof for gravity faces a specific obstruction: the gravitational dressing is the Newtonian field, supported outside every bounded region, and the flat wedge algebra is Type III₁, admitting no interior factor against which the exterior tail could be localised.
We make the obstruction quantitative. For a coherent (Gaussian) dressing the entire deviation K − 2π H_phys is carried by a single functional—the boost-weighted energy of the dressing field outside a finite causal diamond—and we bound it in three steps. Off-diagonal (which-path) matrix elements see only the difference of the two branches' dressings, so the 1/r monopole cancels identically; momentum conservation cancels the dipole, leaving a quadrupole whose exterior weight falls as (d/h)⁴ with the depth h of the source in the diamond—10⁻¹² for a millimetre superposition one metre inside the laboratory light cone; and the finite diamond, unlike the wedge, satisfies Buchholz-Wichmann energy nuclearity, so the split property supplies the finite localisation constant the wedge lacks.
The residual bi-local channel first appears at O(G²); its O(G) part cancels exactly in the Gaussian covariance. The identity is not proven here: the bound is a composition estimate in the Gaussian sector, and the two constants left uncomputed—the interacting localisation constant and the coefficient of the O(G²) residual—are exactly the open content of the conjecture, and exactly what a laboratory discrimination between first- and second-order gravitational decoherence rates would measure.
@article{Sperzel2026ModularObstruction,
author = {Sperzel, Marc},
title = {Bounding the Obstruction to the Modular-Physical Identity in a Finite Causal Diamond},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/modular-identity-obstruction.html}
}
Decoherence
DecoherenceMay 27, 2026
A Mass-Independent Visibility Suppression in the Bose-Marletto-Vedral Experiment from Quantum-Geometric Correspondence
Quantum-Geometric Correspondence predicts that the Bose-Marletto-Vedral (BMV) / QGEM entanglement witness is suppressed by a mass-independent, parameter-free cube law V = exp(-O(1)(d/Δ)³) depending only on the geometric ratio Δ/d, where d is the inter-particle separation and Δ the superposition arm. The suppression arises because the constrained Feynman-Vernon influence functional gives each particle a decoherence rate Γ = Gm²/(ℏΔ) that competes with the entanglement-generating phase; the mass dependence cancels exactly—in sharp contrast with continuous-spontaneous-localisation models and the Diósi-Penrose collapse rate, both of which carry explicit mass scaling.
We derive the suppression for the full family of arm orientations, whose angular dependence is a Legendre P₂(cos θ): at the magic angle cos²θ_m = 1/3 the leading dipole contribution vanishes and the visibility instead follows a quintic, exp[-(9π/7)(d/Δ)⁵]. This yields a sharp dichotomy—observation of the standard BMV signal at QGEM-class parameters would falsify the constrained-influence-functional picture, while mass-independent cubic suppression together with the magic-angle quintic crossover would distinguish QGC from all extant models.
@article{Sperzel2026BMV,
author = {Sperzel, Marc},
title = {A Mass-Independent Visibility Suppression in the Bose-Marletto-Vedral Experiment from Quantum-Geometric Correspondence},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/bmv-prediction.html}
}
DecoherenceJuly 2, 2026
The Gravitational Decoherence Floor: Platform Reach, Finite-Size Scaling Laws, and the Entanglement-Decay Signature
Gravity imposes a decoherence floor on massive quantum systems that no isolation can remove. We show that for a solid object this floor is not the point-mass rate GM²/(ℏd): the Diósi-Penrose branch-pair self-energy of a uniform sphere of radius R displaced by d grows as d² below the crossover d = 2R and saturates, separation-independent, above it. In the saturated regime the floor is Γ_sat = (6/5)GM²/(ℏR) = (6/5)(G/ℏ)(4πρ/3)^(1/3) M^(5/3): at fixed density it depends on mass alone, Γ_sat = k M^(5/3) with k = 1.59×10²⁵ SI for silica.
A 1 pg silica sphere saturates in 0.63 s; the self-consistent testing zone, where the gravitational time falls between environmental and preparation timescales, is 0.8–48 pg for any d ≳ 2R. Conventional qubit platforms are untouched (τ_grav ≥ 10¹⁶ s); levitated spheres with d ≳ 2R are the testing ground, while clamped and phonon-state resonators with sub-radius branch separations are protected by tidal suppression—reversing the naive point-mass verdict.
The same self-energy drives a signature no environmental channel reproduces: entanglement with a distant partner decays at the local-decoherence rate, Γ_AB = Γ_loc. Every prediction is conditional on the framework's first-order-in-G rate; the perturbative G² alternative is ~3×10³⁴ slower. Observation would place gravity in the quantum-to-classical transition; a saturated law of the wrong shape, or a G²-scale coherence lifetime, would falsify it.
@article{Sperzel2026DecoherenceSignatures,
author = {Sperzel, Marc},
title = {The Gravitational Decoherence Floor: Platform Reach, Finite-Size Scaling Laws, and the Entanglement-Decay Signature},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/decoherence-signatures.html}
}
DecoherenceMay 27, 2026
Cosmic Decoherence on de Sitter: Resolving the Naive Infinity
The gravitational decoherence rate Γ = GΔm²/(ℏd) of Quantum-Geometric Correspondence, applied at cosmic scale with Δm the mass of the observable universe and d the Hubble radius, returns Γ ~ 10¹⁰³ Hz—exceeding the causal bound c/R_H ~ H₀ by 121 orders of magnitude. This internal contradiction, logged within the framework as weakness "W2," is the subject of this paper. We show the divergence is an artifact of using the Minkowski graviton propagator at separations d ~ R_H, where it is not valid.
Recomputed on a de Sitter background with Bunch-Davies modes, the coherent-state decoherence functional acquires a physical infrared cutoff at the horizon: super-horizon modes freeze and cease to drive decoherence. The rate gains a closed-form form factor Γ_dS = (GΔm²/ℏd) g(Hd/c) t with g(x) = 1 − (2/π)Si(x), recovering the Minkowski result for Hd/c ≪ 1 and remaining finite and causal at d = R_H. The cosmic rate is governed not by the total mass but by the de Sitter thermal mass per horizon mode; summed over the holographic horizon modes it yields a finite cosmic decoherence rate.
@article{Sperzel2026CosmicDecoherence,
author = {Sperzel, Marc},
title = {Cosmic Decoherence on de Sitter: Resolving the Naive Infinity},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/cosmic-decoherence.html}
}
DecoherenceJune 25, 2026
Modular Spectroscopy: A Spectral Fingerprint of Gravitationally-Induced Decoherence
Quantum-Geometric Correspondence identifies the modular Hamiltonian of an observer's algebra with the physical Hamiltonian, K = 2π H_phys, and from this predicts a first-order-in-G gravitational decoherence rate Γ_dec = E_G/ℏ with E_G = GM²/d. We take that relation as a principle—the Modular-Physical Correspondence (MPC), the sharp form of the thermal-time hypothesis—and ask what it predicts beyond a rate.
Because the principle places every which-path environment in the modular KMS state at the universal Bisognano-Wichmann inverse temperature β_mod = 2π, the fluctuation-dissipation theorem fixes the entire decoherence noise spectrum: S(ω) = η ω coth(πω) with η = 2E_G/ℏ. The spectrum is a white plateau whose zero-frequency value reproduces the static rate E_G/ℏ, crossing over to a linear rise at a knee f_knee ~ E_G/(2πℏ) = GM²/(2πℏd). The floor is uncoolable—it carries the modular temperature, not the laboratory temperature—a property shared with the whole Diósi-Penrose / collapse class. The knee is the MPC-specific signature: it scales as M²/d with no free parameter and, unlike the floor, is independent of the rate coefficient C.
Modular (noise) spectroscopy, sweeping a which-path modulation across ~0.1–100 Hz for femtogram–nanogram masses at micron–millimetre separations, measures the knee directly and discriminates MPC from Diósi-Penrose collapse: above the knee the modulated decoherence rises (~30× the floor at ten times the knee) for MPC and stays flat for a Markovian collapse model. The relation K = 2π H_phys is posited, not proven; the spectrum is a derived consequence, and a measurement of the knee is its decisive test.
@article{Sperzel2026ModularSpectroscopy,
author = {Sperzel, Marc},
title = {Modular Spectroscopy: A Spectral Fingerprint of Gravitationally-Induced Decoherence},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/modular-spectroscopy.html}
}
Cosmology
CosmologyJuly 2, 2026
Holographic Cosmology: Dark Energy and the Arrow of Time from One Horizon Entropy Budget
The Bekenstein-Hawking entropy of the cosmological horizon is one budget, and this paper spends it twice. Equipartition of the horizon's thermal energy against the critical density gives the dark-energy density ρ_DE = α c²H²/G with α = 3Ω_DE/(8π) ≈ 0.082; generalized-second-law saturation at the de Sitter attractor pins the equation of state to w = −1 exactly, indistinguishable from a cosmological constant and consistent with w₀ = −1.03 ± 0.03.
The same budget, applied to the shrinking past horizon, implements the Past Hypothesis: the holographic bound S(t) ≤ S_dS(t) with S_dS(t) → 0 as t → 0 forces low initial entropy without fine-tuning, reads Penrose's 10¹²³ as the horizon's capacity rather than a probability, makes the smallness of ρ_DE in Planck units the reciprocal of that capacity, and orients the Weyl curvature asymmetry along the direction of cosmic decoherence. The two applications are tied by the identity ρ_DE^∞ · S_dS^∞ = 3c⁷/(8G²ℏ), exact in the de Sitter attractor, both sides depending on H_∞ alone.
The limits are stated plainly: α is the measured cosmic entropy budget, fitted rather than predicted; the cosmological constant problem is reparameterized, not solved; and no observable discriminates against ΛCDM at current precision. The gain is conceptual: dark energy and the arrow of time are two readings of one horizon entropy budget.
@article{Sperzel2026HolographicCosmology,
author = {Sperzel, Marc},
title = {Holographic Cosmology: Dark Energy and the Arrow of Time from One Horizon Entropy Budget},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/holographic-cosmology.html}
}
CosmologyJanuary 24, 2026
Emergent Gravity from Entanglement Equilibrium: MOND Phenomenology and Covariant Extension
The MOND acceleration scale a₀ = cH₀/(2π) ≈ 1.08×10⁻¹⁰ m/s² emerges from de Sitter horizon thermodynamics with no adjustable parameters, matching observed values within 10%. Within Quantum-Geometric Correspondence, the de Sitter cosmological horizon establishes a thermal bath at the Gibbons-Hawking temperature, generating extensive (volume-law) entropy in addition to the standard area-law contribution. We postulate that when local entanglement equilibrium—the principle that generalized entropy is stationary under allowed variations—fails at low accelerations, the volume entropy becomes dynamically relevant and produces MOND-like corrections; this failure of local equilibrium is a physical hypothesis, not a derived result.
The same construction gives the Tully-Fisher relation v⁴ = GMa₀ and clear regime separation: general relativity remains exact when ε = g_N/a₀ ≫ 1 (solar system, pulsars) and corrections appear only when ε ≲ 1 (galaxy outskirts), with the volume entropy saturating the Bousso covariant entropy bound. We also sketch Entanglement-Elastic Gravity (EEG), a candidate covariant completion in which an elastic displacement field encoding entanglement strain sources a modified Einstein equation, presented as a schematic proposal rather than a finished theory.
@article{Sperzel2026EmergentGravity,
author = {Sperzel, Marc},
title = {Emergent Gravity from Entanglement Equilibrium: MOND Phenomenology and Covariant Extension},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/emergent-gravity-mond.html}
}
CosmologyMay 30, 2026
The Horizon Clock: A Single de Sitter Frequency Behind the MOND Scale, Cosmic Decoherence, and the Chaos Bound
The MOND acceleration scale a₀ ≈ cH₀/2π and the cosmic gravitational decoherence rate Γ_cos ≈ g(1)H/4π are usually treated as unrelated numerical coincidences with the Hubble rate. We show they are the same quantity. Both are the Gibbons-Hawking temperature of the cosmological horizon written as a frequency—the horizon clock ω_L ≡ k_B T_dS/ℏ = H/2π, which is also the rate of Tomita-Takesaki modular flow of the de Sitter vacuum. The MOND scale is this frequency carried by c (an acceleration), a₀ = c ω_L; the cosmic decoherence rate is the same frequency carried by a pure number.
Consequently the two are locked at a parameter-free ratio, a₀/(c Γ_cos) = 2/g(1) = 5.03, in which every cosmological input cancels. The single non-trivial number g(1) = 1 − (2/π)Si(1) = 0.398 is the sub-horizon fraction of the gravitational self-energy integral evaluated at one Hubble radius—the same ratio that counts the modular "ticks" the universe takes to decohere. We place this in a horizon-clock family, separate what is established in the literature from what is new, and state the falsifiable content honestly.
@article{Sperzel2026HorizonClock,
author = {Sperzel, Marc},
title = {The Horizon Clock: A Single de Sitter Frequency Behind the MOND Scale, Cosmic Decoherence, and the Chaos Bound},
year = {2026},
note = {Quantum-Geometric Correspondence Series},
url = {https://www.marcsperzel.com/papers/html/horizon-clock.html}
}
Lecture Notes
Two self-contained courses on the theory behind the papers: a ground-up volume that assumes only undergraduate physics and builds everything else from scratch, and a compressed six-lecture review for readers who already carry the graduate toolkit.
Foundations Track — a fully self-contained course in fifteen chaptersJul 2026 (first edition draft) · 252 pp
The Quantum-Geometric Correspondence from the Ground Up
Assumes only undergraduate quantum mechanics, special relativity, and calculus. Everything else is built before it is used.
A complete, ground-up course on the theory behind the papers. Part I states the problem: the two best theories in physics contradict each other on where a superposed mass gravitates. Part II builds the full toolkit from scratch — density matrices and decoherence, general relativity, quantum fields and the Unruh effect, black-hole thermodynamics, operator algebras, and modular theory. Part III states the correspondence, lays down its three axioms, derives Einstein’s equations from entanglement equilibrium, and walks with no skipped steps to the gravitational-decoherence rate and the one open construction that separates conjecture from theorem. Part IV takes the theory outdoors: the cosmological sector and the laboratory experiments that will confirm or kill it. Every objection a sharp reader would raise is asked and answered on the spot in numbered question boxes; every claim carries an honest status label (proven / derived / conjectured / assumed); every number is recomputed from G, ħ, c and machine-checked against the repository’s verification harness.
Review Track — the research corpus in 39 dense pagesJun 2026 · 39 pp
The Quantum-Geometric Correspondence: A Six-Lecture Course
For readers who already carry graduate quantum field theory, general relativity, and operator algebras.
The same theory compressed into six lectures in the register of a mathematical-physics monograph: definitions, axioms, theorems and derivations, numbered equations, exact status labels. Covers the kinematic foundations, the three axioms of the Modular Diamond formulation, the gravitational-decoherence mechanism at order G, the operator-algebraic frontier, the cosmological sector, and the phenomenology-and-status ledger. Each lecture ends with a hands-on layer of worked exercises keyed to the repository’s verification scripts.