Research

Double-Pulsar Drag Constrains Finite-Temperature Baths as the Source of Diósi–Penrose Decoherence

A gravitational decoherence rate of first order in Newton's constant, the Diósi–Penrose rate Γ = C·E_G/ħ, cannot come from quantized linearized gravity for an equal-mass spatial superposition, nor, in a Gaussian treatment, from any translation-invariant state with positive energy. The natural remaining source is a bath at finite temperature. We show that such a bath, coupled to the mass density through a Hamiltonian, also drags moving bodies: the noise that decoheres comes with a friction that slows. If the noise has the Diósi–Penrose form and is white and passive at the wavelengths a neutron star resolves, the friction rate is at least (2π/3)(CħG/k_B T_eff) times the density, with T_eff the fluctuation–dissipation temperature. The measured orbital decay of the double pulsar PSR J0737−3039A/B then requires T_eff/C > 3.3×10¹⁶ K. For dissipative Diósi–Penrose models this raises the lower bound on the collapse temperature by a factor 3×10²⁹. The thermal-medium model proposed earlier in this series (C = 1/π) is a bath of this kind and is excluded. What survives is Diósi–Penrose noise without back-action (the infinite-temperature limit), together with baths that behave differently at kilometre scales than in the laboratory. With Gaussian smearing R₀ of the mass density, data allow R₀ > 4.5 Å·C^(1/3) with no upper bound, and C is a free parameter of order one, not a prediction. For a uniform sphere of radius a separated by d ≥ 2a the rate is Γ = C·GM²[6/(5a) − 1/d]/ħ, independent of R₀, so once other decoherence channels are subtracted one visibility measurement determines C. For 1 µg of silica, τ = 65 ps/C. This paper absorbs the earlier linearized no-go. Working paper; not peer-reviewed.

@article{Sperzel2026DoublePulsarDrag,
  author = {Sperzel, Marc},
  title = {Double-Pulsar Drag Constrains Finite-Temperature Baths as the Source of Diósi–Penrose Decoherence},
  year = {2026},
  note = {Quantum-Geometric Correspondence Series},
  url = {https://www.marcsperzel.com/papers/html/g1-decoherence-no-go.html}
}

A Diósi–Penrose Decoherence Rate Breaks Lorentz Invariance

Diósi and Penrose proposed that a mass superposition loses coherence at the rate Γ = C·E_G/ħ, first order in Newton's constant. We ask what an observation of this rate would imply about spacetime symmetry. A companion paper bounds a reciprocal (thermal) source of the rate with the orbital decay of the double pulsar. Here we show that the law of the noise cannot be Lorentz invariant, under stated assumptions: matter is local relativistic quantum field theory with a stable vacuum; the noise excites that vacuum at a finite rate per unit volume; the noise is a classical field, or a Gaussian record fed back through translation-invariant kernels with an exactly Newtonian long-distance limit; its law does not depend on the state; and there are no further degrees of freedom. Noise whose spectrum reaches timelike momenta heats the vacuum without bound unless it singles out a rest frame. Lorentz-invariant noise with purely spacelike support fails at fourth order, where two spacelike transfers add up to a timelike one; independently, a refit of digitized XENONnT data limits such noise to C < 2×10⁻⁸. For a record with feedback, the kernels cannot be Lorentz covariant. A law that reads its time direction from the quantum state is a dynamical aether and is nonlinear over mixtures. The conclusion is not established for infrared-modified Newtonian responses, non-Gaussian or jump records, or non-local couplings. A preferred frame would be visible as a Gaussian cutoff of the spontaneous X-ray spectrum near ħc/R₀, which no covariant noise produces. Working paper; not peer-reviewed.

@article{Sperzel2026DPLorentz,
  author = {Sperzel, Marc},
  title = {A Diósi–Penrose Decoherence Rate Breaks Lorentz Invariance},
  year = {2026},
  note = {Quantum-Geometric Correspondence Series},
  url = {https://www.marcsperzel.com/papers/html/dp-preferred-time.html}
}

Spacelike-Record Gravity: A Candidate Theory of Diósi–Penrose Decoherence and What It Predicts

This is a candidate model, not the canonical theory (version 2, amended October 14). A companion paper showed that a Diósi–Penrose decoherence rate, Γ = C·E_G/ħ, requires noise without back-action whose law is not Lorentz invariant. Here we write down one theory that meets these conditions and work out what it predicts. On an assumed preferred foliation of spacetime (constant-mean-curvature slices), the classical part of geometry is the constraint field sourced by a continuous record of the spacelike part of the energy, stress and momentum of matter and of gravitational radiation; everything else stays quantum. The field depends affinely on the record and reads it through a postulated non-lagging filter in every non-linear term. The record's resolution, R₀ = g(ℓ_P c/H)^½, is built from the Planck length and the expansion rate of the slice, a conjectured rule. The record noise then gravitates at a percent or less, and least decoherence gives C = 1. Stress superpositions decohere at least eight times faster than mass superpositions. Assuming a covariant mean field, the mean is linearized general relativity. With instant reheating, data confine today's R₀ to between 1.2–3.7 µm and 40 µm. The decisive test is a single body: a coherence time above the theory's slowest rate excludes it for every allowed R₀, and a body larger than 440 µm measures C. The model also predicts one kernel for decoherence and the short-range force (testable only near 25–40 µm), null two-sphere entanglement tests and a moving-body rate γ²Γ_rest; the X-ray test of the first version is lost. Starobinsky inflation keeps general relativity's tilt only if the slow-roll potential is dark to the record. The tensions are real: the cosmological constant only decreases under the record's heating, so its initial value is tuned; the resolution rule is conjectured and the filter postulated; the theory assumes a preferred frame (with signalling suppressed by G in that frame) and the Born rule; and its record is undefined on general curved backgrounds. Working paper; not peer-reviewed.

@article{Sperzel2026SpacelikeRecord,
  author = {Sperzel, Marc},
  title = {Spacelike-Record Gravity: A Candidate Theory of Diósi–Penrose Decoherence and What It Predicts},
  year = {2026},
  note = {Quantum-Geometric Correspondence Series},
  url = {https://www.marcsperzel.com/papers/html/spacelike-record-gravity.html}
}

Diósi–Penrose Noise from Relaxing Emergent Geometry: A Thermal Model with C = 1/π

Quantized linearized gravity does not supply a decoherence rate of first order in G for a static, equal-mass spatial superposition. The Diósi–Penrose rate instead requires a source-independent potential noise with a Newtonian spatial correlation and weight at zero frequency. We ask whether that noise can be the equilibrium fluctuation of an emergent, coarse-grained geometry. Summing the fixed-volume area deficits of entanglement-equilibrium balls of size ℓ, and using the canonical fluctuation formula with the lapse, gives at linear order a Gibbs weight for the Newtonian energy functional, with an effective temperature k_B T = 5ħc/(2πℓ). The equal-time kernel is then exactly Diósi's. General relativity propagates these fluctuations rather than relaxing them; relaxation must be added. A candidate rate is the modular mixing of the field that carries the noise; consistency of a single KMS state for statics and dynamics fixes the rate coefficient to C = 1/(πΔ), giving C = 1/π for a dimension-one field. The resulting noise is thermal. Decoherence follows the Diósi–Penrose form with a Yukawa-regularized kernel and smearing length R₀ = ℓ/5, and k_B T R₀ = ħc/(2π). Hard spontaneous emission is suppressed by a Bose factor and bulk heating is below one percent of the Gaussian-smeared value. Confronted with published radiation and heating data, the model survives for R₀ ≳ 3×10⁻¹¹ m; the binding constraint is the softest X-ray search, not heating and not the hard-photon search that excludes the parameter-free model. For a 1 µg mass at 1 mm the decoherence time is about 5 ns, a factor π longer than the Diósi–Penrose value. The results are conditional on four stated assumptions. Working paper; not peer-reviewed.

@article{Sperzel2026ThermalRelaxationDP,
  author = {Sperzel, Marc},
  title = {Diósi–Penrose Noise from Relaxing Emergent Geometry: A Thermal Model with C = 1/π},
  year = {2026},
  note = {Quantum-Geometric Correspondence Series},
  url = {https://www.marcsperzel.com/papers/html/thermal-relaxation-dp.html}
}

Quantized Linearized Gravity Does Not Supply the Diósi–Penrose Decoherence Rate

The Diósi–Penrose proposal assigns a mass superposition a decoherence rate E_G/ħ of first order in Newton's constant. We ask whether quantized linearized gravity supplies that rate for a static, inertial source whose two branches carry equal mass. A rate of this kind has a sharp signature: at fixed geometry its exponent grows as GM²t. We examine the two channels available at linearized order. For the field, the fluctuation–dissipation theorem ties a zero-frequency noise floor to an Ohmic dissipative response. Two-point functions of a free field are supported on the mass shell in any translation-invariant state, and equal branch masses make the source difference vanish as k². The field spectral density therefore rises as ω³, and the floor vanishes at every temperature, including the modular temperature of a causal diamond. For relational clocks, the branch coherence depends on G, M and t only through the product GMt, for any number of clocks in any joint state. A canonical Page–Wootters readout yields a redshift phase at the E_G scale with constant visibility. Corrections of higher order in G cannot restore a first-order term. The exponent E_G t/ħ is reproduced exactly by mass density coupled to white noise with a Newtonian correlation, whose weight lies off the mass shell. Within this scope an observed first-order rate would indicate physics beyond quantized linearized gravity, such as an explicit collapse mechanism. Working paper; not peer-reviewed.

@article{Sperzel2026LinearizedRateNoGo,
  author = {Sperzel, Marc},
  title = {Quantized Linearized Gravity Does Not Supply the Diósi–Penrose Decoherence Rate},
  year = {2026},
  note = {Quantum-Geometric Correspondence Series},
  url = {https://www.marcsperzel.com/papers/html/linearized-rate-no-go.html}
}

From Modular Flow to Fringe Visibility: A Missing Dynamical Step

Identifying modular flow with physical time does not, by itself, determine the decay of interferometric visibility. We examine the additional step needed to infer a gravitational decoherence rate from a modular energy scale. Explicit Type-I controls have nonzero quantum Fisher information with constant or recurrent visibility, including an exactly solvable bounded-below oscillator. They separate generator normalization, distinguishability speed, and the branch-conditioned channel measured by an interferometer. A quantum speed limit bounds orthogonalization frequency from above and supplies no positive lower bound on visibility decay. KMS detailed balance likewise leaves the coupling spectrum undetermined. An exponential law at the gravitational self-energy rate therefore requires a specified state, interaction, readout, and controlled dynamical regime. These controls identify a gap in a proposed derivation; they neither construct the gravitational observer algebra nor exclude a mechanism that supplies the missing dynamics. Working paper; not peer-reviewed.

@article{Sperzel2026ModularVisibilityGap,
  author = {Sperzel, Marc},
  title = {From Modular Flow to Fringe Visibility: A Missing Dynamical Step},
  year = {2026},
  note = {Quantum-Geometric Correspondence Series},
  url = {https://www.marcsperzel.com/papers/html/modular-visibility-gap.html}
}

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, and we separate what the framework derives from what it postulates. The linearized Wheeler–DeWitt constraint entangles a matter superposition with its gravitational dressing, and the branch dressings are distinguishable by a static overlap of first order in G. At linearized order, however, no mechanism in the framework turns this into an irreversible rate. A Diósi–Penrose-type rate Γ_dec = C·GM²/(ħd) therefore enters as a postulate; a companion thermal model fixes C = 1/π conditionally, giving τ_dec ≈ 5.0 ns for 1 µg at 1 mm, π times the Diósi–Penrose value of 1.6 ns. Standard perturbative quantum field theory predicts a G² rate ~10³⁴ times slower, so even an order-of-magnitude measurement is decisive; an observed G¹ rate would signal physics beyond quantized linearized gravity rather than confirm a derivation. Working paper; not peer-reviewed.

@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}
}

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 and the Einstein equations, and yield a thermodynamic dark-energy density ρ_DE = αc²H²/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 = GM²/d: a postulated rate clause of Diósi–Penrose class that quantized linearized gravity does not supply. A thermal model of relaxing emergent geometry gives the conditional coefficient C = 1/π, hence τ_dec = πħd/(GM²) ≈ 4.96 ns for a microgram over a millimetre, a factor π longer than the Diósi–Penrose value (1.6 ns) and some thirty-four orders of magnitude faster than the perturbative G² result. We distinguish what is established from what is not: the energy scale E_G is proven, the operator identity K = 2πH_phys is a conjecture, and the rate and its coefficient are posited, not derived. We give the falsifiability map, summarize the cosmological sector, position the programme against collapse and stochastic-gravity approaches, and list the open problems. Working paper; not peer-reviewed.

@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}
}

Geometric Modular Flow for Gravitons in a Causal Diamond: Dimensional Reduction and the Type II Crossed Product for a Non-Conformal Field

Historical working-paper summary. Read rate-dependent claims with the October 14, 2026 update: the decoherence rate is a stated postulate with a free coefficient, not a derived result.

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}
}

Bounding the Obstruction to the Modular-Physical Identity in a Finite Causal Diamond

Historical working-paper summary. Read rate-dependent claims with the October 14, 2026 update: the decoherence rate is a stated postulate with a free coefficient, not a derived result.

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}
}

What the Bose–Marletto–Vedral Witness Decides: Gravity-Mediated Entanglement Under Quantum Gravity and an Exact Null Under Diósi–Penrose Classical Gravity

The Bose–Marletto–Vedral (BMV) and QGEM proposals aim to witness entanglement generated by gravity between two masses held in spatial superposition. We forecast the witness for four published designs under two hypotheses about the Newtonian interaction: quantum gravity (Q), and a classical, record-sourced Newtonian field with Diósi–Penrose decoherence at coefficient C ≥ 1 and smearing length R₀ (R). Under Q, with which-path dephasing at rate γ per body, the partial-transpose witness is exactly ⟨W⟩ = (1 − p² − 2p|sin(Φ/2)|)/4 with p = e^(−γτ), so entanglement appears if and only if γ is below the entangling rate ω/2. For the Bose et al. design at its stated vacuum (10⁻¹⁵ Pa, 0.15 K) the phase is 0.314 rad, gas collisions give γ = 0.012 s⁻¹, and ⟨W⟩ = −0.061. Under R the two-body state stays separable at all times: the concurrence is exactly zero for R₀ below the bodies' size, and for the Bose geometry for every R₀ up to 0.52 mm. Only the unscreened Bose geometry combines a detectable Q witness with an R null across R's allowed window; the screened designs fall below detection, sit at their own threshold, or leave part of the window open. A decisive run needs a closest approach of at least 157 µm (Casimir–Polder below 10% of gravity), a mass of at least 3.3×10⁻¹⁵ kg, a 100–250 µm arm, and a pressure below 6.6×10⁻¹⁵ Pa at 0.15 K. The single-arm contrast of the same particle already separates Q from R. The required superposition exceeds the current record by six orders of magnitude in size, so the test is not expected this decade. Working paper; not peer-reviewed.

@article{Sperzel2026BMV,
  author = {Sperzel, Marc},
  title = {What the Bose–Marletto–Vedral Witness Decides: Gravity-Mediated Entanglement Under Quantum Gravity and an Exact Null Under Diósi–Penrose Classical Gravity},
  year = {2026},
  note = {Quantum-Geometric Correspondence Series},
  url = {https://www.marcsperzel.com/papers/html/bmv-prediction.html}
}

The Gravitational Decoherence Floor: Platform Reach, Finite-Size Scaling Laws, and the Entanglement-Decay Signature

Historical working-paper summary. Read rate-dependent claims with the October 14, 2026 update: the decoherence rate is a stated postulate with a free coefficient, not a derived result.

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}
}

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}
}

Modular Spectroscopy: A Spectral Fingerprint of Gravitationally-Induced Decoherence

Historical working-paper summary. Read rate-dependent claims with the October 14, 2026 update: the decoherence rate is a stated postulate with a free coefficient, not a derived result.

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}
}

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}
}

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}
}

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}
}