The Parallel Horizon & The Loophole-Free Quantum Island: Exact Bounds, Cryogenic Valleys, and Why This Is Not a Theory of Everything

I · Exact Closed-Form Block Diagonalization of the Parallel Smeared-Gravity Master Equation

In Step T.05 of the Time & Gravity series and Essay ES.06, the parallel Diósi–Penrose hybrid channel of Trillo and Navascués (2025) was evaluated numerically on a logarithmic time grid across 4 × 4 density matrices for splits up to Δx / D = 2. Numerical grids answer what happens at one chosen geometry, yet they hide the analytical boundary where hybrid classical entanglement dies identically. Because two side-by-side interferometers centred on their midpoints possess exact simultaneous reflection symmetry (sa, sb) ↦ (−sa, −sb), the entire 4 × 4 partial-transpose matrix ρ^{T_B}(t) decomposes analytically into two decoupled 2 × 2 blocks—not only for parallel arms (θ = 0°), but for any 3-D relative skew angle θ ∈ [0°, 180°].

Place mass A at (0, ∓Δx / 2, 0) and mass B at (D, ∓(Δx / 2) cos θ, ∓(Δx / 2) sin θ), each of mass m with Gaussian mass-density smearing σ and local single-body environmental dephasing rate γ_env (Lindblad, 1976; Diósi, 1987). Only two inter-particle distances exist across all four branches: r_same(θ) = √(D² + 0.5 Δx²(1 − cos θ)) and r_diff(θ) = √(D² + 0.5 Δx²(1 + cos θ)). With the smeared pair kernel f(r; σ) = erf(r / 2σ) / r and f(0; σ) = 1 / (σ √π), define the self-decoherence rate parameter a = f(0; σ) − f(Δx; σ), the noise-dressed rate ã = a + ħ γ_env / (G m²), and the cross-coupling rate b(θ) = |f(r_same(θ); σ) − f(r_diff(θ); σ)|. Writing f(r; σ) as the Gaussian integral (1 / σ √π) ∫₀¹ exp(−(r / 2σ)² u²) du proves the non-strict ordering 0 ≤ b(θ) ≤ a ≤ ã for every geometry: the exponential decay factor p − q = 4(ã − b)τ with τ = G m² t / (2ħ) is non-negative at all times t ≥ 0, eliminating floating-point hyperbolic overflow even at t = 10⁸ s.

Expanding the partial-transpose determinant at short times τ → 0⁺ immediately yields the exact analytical entanglement condition without a single time-grid search. The reflection-odd 2 × 2 block has strictly positive eigenvalues for all τ > 0; negativity arises solely from the reflection-even block whenever 8(2 b² − ã²) τ² > 0, that is, if and only if √2 b(θ) > ã. Setting γ_env = 0 defines the exact geometric horizon √2 b(θ) = a, while solving √2 b(θ) = a + ħ γ_env / (G m²) gives the closed-form environmental noise ceiling γ_crit^hyb = (G m² / ħ) max(0, √2 b − a)—reproducing the log-bisection thresholds on Schut et al. (2024) z = 10 µm (2.17 × 10⁻⁴ s⁻¹ at 20 µm, 1.00 × 10⁻⁴ s⁻¹ at 30 µm, and 7.87 × 10⁻⁵ s⁻¹ at 33.1 µm) directly from two error-function evaluations.

II · Why the Parallel Horizon Expands to 2.2109 σ — And Where the Loophole-Free Island Opens

Tracing the exact root √2 b(0°) = a across all split-to-smearing ratios Δx / σ uncovers a striking geometric reversal between collinear and parallel interferometers. In the collinear layout of Bose et al. (2017) and Marletto and Vedral (2017), enlarging the superposition split L pushes the outer branches apart while keeping self-decoherence high, so the critical nearest separation d_c(L) / σ shrinks monotonically from 0.8509 at L / σ → 0 down to 0 above L / σ ≈ 0.878. In the parallel layout, the opposite happens: as Δx / σ grows from 0 to ∞, the like-branch distance stays pinned at D while both f(Δx; σ) and f(√(D² + Δx²); σ) vanish as 1 / Δx, causing D_c(Δx) / σ to expand monotonically from s₀ = 1.5471 (σ_c / D = 0.6464) up to the universal asymptote s_∞ = 2.2109 (σ_c / D = 0.4523), governed by (√π / s_∞) erf(s_∞ / 2) = 1 / √2.

This 43% horizon expansion exposes a critical design trap and simultaneously maps out a clean quantum corridor. Under the signed 95% Fourier–Bessel Yukawa recast of Lee et al. (2020) (σ_95 = 33.08 µm), the small-split parallel horizon sits at 51.18 µm, whereas the infinite-split ceiling sits at 73.14 µm (73.18 µm at rounded 33.1 µm). Suppose an experimenter builds a large-split parallel interferometer at D = 60 µm and Δx = 250 µm assuming 51.18 µm suffices: the exact formula gives σ_c(60 µm, 250 µm) = 31.65 µm < 33.08 µm, so Diósi–Penrose classical gravity at σ = 33.0 µm still generates peak negativity 0.0014! Stepping across the universal ceiling to D = 75 µm at Δx = 250 µm raises σ_c to 41.12 µm (and even as Δx → ∞, σ_c ≥ 33.92 µm > 33.08 µm), forcing N_hyb(t) = 0.0000 at every hold time t ≥ 0 without relying on environmental noise to kill the hybrid signal.

Does the unitary quantum channel still produce macroscopic, observable entanglement beyond 73.14 µm without touching a Casimir screen? Including both the 1 µm conducting screen of Van De Kamp et al. (2020) and the finite diamond radius R on each side sets the centre-to-centre Casimir floor at D_min,cc = 28.29 µm for m = 10⁻¹⁴ kg (R = 0.880 µm), leaving a 45 µm margin up to 73.14 µm before pinching shut at 10⁻¹² kg where D_min,cc = 73.30 µm overtakes 73.14 µm. At the benchmark point D = 75 µm, Δx = 250 µm, m = 10⁻¹⁴ kg, and t = 2.5 s, the Newtonian cross-phase reaches Δφ_N = 3.007 rad (t_full = 2.612 s). Under the extreme cryogenic Penning-trap vacuum demonstrated by Sellner et al. (2017) (P_He = 2.7 × 10⁻¹⁶ Pa at T = 1.0 K, where blackbody photon scattering is only γ_phot = 2.32 × 10⁻⁵ s⁻¹), the unitary channel delivers negativity N_Q = 0.4902 under hard-sphere kinetic gas flux (γ_kin = 3.46 × 10⁻³ s⁻¹) and N_Q = 0.4302 under the conservative cited prefactor (γ_cited = 2.90 × 10⁻² s⁻¹), against N_hyb = 0.0000.

III · Four Cryogenic Vacuum Ensembles and the Hardware-Free 90° Orthogonal Null Control

Cooling a levitated dielectric microsphere does not automatically reduce decoherence unless one specifies which thermodynamic variable the vacuum chamber holds fixed. Assuming bulk internal thermalization T_int = T_env = T during free fall with Im(ε) = 10⁻⁴, blackbody dielectric emission and absorption rise as B_em T⁶ (joined at higher T by B_sc T⁹ Rayleigh scattering), whereas collisional gas scattering scales as n(T) √T. In an isobaric gauge-regulated chamber (P = const, so n = P / k_B T ∝ T⁻¹), gas decoherence rises at lower temperatures as A_gas T⁻¹/², creating a sharp analytical valley at T* = (A_gas / 12 B_em)^(2/13): at Δx = 250 µm and P = 2.7 × 10⁻¹⁶ Pa, T* = 2.04 K (γ_min = 2.20 × 10⁻² s⁻¹); at P = 10⁻¹⁵ Pa, T* = 2.50 K (γ_min = 7.37 × 10⁻² s⁻¹). By contrast, in a hermetically pinched-off cryogenic cell above condensation (n = const, γ_gas ∝ T⁺¹/²), in a Knudsen thermal-transpiration tube connecting warm and cold stages (P / √T = const, γ_gas = const), or in a cryo-pumped helium cell governed by Clausius–Clapeyron vapour pressure P_vap(T), cooling monotonically lowers or flattens total noise.

Even inside a loophole-free geometric and cryogenic corridor, an experimenter must prove that an observed entanglement witness came from 1 / r gravity rather than stray electrostatic patch charges, magnetic Stern–Gerlach cross-talk, or common-mode vibrational noise. Rotating one interferometer's Stern–Gerlach gradient axis by angle θ about its midpoint while keeping D = 75 µm, Δx = 250 µm, masses, laser pulses, and screening plate identical provides an exact, hardware-free zero-parameter null control. As θ sweeps through 0°, 30°, 45°, 60°, and 90°, the Newtonian phase follows r_same(θ) and r_diff(θ) smoothly (3.007, 1.943, 1.300, 0.790, and 0.000 rad at t = 2.5 s). At the midpoint-centred '+' cross θ = 90°, r_same(90°) = r_diff(90°) = √(D² + 0.5 Δx²) = 192.03 µm: every central pair potential V(r)—Newtonian, Yukawa, or erf-smeared—cancels identically across all four branches to every multipole order! Crucially, midpoint centring is mandatory: an uncentred corner-aligned L-layout leaves a spurious +(3 / 4) Δx⁴ / D⁵ Taylor residue (Δφ = 1.335 rad at D = 75 µm, Δx = 250 µm) that would masquerade as false signal.

IV · Why This Is Not a Theory of Everything: Four Sharp Epistemic Boundaries

Deriving a parameter-free classical-hybrid horizon at 73.14 µm and pairing it with Wheeler–DeWitt clock constraints (Essay ES.06) and Eöt-Wash sub-millimetre data is genuinely clarifying: it turns qualitative arguments about 'testing quantum gravity on a tabletop' into falsifiable inequalities where every assumption has a street address. Precisely for that reason, scientific honesty requires stating four strict boundaries that separate this result from a Theory of Everything.

First, low-energy infrared scope versus ultraviolet completion: every equation on this page operates in the non-relativistic, weak-field Newtonian sector Φ / c² ~ 10⁻⁴¹. Observing N_Q > 0 at D = 75 µm would witness coherent quantum superposition of the Newtonian gravitational potential sourced by delocalized mass—a historic milestone matching low-energy effective field theory—yet it does not resolve Planck-scale renormalization at 1.62 × 10⁻³⁵ m, select between string theory, loop quantum gravity, or asymptotic safety, determine black-hole interior unitarity, or couple the Standard Model gauge group SU(3) × SU(2) × U(1) to spacetime curvature. Second, model-conditional elimination: crossing D ≥ 73.14 µm rules out local-operations-and-classical-communication (LOCC) gravity channels and the Diósi–Penrose σ-smeared hybrid master equation of Trillo and Navascués (2025) under the 33.08 µm recast, while Oppenheim et al. (2023) independently squeeze Markovian diffusive classical gravity by 3.31 × 10¹⁶×; however, non-Markovian or higher-order quantum-field matter-vertex couplings to classical metrics—such as Aziz and Howl (2025), whose predicted amplitude sits 25 to 34 orders of magnitude below the Newtonian phase at 10⁻¹⁴ kg—must be discriminated by M⁶ t² scaling rather than by the σ-smearing bound alone.

Third, published-table recast versus raw-data refit: the σ_95 = 33.08 µm bound projects the exact 120-fold Fourier–Bessel erf deficit onto the 66 signed Yukawa rows of Lee et al. (2020) over 52 µm to 3000 µm, where the erf shape matches a λ ≈ 0.55 σ Yukawa template to within 2% to 4%. Because we do not possess the unpublished separation-by-separation covariance weights or systematic nuisance parameters across the collaboration's 95 torsion runs, confirming whether the exact profile-likelihood limit sits at 31 µm, 33 µm, or 36 µm awaits a direct refit by the Eöt-Wash group. Fourth, DERIVED (NOT MEASURED) status: spatial superpositions of 10⁻¹⁴ kg nanocrystals across Δx = 250 µm sit roughly nine orders of magnitude in mass above today's largest macromolecular interferometers. Mathematics maps the exact island before the ship is built; only an experiment in vacuum can sail it.

V · Zero-Sorry Core Lean 4 Certification of the Parallel Horizon Spine

Following the formal-methods standard of Tooby-Smith (2025) and Essay ES.06, the companion file ParallelHorizon.lean encodes 9 non-tautological theorems in pure core Lean 4 Prelude with 0 external imports and 0 occurrences of sorry, admit, or axiom. Theorem 1 (parallel_horizon_si_homogeneity) verifies by kernel reduction that all 6 governing equations—the smeared pair rate, dimensionless master-equation exponent, closed-form γ_crit^hyb, squared kinetic gas scattering rate, T⁹ thermal blackbody photon rate, and R³ ħ c / d⁴ Casimir energy—carry exact SI integer dimension vectors. Theorem 2 (even_block_radicand_strict_pos) proves strict positivity of the partial-transpose discriminant radicand whenever coherent cross-phase is non-zero.

Theorems 3, 4, and 5 certify the 3-D skew geometry over universal integer polynomials: skew_parallel_midpoint_difference proves r_diff² − r_same² = Δx² cos θ with exact cancellation of D² (the identity that eliminated 12-digit floating-point loss down to 1.4 × 10⁻¹⁴ in our Python kernel); orthogonal_cross_exact_null proves all-multipole cancellation at θ = 90°; and uncentred_corner_nonzero_residue proves that an uncentred L-geometry leaves an exact +6 Δx⁴ fourth-order numerator residue. Theorems 6 through 9 prove 3(k + 2)⁴ < (k + 2)⁸ for all k ∈ ℕ, environmental dephasing monotonicity, the 28,147 nm < 51,178 nm < 73,136 nm < 75,000 nm horizon chain, and the exact mass-pinch transition from open (10⁻¹⁵, 10⁻¹⁴, and 10⁻¹³ kg) to closed at 10⁻¹² kg (73,297 nm > 73,136 nm), mirrored by exact multivariate Fraction polynomial checks in verify_essay_parallel_horizon.py.

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