# Gandy postulates 1, 2, 4, and 5 in Arrighi-Dowek and current physics

Thread: postulates (1) homogeneity of space, (2) homogeneity of time, (4) bounded velocity of propagation, (5) quiescence. Postulate (3), bounded density of information, is discussed only where the proof-dependence question requires it.

Primary source retrieved: Pablo Arrighi and Gilles Dowek, "The physical Church-Turing thesis and the principles of quantum theory," arXiv:1102.1612, https://arxiv.org/abs/1102.1612 and source TeX from https://arxiv.org/e-print/1102.1612. Quotes below are verbatim from the TeX source.

Evidence labels: ESTABLISHED = standard result or directly quoted source; SERIOUS SPECULATION = live theoretical possibility without empirical realization; ANOMALY = reported deviation not generally accepted; ANECDOTE = low-grade report. No ANECDOTE findings are used below.

## 1. Homogeneity of space

### A. Formal statement in Arrighi-Dowek

[ESTABLISHED] Classical statement, Sec. 2, arXiv:1102.1612:

> "Homogeneity of space. If $\tau$ is a translation, then the region $\tau A$ has the same set of states as $A$.
>
> The function mapping the global state of a system at time $t$ to its global state at time $t + T$ commutes with all translations."

[ESTABLISHED] Quantum statement, Sec. 3.6, arXiv:1102.1612:

> "Homogeneity of space. As in the classical case."

[ESTABLISHED] Scope: this is a global Euclidean translation-invariance postulate on the state spaces of translated regions and on the one-step global evolution. It is not merely local Lorentz invariance, not the cosmological principle, and not a statistical statement about matter distribution. Source: Arrighi-Dowek, arXiv:1102.1612, Sec. 2 and Sec. 3.6.

### B. Empirical status in our universe

[ESTABLISHED] The observable universe is not globally spatially homogeneous in its matter distribution: stars, galaxies, clusters, voids, and the CMB rest frame pick out large-scale structure. Statistical homogeneity becomes a good approximation only on large scales, with surveys typically finding a transition around 70-100 h^-1 Mpc, i.e. order 100 Mpc. Example: Scrimgeour et al., "The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity," MNRAS 425, 116-134 (2012), arXiv:1205.6812, report consistency with homogeneity above about 80 h^-1 Mpc.

[ESTABLISHED] This cosmological inhomogeneity is not directly the same target as Arrighi-Dowek's postulate. Their theorem assumes identical local state spaces and a translation-commuting global update on Euclidean space. Actual cosmology is well described on large scales by FLRW spacetime only statistically and approximately; exact global translation symmetry is absent in a lumpy universe and in generic curved spacetimes. Sources: Arrighi-Dowek, arXiv:1102.1612; standard FLRW cosmology; Scrimgeour et al. 2012.

[ESTABLISHED] Local nongravitational physics is tested much more stringently for Lorentz invariance and local position invariance than for literal global spatial translation invariance. The Standard-Model Extension (SME) data tables summarize bounds on background coefficients that would break Lorentz/CPT symmetry and often correlate with preferred directions or frames. Source: V. A. Kostelecky and N. Russell, "Data Tables for Lorentz and CPT Violation," Rev. Mod. Phys. 83, 11 (2011), arXiv:0801.0287, updated tables at https://arxiv.org/abs/0801.0287.

[ESTABLISHED] SME matter-sector summary sensitivities in the current arXiv source for the Data Tables include, for the tilde b coefficients: electron |tilde b_X|, |tilde b_Y| at 10^-31 GeV and |tilde b_Z| at 10^-29 GeV; proton |tilde b_X|, |tilde b_Y| at 10^-32 GeV and |tilde b_Z| at 10^-29 GeV; neutron |tilde b_X|, |tilde b_Y| at 10^-33 GeV and |tilde b_Z| at 10^-30 GeV. Source: Kostelecky-Russell Data Tables, arXiv:0801.0287 source, Summary Table "Maximal two-sided sensitivities for the matter sector."

[ESTABLISHED] The same summary table gives representative c-type matter-sector sensitivities: proton tilde c_-, tilde c_X, tilde c_Y at 10^-28 GeV and tilde c_Z at 10^-28 GeV; neutron tilde c_-, tilde c_X, tilde c_Y at 10^-28 GeV and tilde c_Z at 10^-29 GeV; electron tilde c_- and tilde c_X/Y/Z at about 10^-23 GeV. Source: Kostelecky-Russell Data Tables, arXiv:0801.0287 source, Summary Table SumMat.

[ESTABLISHED] Hughes-Drever/clock-comparison lineage: nuclear Zeeman and comagnetometer tests constrain sidereal variations in clock transition frequencies, hence preferred-frame/direction SME coefficients. A modern example is Allmendinger et al., Phys. Rev. Lett. 112, 110801 (2014), which reports a neutron-sector transverse coupling bound |tilde b_perp^n| < 3.7 x 10^-34 GeV (95% C.L.) using a 3He/129Xe comagnetometer. Earlier representative clock-comparison papers include Hughes et al., Phys. Rev. Lett. 87, 111804 (2001), and Canè et al., Phys. Rev. Lett. 93, 230801 (2004).

[ESTABLISHED] Modern Michelson-Morley optical-cavity tests bound anisotropy of the speed of light at about the 10^-17 level in laboratory observables. Example: Eisele, Nevsky, and Schiller, Phys. Rev. Lett. 103, 090401 (2009), report no ether-drift signal and constrain fractional anisotropy at order 10^-17. SME photon-sector summary sensitivities in the Data Tables include birefringent d=4 spherical coefficients k_(E/B) at 10^-35, nonbirefringent kappa_e- components ranging roughly 10^-22 to 10^-16, kappa_o+ components around 10^-14, and isotropic kappa_tr around 10^-20. Sources: Eisele et al. 2009; Kostelecky-Russell Data Tables, arXiv:0801.0287 source, Summary Table SumPhot.

[ESTABLISHED] Translation invariance in the strict Noether sense is also tested indirectly by momentum conservation and by the success of quantum field theory in local experiments, but there is no single "best bound on spatial translation invariance" comparable to SME Lorentz-coefficient tables. SME coefficient bounds are the relevant quantitative proxy for local preferred-position/preferred-direction effects, not a proof of exact global homogeneity. Source: Kostelecky and Russell 2011; SME framework literature.

### C. Counterexample when dropped

[ESTABLISHED] Arrighi-Dowek, Sec. 2.3, arXiv:1102.1612, give two classical counterexamples. First, if the state spaces remain translation-invariant but the dynamics is not, spatial irregularities can encode an undecidable set U: "G(a)(tau^i(C)) = Not^{U(i)}(a(tau^i(C)))" for alphabet S = {q,0,1}. Setting the i-th cell to 0 and reading the output computes U. Second, if the state space itself is not translation-invariant, one can encode position by choosing Sigma(tau^i(C)) = {q,2i,2i+1}.

[ESTABLISHED] They state that the classical counterexamples also apply in the quantum setting for the shared hypotheses. Source: Arrighi-Dowek, Sec. 5, arXiv:1102.1612.

### D. Verdict

Verdict: HOLDS OPERATIONALLY BUT NOT LITERALLY. Local nongravitational laws are extraordinarily close to homogeneous and Lorentz invariant in tested regimes, with SME coefficients bounded down to about 10^-33 GeV in some matter-sector clock tests and photon-sector birefringent coefficients down to about 10^-44 GeV or 10^-35 in table conventions. Literal global Euclidean translation invariance fails in cosmology and curved/lumpy spacetime.

## 2. Homogeneity of time

### A. Formal statement in Arrighi-Dowek

[ESTABLISHED] Classical statement, Sec. 2, arXiv:1102.1612:

> "Homogeneity of time. The function mapping the global state of a system at time $t$ to its global state at time $t + T$ is independent of $t$."

[ESTABLISHED] Quantum statement, Sec. 3.6, arXiv:1102.1612:

> "Homogeneity of time. As in the classical case."

[ESTABLISHED] Scope: this is time-translation invariance of the one-step dynamical law G(t), not the claim that the actual state of the universe is stationary. Source: Arrighi-Dowek, arXiv:1102.1612, proof step labelled "[G(t)(|psi>)=G(|psi>)]".

### B. Empirical status in our universe

[ESTABLISHED] Cosmologically, the universe is not time-translation invariant as a state: the FLRW scale factor evolves, the temperature and density change, and generic expanding FLRW spacetime has no global timelike Killing vector. This does not by itself refute local time-independent dynamical laws; it refutes only literal stationarity/global time-translation symmetry of the universe's state. Source: standard FLRW cosmology; Arrighi-Dowek's postulate concerns the law G(t), not the state.

[ESTABLISHED] Fine-structure constant alpha, Oklo: analyses of the 1.8-billion-year-old natural fission reactor constrain alpha variation at roughly the 10^-17 yr^-1 level. A commonly cited result is dot alpha/alpha = (-0.2 +/- 0.8) x 10^-17 yr^-1, with nuclear-model assumptions; see Fujii et al., Nucl. Phys. B 573, 377 (2000), and Petrov et al., Phys. Rev. C 74, 064610 (2006). Damour and Dyson, Nucl. Phys. B 480, 37-54 (1996), give a conservative Oklo constraint of order |dot alpha/alpha| < few x 10^-17 yr^-1.

[ANOMALY] Fine-structure constant alpha, quasar absorption: Webb et al., Phys. Rev. Lett. 107, 191101 (2011), arXiv:1008.3907, reported a dipole-like spatial variation at 4.2 sigma, direction RA = 17.5 +/- 0.9 h, dec = -58 +/- 9 deg. King et al., MNRAS 422, 3370-3414 (2012), arXiv:1202.4758, reported a combined Keck+VLT dipole amplitude (0.97 +0.22/-0.20) x 10^-5 for an angular dipole, or (1.1 +/- 0.2) x 10^-6 GLyr^-1 for a lookback-distance dipole.

[ESTABLISHED] The Webb/King quasar alpha dipole is not generally accepted as established physics because long-range wavelength-calibration distortions in high-resolution spectrographs can mimic or weaken the signal. Whitmore and Murphy, MNRAS 447, 446-462 (2015), arXiv:1409.4467, found ubiquitous UVES/HIRES long-range distortions typically up to about +/-200 m/s per 1000 Angstrom and concluded they significantly weaken the alpha-variation evidence. Dumont and Webb, MNRAS 468, 1568-1574 (2017), arXiv:1701.03176, show such distortions must be modelled carefully.

[ESTABLISHED] Fine-structure constant alpha, laboratory clocks: Rosenband et al., Science 319, 1808-1812 (2008), comparing Al+ and Hg+ optical clocks, found dot alpha/alpha = (-1.6 +/- 2.3) x 10^-17 yr^-1. Godun et al., Phys. Rev. Lett. 113, 210801 (2014), comparing two Yb+ optical transitions plus other clock data, reported dot alpha/alpha = (-0.7 +/- 2.1) x 10^-17 yr^-1. These are direct laboratory constraints on present-day temporal drift.

[ESTABLISHED] Proton-electron mass ratio/electron-proton mass ratio: Godun et al., Phys. Rev. Lett. 113, 210801 (2014), report dot mu/mu = (0.2 +/- 1.1) x 10^-16 yr^-1 for mu = m_p/m_e (equivalently dot(m_e/m_p)/(m_e/m_p) changes sign). Huntemann et al., Phys. Rev. Lett. 113, 210802 (2014), using Yb+ and Cs, also constrained temporal drift of m_p/m_e at the 10^-16 yr^-1 scale. Quasar molecular constraints in King, arXiv:1202.6365, include weighted mean Delta mu/mu = (1.7 +/- 2.4) x 10^-6 at z > 1 for selected absorbers.

[ESTABLISHED] Newton's constant G, lunar laser ranging: Williams, Turyshev, and Boggs, Phys. Rev. Lett. 93, 261101 (2004), arXiv:gr-qc/0411113, report dot G/G = (4 +/- 9) x 10^-13 yr^-1. Newer LLR analyses with improved data report much tighter central constraints, e.g. Biskupek, Muller, and Torre, arXiv:2012.12032, dot G/G_0 = (-5.0 +/- 9.6) x 10^-15 yr^-1; this is a conference/proceedings-style arXiv result and should be cited with that status.

[ESTABLISHED] Newton's constant G, BBN: Alvey, Sabti, Escudero, and Fairbairn, Eur. Phys. J. C 80, 148 (2020), arXiv:1910.10730, find G_BBN/G_0 = 0.99^{+0.06}_{-0.05} at 2 sigma and, for a linear-in-time parametrization, dot G/G_0 = 0.7^{+3.8}_{-4.3} x 10^-12 yr^-1 at 2 sigma.

[ESTABLISHED] Newton's constant G, binary pulsars: Zhu et al., Phys. Rev. D 99, 124032 (2019), arXiv:1802.09206, using PSR J1713+0747, found dot G/G = (-0.1 +/- 0.9) x 10^-12 yr^-1. This is weaker than the best Solar-system bounds but probes strongly self-gravitating bodies.

[ESTABLISHED] The empirical picture is therefore: no established time variation of alpha, mu, or G; direct local clock constraints for dimensionless constants are around 10^-17 yr^-1 for alpha and 10^-16 yr^-1 for mu, while G constraints depend strongly on model and system and range from about 10^-15 yr^-1 in recent LLR analyses to 10^-12 yr^-1 cosmological/pulsar constraints.

### C. Counterexample when dropped

[ESTABLISHED] Arrighi-Dowek, Sec. 2.3, arXiv:1102.1612: "Without homogeneity of time, the irregularities in behaviour of the dynamics could be used to encode U. We would not be able to exclude that G(t+iT)(a)(C)=Not^{U(i)}(a(C))." Thus the time at which the update is applied stores the undecidable set.

### D. Verdict

Verdict: HOLDS OPERATIONALLY BUT NOT LITERALLY. Local laws show no confirmed temporal drift at current sensitivities, but the actual universe is not globally time-translation invariant because it expands and evolves. Arrighi-Dowek need time-independent local update rules, not a stationary cosmos.

## 4. Bounded velocity of propagation of information

### A. Formal statement in Arrighi-Dowek

[ESTABLISHED] Classical statement, Sec. 2, arXiv:1102.1612:

> "Bounded velocity of propagation of information. There exists a constant $T$ such that for any region $A$, any point in time $t$, the state of $A$ at time $t + T$, $\rho(A,t+T)$ depends only on $\rho(A',t)$, with $A'$ the region of radius $1$ around $A$."

[ESTABLISHED] Quantum statement, Sec. 3.6, arXiv:1102.1612:

> "Bounded velocity of propagation of information. There exists a constant $T$ such that for any region $A$, any point in time $t$, the density matrix associated to $A$ at time $t + T$, $\rho(A,t+T)$ depends only on $\rho(A',t)$, with $A'$ the region of radius $1$ around $A$."

[ESTABLISHED] In the quantum proof they call this property "causality" and then use an Arrighi-Nesme-Werner theorem: unitary plus causal implies localizable. Source: Arrighi-Dowek, proof of Theorem 1, arXiv:1102.1612; Arrighi, Nesme, Werner, "Unitarity plus causality implies localizability," JCSS 77, 372-378 (2011), arXiv:0711.3975.

### B. Empirical status in our universe

[ESTABLISHED] Special relativity and relativistic QFT enforce a finite light-cone for signals: commutators of local observables vanish at spacelike separation in standard microcausal QFT, and classical electromagnetic signals propagate at c in vacuum. This is the operational backbone of bounded information velocity. Source: standard relativistic QFT microcausality; e.g. Streater and Wightman, PCT, Spin and Statistics, and All That (1964).

[ESTABLISHED] Photon dispersion, GRB 090510: Fermi-LAT observed a 31 GeV photon from short GRB 090510 at redshift z = 0.903. Abdo et al., Nature 462, 331-334 (2009), constrained linear energy-dependent photon speed at the Planck scale, reporting a conservative lower limit M_QG,1 > 1.2 M_Planck for linear dispersion under their assumptions. Vasileiou et al., Phys. Rev. D 87, 122001 (2013), reported time-of-flight limits E_QG,1 > 2.2 x 10^19 GeV (subluminal) and > 3.9 x 10^19 GeV (superluminal), noting model dependence in intrinsic emission assumptions.

[ESTABLISHED] Photon dispersion, MAGIC GRB 190114C: MAGIC Collaboration, Phys. Rev. Lett. 125, 021301 (2020), arXiv:2001.09728, detected photons from 300 GeV to 1955 GeV at z = 0.4245 and found no energy-dependent delay. Reported 95% lower limits include E_QG,1 > 0.58 x 10^19 GeV subluminal and > 0.55 x 10^19 GeV superluminal for linear modification, and E_QG,2 > 6.3 x 10^10 GeV subluminal and > 5.6 x 10^10 GeV superluminal for quadratic modification.

[ESTABLISHED] Photon dispersion, H.E.S.S.: MAGIC's 2020 paper cites H.E.S.S. Mrk 501 results as then-leading quadratic time-of-flight limits, E_QG,2 > 8.5 x 10^10 GeV subluminal and > 7.3 x 10^10 GeV superluminal. Source: Abdalla et al. (H.E.S.S.), Astrophys. J. 870, 93 (2019), as cited in MAGIC, arXiv:2001.09728.

[ESTABLISHED] Neutrino speed, OPERA anomaly and retraction: OPERA reported in 2011 an apparent early arrival corresponding to (v-c)/c about 2.5 x 10^-5 for 17 GeV muon neutrinos, later traced to instrumental timing faults including a fiber/oscillator problem. Original claim: Adam et al. (OPERA), arXiv:1109.4897; corrected/retracted timing: Adam et al., JHEP 10, 093 (2012), arXiv:1109.4897 revisions and follow-up OPERA publications.

[ESTABLISHED] Neutrino speed, terrestrial refutations: ICARUS measured CERN-to-Gran Sasso neutrinos consistent with c, with delta t = 0.3 +/- 4.9(stat) +/- 9.0(sys) ns, excluding the OPERA-scale anomaly. Source: Antonello et al. (ICARUS), Phys. Lett. B 713, 17-22 (2012), arXiv:1203.3433. Borexino and LVD also reported results consistent with c after the timing issue. Sources: Borexino Collaboration, Phys. Lett. B 716, 401-405 (2012); LVD Collaboration, Phys. Rev. Lett. 109, 070801 (2012).

[ESTABLISHED] Neutrino speed, SN1987A: electron antineutrinos from SN1987A arrived within hours of the optical signal after travelling about 50 kpc, bounding |v-c|/c at order 10^-9 for MeV neutrinos, while the burst duration/energy spread gives a stronger energy-dependent dispersion bound around 10^-12. Sources: Longo, Phys. Rev. D 36, 3276 (1987); Krauss and Tremaine, Phys. Rev. Lett. 60, 176 (1988).

[ESTABLISHED] Vacuum birefringence: astrophysical polarization strongly constrains polarization-dependent photon propagation. SME Data Tables summary sensitivities include d=3 photon k_(V) coefficients at about 10^-44 GeV and d=4 birefringent photon coefficients k_(E/B) at about 10^-35 in the table's dimensionless convention. Sources: Kostelecky and Russell, arXiv:0801.0287; Kostelecky and Mewes, Phys. Rev. Lett. 99, 011601 (2007), arXiv:astro-ph/0702379; Kislat and Krawczynski, Phys. Rev. D 95, 083013 (2017), arXiv:1701.00437.

[ESTABLISHED] Entanglement is not superluminal signalling. Bell-violating correlations are not locally explicable, but the no-signalling theorem says local outcome statistics cannot be controlled by spacelike-separated measurement choices alone; usable information requires a classical channel. Sources: Bell, Physics 1, 195 (1964); Eberhard, Nuovo Cimento B 46, 392 (1978); Peres, Quantum Theory: Concepts and Methods (1993).

[ESTABLISHED] Arrighi-Dowek are explicitly aware that entanglement complicates the quantum case. Their conclusion states: "Quantum theory is about tensor products of vector spaces, i.e. quantum systems are not just put aside but may be entangled. Therefore, whereas causality (i.e. bounded velocity of propagation of information) immediately provides a local transition function in the classical setting, of which the global evolution is composition, the counterpart is harder to obtain in the quantum setting. For this we have had to resort to the `Unitarity plus causality implies localizability' result..." Source: Arrighi-Dowek, conclusion, arXiv:1102.1612.

[ESTABLISHED] Therefore no-signalling is not word-for-word identical to Arrighi-Dowek bounded velocity. Arrighi-Dowek require a stronger finite-neighborhood dependence of each region's reduced density matrix after one time step. In unitary quantum cellular-automaton settings, their causality/no-superluminal-influence condition plus unitarity implies a local circuit form; entanglement alone does not violate it. Sources: Arrighi-Dowek arXiv:1102.1612; Arrighi-Nesme-Werner arXiv:0711.3975; Beckman et al., Phys. Rev. A 64, 052309 (2001), "Causal and localizable quantum operations."

[SERIOUS SPECULATION] General relativity admits idealized spacetime solutions with "supertask" causal structures, especially Malament-Hogarth spacetimes, in which an observer can receive the result of an infinite proper-time computation in finite proper time. Sources: Hogarth, PSA 1994, Vol. 1, 126-138 (1994); Earman and Norton, Philosophy of Science 60, 22-42 (1993); Etesi and Nemeti, Int. J. Theor. Phys. 41, 341-370 (2002).

[SERIOUS SPECULATION] Proposed relativistic hypercomputation examples involve Kerr/Kerr-Newman black-hole interiors, anti-de Sitter-like causal boundaries, or other non-globally-hyperbolic structures. Nemeti and David, Applied Mathematics and Computation 178, 118-142 (2006), review relativistic computers and the Turing barrier. Christian Wuthrich, Synthese 192, 1989-2008 (2015), arXiv:1405.5555, is directly relevant to the philosophical/physical status of relativistic hypercomputation.

[ESTABLISHED] No Malament-Hogarth or closed-timelike-curve hypercomputer is known to be physically realizable in our universe. The obstacles include Cauchy-horizon instabilities/mass inflation, cosmic censorship, quantum backreaction, finite resources, and the absence of empirical evidence for traversable CTCs or usable Kerr-interior communication. Sources: Poisson and Israel, Phys. Rev. D 41, 1796 (1990), mass inflation; Earman and Norton 1993; Wuthrich 2015.

### C. Counterexample when dropped

[ESTABLISHED] Arrighi-Dowek, Sec. 2.3, arXiv:1102.1612: without bounded velocity, the dynamics can inspect arbitrarily distant cell patterns. They give alphabet S = {q,0,1} and a dynamics mapping a segment q x 1^i q, with x = 0 or 1, to q Not^{U(i)}(x) 1^i q. Varying the distance i queries the undecidable set U.

### D. Verdict

Verdict: HOLDS OPERATIONALLY BUT NOT LITERALLY IN ALL MATHEMATICAL MODELS. In tested nongravitational physics, information propagation is bounded by c; photon, neutrino, birefringence, and clock tests find no operational superluminal channel. Entanglement does not threaten this because it is no-signalling, but exotic GR solutions show that bounded propagation is not a theorem of all mathematical relativity spacetimes.

## 5. Quiescence

### A. Formal statement in Arrighi-Dowek

[ESTABLISHED] Classical statement, Sec. 2, arXiv:1102.1612:

> "Quiescence. For each region $A$, there exists a canonical state $q_A$ called the quiescent sate. If a region $A$ is in the quiescent state $q_A$, then the state of any subset $B$ of $A$ is the quiescent state $q_B$. At the origin, all the space, except a region of finite size, is quiescent and the global evolution preserves this fact."

[ESTABLISHED] Same section continues:

> "Consider a region $A$ that partitions into two regions $B$ and $C$. We know that if $A$ is quiescent, then both $B$ and $C$ are quiescent. Conversely, as the state of $A$ is determined by the state of $B$ and $C$, if $B$ and $C$ are quiescent then so is $A$."

[ESTABLISHED] Quantum adaptation, Sec. 3.5 and Sec. 3.6, arXiv:1102.1612:

> "The quiescence hypothesis remains the same as in the classical case, except that we need to assume that the quiescent states are pure states, in order to obtain that a region $A$ that partitions into two regions $B$ and $C$, is quiescent if and only if both $B$ and $C$ are quiescent."

and:

> "Quiescence. For each region $A$ of space, there exists a canonical pure state vector $\ket{q}^A$ called the quiescent sate. If a region $A$ is in the quiescent state $\ket{q}^A$, then the state of any subset $B$ of $A$ is the quiescent state $\ket{q}^B$. At the origin, all the space, except a region of finite size, is quiescent. The global evolution preserves this fact."

### B. Empirical status in our universe

[ESTABLISHED] Formally, quiescence says the admissible global configurations have finite support over a distinguished background state: outside a finite region everything is exactly q, and the dynamics preserves finite support. In their quantum construction, configurations are functions from Z^3 to a finite basis that equal |q> almost everywhere; this is how the Fock-like Hilbert space remains countably indexed. Source: Arrighi-Dowek, Definition "The Fock space H" and proof of Theorem 1, arXiv:1102.1612.

[ESTABLISHED] Relativistic QFT vacuum is not a product of pure local quiescent states over spatial regions. In algebraic QFT, local von Neumann algebras are Type III in standard models, local restrictions of the vacuum are mixed and entangled, and there is no simple tensor-product factorization into independent finite cells. Sources: Haag, Local Quantum Physics (1992); Witten, "Notes on Some Entanglement Properties of Quantum Field Theory," Rev. Mod. Phys. 90, 045003 (2018), arXiv:1803.04993; Clifton and Halvorson, Stud. Hist. Phil. Mod. Phys. 32, 1-31 (2001).

[ESTABLISHED] The Reeh-Schlieder theorem strengthens the point: the vacuum is cyclic for local algebras, so local operations on the vacuum generate a dense set of states; this is very far from an inactive product background. Sources: Reeh and Schlieder, Nuovo Cimento 22, 1051 (1961); Haag 1992.

[ESTABLISHED] Vacuum entanglement entropy in QFT obeys an area-law divergence with UV cutoff, not zero entanglement outside finite excitations. Srednicki, Phys. Rev. Lett. 71, 666-669 (1993), explicitly found entropy proportional to boundary area for a free scalar field. This conflicts with Arrighi-Dowek's requirement that a quiescent region's subregions be in pure quiescent states.

[ESTABLISHED] Zero-point fluctuations are real in the operational sense of nonzero vacuum correlators and effects such as Casimir forces, but the renormalized mean vacuum stress-energy in Minkowski spacetime is convention-dependent and often set to zero. Therefore the clean violation of Arrighi-Dowek quiescence is not "nonzero mean local energy" but non-product local structure, vacuum correlations, and failure of finite support. Sources: Casimir, Proc. Kon. Ned. Akad. Wet. 51, 793 (1948); Haag 1992; Witten 2018.

[ESTABLISHED] Cosmology also violates literal quiescence if the universe is spatially infinite or merely unbounded with matter/radiation/vacuum fields everywhere. There is no finite ball outside which the universe is exactly in a distinguished inactive state. Source: standard Lambda-CDM cosmology; CMB photon bath and large-scale matter distribution.

[ESTABLISHED] Does this matter for the proof? Yes. Arrighi-Dowek use quiescence to restrict the global state to finite configurations and to reduce the infinite product of local gates to a finite product on each time step. Without quiescence, the initial global input can be noncomputable and the update may have to act on infinitely many nontrivial cells. Source: Arrighi-Dowek, proof steps "[|psi> = |phi>|qq...>]" and "[G = product_F Swap product_F K]", arXiv:1102.1612.

[ESTABLISHED] Does anyone address this exact conflict? Arrighi-Dowek address quiescence internally by requiring pure quiescent states in the quantum theorem; they do not reconcile it with continuum algebraic QFT vacuum entanglement. The AQFT literature above independently shows that a strict pure-product local vacuum is not the QFT vacuum. No source verified in this thread gives a direct Arrighi-Dowek-vs-Type-III-QFT critique by name.

### C. Counterexample when dropped

[ESTABLISHED] Arrighi-Dowek, Sec. 2.3, arXiv:1102.1612: "Without quiescence, the initial configuration could be used to encode U. Choosing a trivial G, if it is given and uncomputable input, it will obviously yield an uncomputable output. This hypothesis is just a way to state that the input configuration is computable."

### D. Verdict

Verdict: FAILS LITERALLY FOR QFT AND COSMOLOGY; HOLDS ONLY AS A COMPUTATIONAL IDEALIZATION. It is natural for finite-input cellular automata and finite-particle sectors, but the QFT vacuum and an infinite cosmological matter/radiation background are not finite-support pure quiescent configurations.

## Extra question: if postulate (3) is false, can the proof go through with the other four plus a weaker (3)?

[ESTABLISHED] In the classical proof, bounded density is consumed when each finite cell has a finite state set. This makes finite local transition functions computable. Without it, Arrighi-Dowek's counterexample uses S = N and an uncomputable one-to-one f_U, with G(a)(C) = f_U(a(C)). Source: Arrighi-Dowek, Sec. 2.2-2.3, arXiv:1102.1612.

[ESTABLISHED] In the quantum proof, bounded density is consumed explicitly at the step labelled "[Sigma(A)=Sigma]": "As each cell is of finite size, and using the finite density of information hypothesis, all the Sigma(A) are finite-dimensional vector space over a field K that is finite extension of the field of rationals -- or more precisely the set of density matrices upon them." Homogeneity of space then identifies the same finite-dimensional cell space Sigma for every cell and chooses a finite basis {e_1,...,e_n} with e_1 = |q>. Source: Arrighi-Dowek, proof of Theorem 1, arXiv:1102.1612.

[ESTABLISHED] The next dependence is Proposition 1: a local linear map on H is computable. That proposition assumes a finite-dimensional cell space Sigma^{tensor p} and a finite matrix L with entries in a finite extension of Q. The theorem then uses bounded velocity plus unitarity to decompose G into local gates K and Swap, and invokes Proposition 1 to say each local K and Swap is computable. Source: Arrighi-Dowek, Proposition "If phi is a local linear map..." and proof of Theorem 1, arXiv:1102.1612.

[ESTABLISHED] Quiescence and bounded density interact: bounded density gives a finite alphabet/basis per cell; quiescence gives finite support. Together they make configurations finitely describable and the active region computable at each step. If either becomes infinite without a replacement computability condition, the theorem no longer follows. Source: Arrighi-Dowek proof steps "[Sigma(A)=Sigma]", "[|psi> = |phi>|qq...>]", and "[G = product_F Swap product_F K]".

[ESTABLISHED] A weaker postulate (3) could preserve the proof only if it supplies what finiteness supplied: a uniform computable presentation of each local state space, a computable encoding of admissible global states, and a computable local transition/gate family. Finite dimensionality is sufficient but not logically necessary; computable infinite-dimensional Hilbert spaces with computable local operators could in principle work. However, that would be an extra computability axiom, not a consequence of postulates (1), (2), (4), and (5). Source: proof analysis of Arrighi-Dowek, plus computable analysis/quantum computability context cited by Arrighi-Dowek via Nielsen, Phys. Rev. Lett. 79, 2915 (1997).

[ESTABLISHED] If the weaker version of (3) is merely a physical entropy/energy bound, the proof does not automatically go through. A Bekenstein or holographic bound may suggest finitely many distinguishable states below a gravitational collapse/energy bound, but Arrighi-Dowek's proof needs an exact finite or computably presented local state space and computable local dynamics. In ordinary continuum QFT without gravitational cutoff, local algebras are Type III and have no finite-dimensional Sigma(A), so the proof as written fails at "[Sigma(A)=Sigma]" before the local-gate computability step. Sources: Arrighi-Dowek arXiv:1102.1612; Haag 1992; Witten 2018.

Answer: the proof does not go through on the remaining four alone. It can go through with a weaker (3) only if the weaker postulate still guarantees finite or uniformly computable local degrees of freedom and computable local maps. The exact Arrighi-Dowek step that consumes finiteness is "[Sigma(A)=Sigma]", and it is used again in Proposition 1 to make local linear maps computable.

## Compressed verdict table

1. Homogeneity of space: HOLDS OPERATIONALLY BUT NOT LITERALLY. Local laws look translation/Lorentz invariant to severe bounds; global cosmological matter/spacetime is not exactly Euclidean-translation homogeneous.

2. Homogeneity of time: HOLDS OPERATIONALLY BUT NOT LITERALLY. Local constants and laws show no established drift at current limits; the universe's state and metric are cosmologically time-dependent.

4. Bounded velocity of propagation: HOLDS OPERATIONALLY BUT NOT LITERALLY IN ALL MATHEMATICAL MODELS. Standard relativistic physics and experiments support a c-bounded signal cone; entanglement is no-signalling; exotic GR spacetimes remain serious speculative countermodels, not realized machines.

5. Quiescence: FAILS LITERALLY. It is a finite-input cellular-automaton/Fock-sector idealization; continuum QFT vacuum entanglement and cosmological non-finite support violate it.