In the Type II algebras of semiclassical gravity, an observer's clock with energy bounded in a band regularizes the observables of a static patch, and entropies of the resulting algebras reproduce generalized entropy. What has not been computed in that program is the mutual information budget of the observer: how many bits of clock-field correlation the observer's time-bandwidth resources permit. We prove the regularized answer: for any solution of the constraint (H_fields + H_clock)|Ψ⟩ = 0 with clock spectrum confined to bandwidth W and ticks resolved over duration T, the clock-field mutual information obeys I ≤ 2 log₂(1 + ⌊WT/2π⌋) with equality if and only if the clock is the perfect lattice clock the state that, in the de Sitter algebra of Chandrasekaran-Longo-Penington-Witten, is the maximum-entropy state describing the empty static patch. Rigidity holds: any clock with d orthogonal equally spaced ticks carries at least 2 log₂ d bits, so perfect resolution and budget saturation are the same property. Four components of the statement are measured on IBM quantum hardware: an observer ladder (the same global state read by observers of increasing resolution yields 0 → 1.7 → 3.4 → 5.2 bits, and exactly 0 for a misaligned observer); a continuous observer-frame sweep following I = 1 - h((1 + V cos θ)/2) with R² = 0.998 the ledger invariance full restoration of the correlation under the co-rotation U_A ⊗ U_B at every angle, flat to within shot noise; and the rigidity relation itself tick overlap ε and information I measured by independent circuits at seven detunings, with the forbidden region I < 2 - 2 log₂(1 + ε) empty at every point. The lift of the regularized theorem to the Type II crossed product, where empty de Sitter is conjectured to realize the perfect clock exactly, is stated as a sharp open conjecture. Statements are labelled [P] proved, [N] measured, [C] conjecture.
In the Type II algebras of semiclassical gravity, an observer's clock with energy bounded in a band regularizes the observables of a static patch, and entropies of the resulting algebras reproduce generalized entropy. What has not been computed in that program is the mutual information budget of the observer: how many bits of clock-field correlation the observer's time-bandwidth resources permit. We prove the regularized answer: for any solution of the constraint (H_fields + H_clock)|Ψ⟩ = 0 with clock spectrum confined to bandwidth W and ticks resolved over duration T, the clock-field mutual information obeys I ≤ 2 log₂(1 + ⌊WT/2π⌋) with equality if and only if the clock is the perfect lattice clock the state that, in the de Sitter algebra of Chandrasekaran-Longo-Penington-Witten, is the maximum-entropy state describing the empty static patch. Rigidity holds: any clock with d orthogonal equally spaced ticks carries at least 2 log₂ d bits, so perfect resolution and budget saturation are the same property. Four components of the statement are measured on IBM quantum hardware: an observer ladder (the same global state read by observers of increasing resolution yields 0 → 1.7 → 3.4 → 5.2 bits, and exactly 0 for a misaligned observer); a continuous observer-frame sweep following I = 1 - h((1 + V cos θ)/2) with R² = 0.998 the ledger invariance full restoration of the correlation under the co-rotation U_A ⊗ U_B at every angle, flat to within shot noise; and the rigidity relation itself tick overlap ε and information I measured by independent circuits at seven detunings, with the forbidden region I < 2 - 2 log₂(1 + ε) empty at every point. The lift of the regularized theorem to the Type II crossed product, where empty de Sitter is conjectured to realize the perfect clock exactly, is stated as a sharp open conjecture. Statements are labelled [P] proved, [N] measured, [C] conjecture.
In de Sitter static patch holography, coupling bulk quantum fields to an observer's dynamical clock yields a type II1 von Neumann algebra M+ of gauge-invariant observables. However, the unique tracial state on this algebra fails to reproduce semiclassical bulk correlation functions. We resolve this issue by introducing a family of heavy-clock states, representing an observer reference frame with heavy inertia. We prove that in the heavy-clock limit, n-point correlation functions of gauge-invariant dressed observables reproduce the bulk quantum field correlation functions on de Sitter spacetime. Furthermore, we investigate a generalized observer Hamiltonian Hobs=∣π∣ that incorporates both forward-running (π≥0) and backward-running (π<0) clocks. The resulting von Neumann algebra M of gauge-invariant observables remains a type~II1 factor and is isomorphic to M+⊗M2(C), where M2(C) is the algebra of 2×2 complex matrices transforming between the two clock orientations. Time-reversal symmetry is implemented in this setting as an anti-unitary operator that exchanges the two clock orientations, yielding a type~II1 von Neumann algebra F of gauge-invariant, time-reversal-invariant observables. We generalize the heavy-clock state construction to both M and F, showing that n-point correlation functions calculated in heavy-clock states match the corresponding semiclassical bulk correlation functions. Finally, by coupling Einstein gravity to the Brown--Kuchař dust field, we derive the observer Hamiltonian Hobs=∣π∣ from a perturbative reduction along the observer's worldline in the G→0 limit. This provides a field-theoretic foundation for the observer model developed in this work.
If gravity is information, then the gravitational potential must be an amount of information written at a boundary — and it must be written in a way that does not depend on what is on the other side. That is the informational form of the equivalence principle: a potential is universal, indifferent to the composition of what it acts on. This paper puts that requirement on a quantum processor and measures it.This is a follow-up to Gravity as Approach to a Rate–Resolution Bound (doi:10.5281/zenodo.21808410), and reports the hardware campaign that paper did not contain. In the parity–entanglement programme a crossing write changes a region's bias by ⟨P⟩ → ⟨P⟩|cos 2θ|, so that the amount of record it fixes, δS₂ = −ln|cos 2θ|, has the bias cancelled out of it: it is independent of the state of the region written on, and under Φ = −c²δS₂ it is the Newtonian potential, with θ = v_esc/2c. That independence, and not the mere existence of a correlation between regions, is what separates a gravitational potential from an arbitrary entanglement flow, and it is directly testable. We test it with an estimator that is decay-free by construction — the ratio R_ij(θ) = ⟨P⟩_i(θ)/⟨P⟩_j(θ) of two states carried through identical circuits, in which any depolarising factor cancels identically — and we test it against a deliberately introduced violation. On ibm_kingston, across fifteen independent measurements in two jobs, the sector-exchange write shows no drift: χ² = 12.77 on 15 degrees of freedom, p = 0.62, largest deviation 2.3σ. In the same jobs, and with the same estimator, a control write that is state-dependent by construction is resolved at p = 1.1×10⁻²⁰, and its drift is reproduced by a closed form with no fitted parameter at χ²/dof = 0.52, the largest predicted effect (16.74%) being recovered as 16.59%. The conserved membrane ⟨Z_A Z_B⟩ supplies a second, independent discriminator, dropping by 0.007 in the signal arm and by 0.071 in the control. We also report a measured correction to Eq. (3) of the parent paper: the identity ⟨P⟩ = Tr ρ_A² = e^−S₂ holds in the Gaussian sector but not in the qubit sector, where Tr ρ_A² = (1+⟨P⟩²)/2, confirmed tomographically to 0.3%. Exact density-matrix propagation shows the estimator cancels depolarising, dephasing, bit-flip and coherent write-angle error to machine precision, so a miscalibrated write can manufacture neither the null nor a violation; amplitude damping is the sole surviving channel, at 0.11γ. Every number is re-derived offline from the deposited counts by the accompanying script. Statements are labelled [P] proved, [K] known, [N] numerical or measured, [F] formal.
The Parity-Entanglement Bound (PEB) fixes the clock-system mutual information of a ℤ₂ constrained bipartite pure state at exactly two bits. We show that this invariant functions as the zero point of a metric geometry built from information. Six results are given. (i) The deviation Π = 1 - 1/2 I is the unique local-unitary (LU) invariant of the minimal state (pure, ℤ₂, bipartite); relaxing each hypothesis adds a definite number of further axes, which we count numerically (mixedness +1, ℤn+(n-2), tripartite +2; a continuous gauge group opens a continuous axis), (ii) The correlation quantity d = -ln κ is non-negative, symmetric, and vanishes in the coincidence limit; in the gapped regime, at separations above a floor r*(m), it obeys the triangle inequality and is a metric. (iii) The validity window is mapped: the floor r*(m) grows with the gap and the window closes entirely once the correlation length falls below the lattice scale (m ≥ 1.7 in 1D, ≥ 2 in 2D, lattice units); in the opposite, gapless limit the inequality survives with growing slack (0 violations atm m = 0.02) and an asymptotic argument establishes it analytically in the logarithmic regime. Non-Gaussianity per se does not damage the inequality: on a φ⁴ chain the observed violations are reproduced quantitatively by a free field of matched effective gap. (iv) Local noise acts on d as an exactly additive per-region dilation, d → d + δA + δB with δ ≥ 0; pure dephasing leaves d exactly invariant, and local noise can only increase the triangle slack, so the metric is noise-robust by construction. (v) The PEB value equals a combinatorial crossing number of stabiliser generators, hence is independent of system size, clock size, and clock position; it is formally analogous to a topological entanglement entropy. (vi) Following the Dirac-equation construction [6], the two-dimensionality dim ℋC = 2 underlies a distance/time split. The construction is verified by exact and Monte-Carlo simulation (geodesic locking 60/60 across five seeds; zero triangle violations above the floor in more than 1.6 × 10⁴ triples spanning gapped, near-critical, interacting, and noisy regimes) and, as a proof of concept, by a state-preparation study on quantum hardware (IBM FEZ). Formal correspondences to gravitational behaviour geodesic bending in a non-uniform gap and a volume-to-area transition in the crossing count are recorded in Sec. no gravitational derivation is claimed. Statements are labelled [P] proved, [K] known-with-citation, [N] numerical, or [0] open.
Typicality arguments for emergent spacetime rely on the Lubkin-Page bounds, which show that generic quantum states have vanishing correlations between subsystems. These bounds assume a tensor-product Hilbert space (a Type~I von~Neumann algebra), but the observable algebras in quantum field theory and quantum gravity are generically Type~II or Type~III, raising the question of whether the bounds survive. We prove that they do for all Type~II von~Neumann factors. For the hyperfinite Type~II1 factor with a tripartite decomposition R≅A⊗B⊗E, the mutual information between subsystems A and B vanishes as O((dAdB/dE)2) in finite-dimensional approximations, provided dAdB≤dE (Theorem~1). For Type~II∞ factors, including the gravitational algebras constructed via the crossed-product method by Witten and by Chandrasekaran, Longo, Penington, and Witten, the bound acquires an additional exponential suppression controlled by the Bekenstein-Hawking entropy (Theorem~2). We identify the obstructions to extending the result to Type~III factors and discuss the open question of whether the commutant of the observable algebra can serve as a natural thermal bath that tightens the bound further.
I propose that spacetime is a discrete, lossy rendering of a continuous, non-local information space, the Hidden Informational Dimension (HID). The rendering occurs at a granularity set by Planck’s constant, and each processing tick transmits the relational information of a joint measurement with fidelity equal to the overlap of the questions asked. The resulting dressing law, Eobs(a, b) = −sgn(a·b) |a·b|p, reproduces the complete quantum singlet correlator at unit dressing exponent and retrodicts the Tsirelson bound as B = 4 · 2−1/2 = 2√2. The tick law is itself derived from the granularity postulate up to a single frame-selection axiom: its covariant rival and its dispersive alternative are excluded by existing experiments, the latter at ∼200σ, certifying the processing sequence as a deterministic clock. The exponent is identified with the local processing rate—the gravitational time-dilation factor, augmented by a cosmological entropy term—so the framework contains no free parameters beyond a single gauge choice: the processing foliation is identified with the comoving cosmic rest frame, anchoring the exponent to unity at the cosmic mean potential of the present epoch. A rendered universe is quantum while 1 ≤ p ≤ 2; at horizons p → 0 and the bare no-signalling statistics (B = 4) are exposed; at the entropic limit p → ∞ and correlation dies. The accompanying strong-gravity violation of information causality is quantified and quarantined: a final-station composition theorem forbids its export, and the collapse of communication complexity is confined to r < 1.52 rs, just outside the photon sphere. The theory therefore predicts a running Tsirelson bound, B(p) = 4 · 2−p/2, together with a characteristic deformation of the correlation curve that is pinned at aligned and conjugate settings and maximal near cos θ = e−1—a signature no noise process can imitate—and an absolute, one-sided offset of (1–6) × 10−5 above 2√2 in every terrestrial laboratory, which sits in the potential well of local large-scale structure. Information loss is priced by Landauer’s principle—each CHSHoptimal readout erases exactly half a bit—gravitation is embedded as processing-rate modulation, and the cyclic waveform of the universe is identified with the sign-extended “necklace” instantons of the minisuperspace no-boundary path integral. A staged experimental programme, from tabletop Tsirelson-saturation tests to a solar-orbit Bell mission, is proposed.
Operational theorem package: R′ (resolution), T′ (common-mode invariance), C′ (sub-bin curvature irrelevance), D′ (near-null registrar trivialization), H′ (horizon implementability) This record contains the final manuscript and supporting materials for a strictly operational analysis of quantum measurement in spacetime. Following the introduction of Quantum Measurement Timeline Framework (QMTF), in which a measurement is defined as a POVM statement on a declared spacetime target frameD* = R* × T* at finite resolution (Δx, Δt). An event is the realization of a single POVM effect E_U on one spacetime bin U; all dynamical relations among alternatives—including gravitational influence—belong to pre-event evolution. Within this orthodox setting, we prove a five-part operational theorem package: R′ (resolution). Finite operational resolution is derived, not assumed, from the metrological chain quantum Fisher information → Bures/Fidelity → total variation distance, yielding explicit Δx_min and Δt_min at any discrimination threshold ε_disc. T′ (common-mode invariance). Uniform time dilation or global phases are probability-irrelevant for any POVM; only differential phases can affect outcome statistics. C′ (sub-bin curvature irrelevance). Curvature variations confined below the declared resolution cannot change single-bin event probabilities beyond ε_disc. After Hadamard-controlled spatial and temporal smearing, such effects are common-mode or strictly sub-threshold. D′ (near-null registrar trivialization). With finite information rate, the boundTVD ≤ √(F / 8) · τ implies that no nontrivial discrimination is possible as the registrar’s proper-time budget τ → 0. H′ (horizon implementability). An exterior laboratory cannot implement POVM effects localized strictly in the interior algebra of a black-hole spacetime; interior bins have zero operational support for exterior observers. Together, these results close “gravity at the event” in the single-run semiclassical domain: gravity can only shape pre-event phases and support via proper time and cannot appear as an event-internal degree of freedom at any finite operational resolution. Each theorem is stated with explicit inequalities and clean falsifiers (sub-resolution curvature tests, near-null discrimination limits, and exterior-only interior implementability), ensuring refutability. The framework uses no assumptions beyond standard quantum mechanics (POVMs and instruments) and general relativity (proper time and locality) and is consistent with interferometric observations such as COW neutron interferometry and Kasevich–Chu atom interferometers, where gravity appears exclusively through differential pre-event phases. Keywords: POVM measurement; operational quantum measurement; quantum Fisher information; distinguishability bounds; Unruh–DeWitt detector; Hadamard states; proper time; semiclassical quantum mechanics and general relativity; algebraic quantum field theory; interferometry, QMTF. Author: R. Necsanu ORCID: 0009‑0009‑1281‑9180 Reference record (QMTF): Necsanu (2026), Zenodo 10.5281/zenodo.18262158