Most proposed table-top tests of gravity in the quantum regime use equal-amplitude spatial superpositions and seek either gravitationally induced entanglement or collapse-motivated loss of interference. This paper develops a complementary theory-classification framework in which branch weights are scanned as operational control parameters. A massive source is prepared in a two-branch state\[\ket{\psi_p}=\sqrt p\ket{L}+\sqrt{1-p}\ket{R},\]or, more generally, in an \(m\)-branch state\[\ket{\psi}=\sum_{a=1}^{m}\sqrt{p_a}e^{i\theta_a}\ket{a}.\]The elementary two-branch second cumulant \(p(1-p)(\Delta q)^2\) is treated as a baseline branch-resolution law, not as a quantum-gravity signature. The main object is an enlarged witness geometry combining mean response, branch-induced covariance and higher cumulants, operational recoverability, and excess-noise or irreversibility coordinates. The \(m\)-branch extension yields simplex and covariance-rank constraints; the \(n\)-particle extension distinguishes independent ensembles from collectively entangled branch states by \(n\) versus \(n^2\) scaling. Recoverability is formulated as a resource-theoretic witness convention: adaptive recovery may use probe instruments and classical feed-forward but may not introduce an independent branch-basis coherence resource; a separate joint-recovery functional captures coherent erasure using accessible source-probe correlations. Worked models illustrate deterministic expectation-value gravity, coherent branch-mediated interaction, effective irreversible branch-basis dephasing, and noisy classical measurement-feedback dynamics. A finite-record, calibrated measurement-feedback bound shows that, within that simulator class, reproducing nontrivial branch-conditioned response requires record distinguishability; the associated record overlaps bound adaptive recovery of source coherence, with response-calibrated corollaries when the feed-forward range is bounded. The resulting framework is not a new microscopic theory of gravity. It is a model-discrimination scheme for locating gravitational response models in a witness space generated by branch-weight, branch-number, and ensemble-size scans.
Most proposed table-top tests of gravity in the quantum regime use equal-amplitude spatial superpositions and seek either gravitationally induced entanglement or collapse-motivated loss of interference. This paper develops a complementary theory-classification framework in which branch weights are scanned as operational control parameters. A massive source is prepared in a two-branch state[\ket{\psi_p}=\sqrt p\ket{L}+\sqrt{1-p}\ket{R},]or, more generally, in an (m)-branch state[\ket{\psi}=\sum_{a=1}^{m}\sqrt{p_a}e^{i\theta_a}\ket{a}.]The elementary two-branch second cumulant (p(1-p)(\Delta q)^2) is treated as a baseline branch-resolution law, not as a quantum-gravity signature. The main object is an enlarged witness geometry combining mean response, branch-induced covariance and higher cumulants, operational recoverability, and excess-noise or irreversibility coordinates. The (m)-branch extension yields simplex and covariance-rank constraints; the (n)-particle extension distinguishes independent ensembles from collectively entangled branch states by (n) versus (n^2) scaling. Recoverability is formulated as a resource-theoretic witness convention: adaptive recovery may use probe instruments and classical feed-forward but may not introduce an independent branch-basis coherence resource; a separate joint-recovery functional captures coherent erasure using accessible source-probe correlations. Worked models illustrate deterministic expectation-value gravity, coherent branch-mediated interaction, effective irreversible branch-basis dephasing, and noisy classical measurement-feedback dynamics. A finite-record, calibrated measurement-feedback bound shows that, within that simulator class, reproducing nontrivial branch-conditioned response requires record distinguishability; the associated record overlaps bound adaptive recovery of source coherence, with response-calibrated corollaries when the feed-forward range is bounded. The resulting framework is not a new microscopic theory of gravity. It is a model-discrimination scheme for locating gravitational response models in a witness space generated by branch-weight, branch-number, and ensemble-size scans.
Varying the probabilities of the branches in a massive spatial superposition provides a controlled way to test gravitational response models. We develop this comparison for fixed source configurations and calibrated probe measurements. The between-branch covariance obeys a pairwise-distance law on the probability simplex and has rank at most one less than the number of occupied branches. Higher cumulants and ensemble-size scans constrain the same response model, although their statistics can also be produced by classical branch mixtures. Coherence recovery adds information when its permitted operations and accessible records are specified independently. For a branch-preserving measurement that produces an irreversible classical record and drives a common feed-forward response, we derive an upper bound on recovered source coherence in terms of record overlaps. A calibrated response diameter converts this into a response--recovery inequality. A second bound relates recovery to the conditional response variances and remains applicable to unbounded feedback with finite second moments. An explicit binary model saturates both inequalities. Coherent momentum transfer provides a contrasting example in which the probe statistics remain branch resolved while an eraser measurement restores source coherence. These comparisons require separate measurements of conditional probe noise, trajectories, and calibration parameters. The bounds apply to the stated classical-record channel; their application to a gravitational theory requires a derivation of that channel from the theory. Branch-weight scans therefore support quantitative tests of specified models without providing a theory-independent criterion for quantized gravity.
I register the prediction that no experiment will witness gravitationally induced entanglement between two masses, each in two branches, while their entangling phase stays below 2 rad, at any constant rate at which each mass's environment records its position. Under gravity whose source in each description is a function of the system's reduced state, conditioned on every record that exists, two masses that nothing records stay in a product state at every entangling phase. Recorded masses stay unentangled while the phase is below 4 rad with each mass's share counted once, or below 2 rad with each mass taking its full potential. Every published parameter set computed here lies below both, between 0.0150 rad and 0.594 rad. A witness of gravitationally induced entanglement reported at 5 standard deviations in any such experiment excludes the prediction on its own, whatever the recording rate. Quantized gravity entangles the masses only while each is recorded more slowly than a threshold set by the experiment's parameters, 0.0623 /s at Bose et al.'s, so a null result counts against it only where an experiment establishes its recording rate below that threshold. The source in the description occupying branch A is n_r^A = n_A + f(η) (n_B − n_A)/2, where n_A and n_B are the branches' densities and η is the overlap of the mass's two branch records, e^(−Γ t) at recording rate Γ. The product state is derived; the weight f(η) = η is derived from the stated condition that the source depend affinely on the coherence; the phase bound is computed in a two-branch open-system model and rests on the stated two-body form. Nothing is fitted. The statistic is the reported witness with the recording rate the experiment establishes, read against both predictions at its own parameters; the chance of a false exclusion from statistical fluctuation alone is 2.9 × 10⁻⁷ for each experiment.
Tests of gravity-mediated entanglement aim to determine whether two massive objects in spatial superposition can become entangled through their mutual gravitational interaction. In an ideal model, the two objects are often treated as pointlike branches, but real wave packets have a finite width and are also exposed to dephasing and residual non-gravitational forces. We study these effects in a two-branch gravitational cat-state model with Gaussian wave packets. The Newtonian interaction is averaged over the finite wave-packet width, and spatially correlated dephasing is included phenomenologically. We then ask a practical question: when does the generated state remain useful rather than merely entangled? To answer this, we examine Bell violation, teleportation fidelity, coherent information, Fisher information, and state distinguishability. We find finite parameter windows where the same state is Bell-nonlocal and teleportation-useful, but these windows shrink as the wave packets become broader or the dephasing becomes stronger. We also show that a residual electrostatic branch interaction can coherently cancel the gravitational nonlocal phase and strongly suppress the operational signal. The results identify finite-width and electrostatic tolerance boundaries that should be considered in future tests of gravity-mediated entanglement.
Pedagogical Guide, Foundations Audit, Uniqueness & Substitutability Analysis, Literature Comparison, and Reading Guidance : A Pedagogical Guide to Understanding the OT–GKSL Architecture This paper develops a comparative interpretation of two recent Oxford research directions through the certified source–readout architecture of the Einstein-locked OT–GKSL framework: the experimental extraction of classical ticks from a quantum clock studied by Wadhia, Meier, Ares and collaborators, and the post-selected repulsive gravitational signal proposed by Saldanha, Marletto and Vedral. The comparison is structural rather than mechanistic. OT–GKSL starts from open quantum-state dynamics rather than from a primitive spacetime geometry. Its native objects are density states, the complete GKSL generator, and, in the detailed-balance dissipative sector, a positive quantum optimal-transport geometry. Classical events, clocks, Lorentzian geometry and gravitational observables are reconstructed only through stable physical records, event individuation, calibration and certification. The relevant hierarchy is ρ → ω(ρ) = M(ρ) → [ρ]evt → Xᵃ → s → eᵃ → g_ro, with the native OT geometry and the Lorentzian readout metric kept strictly distinct. The native process carries a directed semigroup ordering, but not a primitive Lorentzian causal structure. For temporal readout, the framework distinguishes native entropic ordering, a clock-like record, and calibrated classical time: t_ent ≠ X⁰ ≠ t_ro. This separation provides a natural operational language for the quantum-clock experiments, where microscopic clockwork dissipation and the energetic/informational burden of extracting a classical record are experimentally distinguishable. The associated OT–GKSL reporting ledger is C_tick = C_nat + C_ext + C_cert, where C_nat denotes the native clockwork contribution, C_ext the extraction and amplification burden, and C_cert the additional cost of maintaining a stable, calibrated and auditable record. This is a reporting ledger, not a universal microscopic thermodynamic identity. The corpus explicitly identifies the experimentally realized distinction clockwork cost ≠ classical-record extraction cost. For the gravitational proposal, each branch of the Saldanha–Marletto–Vedral protocol remains individually attractive, δ_A > 0, δ_B > 0, while post-selection and coherent amplitude interference can produce an effective conditional shift δ_eff = (β δ_B − α δ_A)/(β − α), which may be negative. The paper emphasizes the distinction between the unconditional joint dynamics and the conditioned measurement record. In particular, for complementary post-selection outcomes f and f̄, P_f ⟨p⟩_f + P_f̄ ⟨p⟩_f̄ = α² δ_A + β² δ_B, for a probe initially centered at zero momentum in the ideal branch-kick model. Thus a negative conditional momentum shift is not by itself equivalent to a negative gravitational coupling, a repulsive force on every branch, or a complete negative momentum balance for the total source–probe–apparatus system. Within OT–GKSL, the post-selection instrument is treated as an imported quantum-measurement structure rather than as an independently derived prediction of the framework. Measurement is represented as a directed certified readout between physical subsystems, A ← B, where the record-bearing subsystem must satisfy stability, accessibility, rank and bridge conditions. Classicalization is required only for the declared record degrees of freedom and does not imply complete dephasing of all quantum degrees of freedom. The resulting mirror is therefore: Ares/Wadhia:native clockwork → detector interaction → classical record → calibrated temporal estimate, Saldanha–Marletto–Vedral:coherent source–probe dynamics → measurement/post-selection → conditional record → gravitationally interpreted probe observable. The common lesson is not that the two systems share a physical mechanism, nor that either experiment confirms OT–GKSL. Rather, both highlight the importance of distinguishing native quantum dynamics from the physical process by which a classical statement becomes accessible. OT–GKSL extends this distinction to spacetime itself: classical Lorentzian geometry is treated as a post-constructed, certified readout rather than as the primitive background of the native dynamics. The paper explicitly identifies the remaining open bridge problem for the gravitational protocol: embedding the SMV preparation into a native OT–GKSL source state, deriving the corresponding gravitational branch interaction from the certified source/readout construction, and embedding the post-selection and probe records into the framework. A complete source–probe–apparatus energy–momentum closure is outside the scope of the present comparative study. 1. Foundations of the Architecture: Foundations |GKSL/Lindblad ; Carlen–Maas ; Jacobson ; Sakharov ; Donoghue ; Lovelock) Establishes the core Einstein-locked OT/GKSL architecture for certified geometric readout and coherence-dependent gravitational sourcing. Optimal-Transport Gravity Trilemma | Identifies the certified operational boundary of geometric readout by proving the fundamental trade-off between temporal resolution, coframe stability, and bridge fidelity. 2. Emergence and Recovery of Classical Physics: Exact Reduced OT/GKSL Equations | Mori–Zwanzig/projection operators ; effective field theory ; Carlen–Maas ; Wilsonian reduction / Demonstrates the controlled recovery of classical Newtonian and gravitational sectors as exact non-linear reductions of the native OT/GKSL state dynamics. Certified Einstein Non-Linear Readout | Lovelock ; Bianchi identities ; Donoghue EFT ; Jacobson thermodynamic gravity// Develops the full non-linear Einstein-locked readout closure for the metric sector. Non-Linear Dynamics and Readout | Dynamical systems, center manifold/effective reduction ; quantum Markov semigroups ; non-linear open-system reductions // Explores the exact reduced non-linear evolution on collective state manifolds. The Seeley–DeWitt Bridge | Seeley–DeWitt heat-kernel ; Vassilevich // Formalizes the operational connection between native state dynamics and the effective classical readout. The SDW Bridge: Composite Brout–Englert–Higgs Dynamics, Spectral Separation, and the Emergent Graviton | Formalizes the emergence of the Brout-Englert-Higgs composite scalar and the spin-2 graviton via the Seeley-DeWitt expansion, strictly preserving the Einstein-Lock. Bridge between QCD and OT/GKSL Readout | Wilson lattice gauge theory ; Gross–Wilczek–Politzer asymptotic freedom ; Kogut–Susskind Hamiltonian lattice gauge theory // Connects the Optimal Transport / GKSL framework to Quantum Chromodynamics, exploring the constitutive bridge and effective low-energy dynamics. 3. The Certified Boundary and Structural Limits: Certified Spacetime Readout on Finite Support: A Unified Temporal and Geometric Boundary | Decoherence / Quantum Darwinism ; quantum reference frames ; finite information bounds ; Jacobson // Unifies the temporal and geometric branches of classical readout into a single certified spacetime problem. Introduces the unified spacetime readout burden and derives the central unified certified-budget inequality, proving that temporal precision, geometric coframe nondegeneracy, and bridge compatibility draw from the same finite entropic and informational resources and cannot be made simultaneously ideal. Certified Causality, Locality, Nonlocality, and Relativity in the Einstein-Locked OT/GKSL Framework | Algebraic QFT/locality ; operational quantum theory ; quantum reference frames ; relativistic causality tests // Determines the exact status of causality, locality, nonlocality, and the principle of relativity within the Einstein-locked OT/GKSL architecture. Shows that causal-local spacetime semantics is a certified readout property rather than a primitive native axiom; proves a patchwise gluing theorem for certified local causal structure; and derives a unified finite-budget inequality showing that temporal precision, geometric certification, bridge admissibility, and overlap compatibility all compete for a single residual causal-local headroom on finite effective support. Entropic Tick Cost and Certified Temporal Readout in the Einstein-Locked OT/GKSL Framework | Demonstrates that classical ticks are finite-resource readout objects extracted from native entropic ordering, rather than primitive background parameters. Decomposes the entropic tick cost into native, extraction, and certification branches, and derives a theorem-level certified temporal budget inequality connecting temporal resolution, finite effective support, and certification margins. Entropic Tick Cost & Spectral Budget | Page–Wootters time ; thermal time hypothesis ; quantum clocks ; Salecker–Wigner bounds // Establishes a theorem-strength certified boundary for classical spacetime by proving a fundamental trade-off between entropic tick resolution, coframe stability, and finite informational budget. Toy Certified Pipeline from Optimal Transport QCD | Provides a protocol-level implementation and scaling model for certified bridge margins. Certified Spectral Boundary from Heat-Kernel Budgets and Entropic Transport in the Einstein-Locked OT/GKSL Framework | Heat-kernel spectral budgets; entropic OT/GKSL transport; certified spectral boundary; Einstein-locked readout. Develops a spectral-geometric control layer for the OT/GKSL framework, where the native heat trace bounds finite spectral resources, the cutoff gap defines a certification margin, and entropic transport controls the drift of readout-support budgets without inducing a state-dependent Einstein–Hilbert kinetic term. Correlation Separation in the Einstein-Locked OT/GKSL Framework | Establishes a theorem-level distinction between native, readout, and causal-local correlations, and reframes t
Proposals for observations in closed quantum gravity must specify how unresolved microscopic alternatives become measurement outcomes. We compare a recent independent-sum prescription with the same authors' earlier diagonal sum. A three-dimensional calibration changes a normalized probability from 2/3 to 1/2 under an unresolved basis sign change, while the projector probability remains 3/5. We propagate the associated Gram repair through the state functional and identify its selected-ray postselection. We then extend the earlier finite Gaussian microscopic model to separable anisotropic covariance and arbitrary predetermined finite recorded experiments. Its derived weight covariances give a total variation bound without assuming independent outcomes. Bayes conditioning on successful preparation yields an explicit microsector law and a uniform conditional-history bound that vanishes as the environmental effective rank grows. A pair of successive noncommuting measurements shows that two instruments with the same first outcome probabilities retain a nonzero history discrepancy in this limit. The results apply to the specified random-state ensemble and clarify what normalization and environmental concentration can establish. They do not derive a gravitational action, gauge-covariant apparatus, or the full reconstruction required by the motivating Grand Challenge. This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, authorized its preparation and publication and is responsible for it.
Papers I and II derived gravitational geometry and second-order dynamics from finite accessibility without invoking measurement, observation protocols, or collapse as primitive notions. This paper extends the operational program to the measurement sector, deriving the structure of record formation and probabilistic weights from the same finite accessibility and compositional constraints. Finite accessibility restricts admissible operations on a subsystem to a proper operator subalgebra. Because bounded certification excludes operators requiring nonlocal coordination or arbitrarily fine phase resolution, the accessible algebra is strictly smaller than the full operator algebra. When a system becomes correlated with orthogonal environment sectors, off-diagonal operators between those sectors fall outside the accessible algebra. The reduced state is therefore indistinguishable, under all admissible operations, from its diagonal ensemble. Definiteness emerges as an algebraic consequence of sector restriction, not as a collapse postulate. The weights of these sectors are fixed uniquely by compositional consistency. Phase invariance, multiplicativity under independent refinement, and stability under record coarse-graining constrain any admissible assignment of sector weights to satisfy a functional equation whose only continuous, non-contextual solution is quadratic in amplitude. The Born rule therefore arises as the unique invariant measure compatible with finite accessibility. Irreversibility follows from action scaling: reversal requires operators outside the admissible tensor algebra and incurs action cost proportional to the number of correlated subsystems. Since admissible operations are bounded by the action gap, macroscopic record reversal is excluded. Measurement introduces no new physical primitives beyond finite accessibility, locality, and compositional stability. Record formation, classical definiteness, and Born probabilities are structural consequences of algebraic restriction under finite accessibility. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** Born rule derivation, measurement problem, quantum statistics, finite accessibility, operator subalgebra, record formation, definiteness emergence, wave function collapse, compositional consistency, accessible state spaces, action gap, irreversibility, sector restriction, quantum foundations, algebraic quantum mechanics, decoherence