Multipartite Witnesses, Scaling Tomography & Active Clocks: Separating Quantum Gravity from Classical Hybrids Beyond Two Bodies

I · Exact 8 × 8 Equilateral-Prism Dynamics & The Position-Pointer Mermin Law

In Essays ES.07 and ES.08, stepping two parallel Stern–Gerlach matter-wave interferometers out to center-to-center gap D = 75 µm at superposition split Δx = 250 µm and mass m = 10⁻¹⁴ kg (Bose et al., 2017; Marletto & Vedral, 2017; Schut et al., 2024) isolated a clean two-qubit island above the 73.14 µm smeared-gravity ceiling. Yet bipartite entanglement alone tests only pairwise correlations. Placing 3 identical interferometers at the vertices of an equilateral triangle of side D = 75 µm in the (x, y) plane—each split by ±Δx / 2 = ±125 µm along z—promotes the Hilbert space to an 8 × 8 three-qubit register where every pair shares identical like-branch separation r_same = 75.00 µm and unlike-branch separation r_diff = √(D² + Δx²) = 261.01 µm, accumulating pairwise entanglement frequency ω_pair = 2(G m² / ħ)(1 / D − 1 / r_diff) = 1.20275 rad/s and reaching Δφ(t_π) = π at hold time t_π = 2.6120 s.

Preparing |+⟩_X^⊗3 = 2⁻³/²(|0⟩ + |1⟩)^⊗3 and evolving under the Ising gravitational propagator U(t) = exp[i(Δφ(t) / 4)(Z₁Z₂ + Z₂Z₃ + Z₁Z₃)] alongside single-particle spatial-branch localization η = exp(−γ_loc t) uncovers a striking operator asymmetry in the Mermin–Ardehali–Belinskii–Klyshko observable M₃ = X₁X₂X₃ − X₁Z₂Z₃ − Z₁X₂Z₃ − Z₁Z₂X₃. Because simultaneous branch inversion (z₁, z₂, z₃) ↦ (−z₁, −z₂, −z₃) leaves z₁z₂ + z₂z₃ + z₁z₃ invariant, X₁X₂X₃ commutes identically with U(t): it acquires 0 gravitational phase and suffers full three-body dephasing η³. Conversely, in −X₁Z₂Z₃, the spatial-branch operators Z₂ and Z₃ commute with position localization while X₁ flips z₁ ↦ −z₁: averaging z₂z₃ cos[½ Δφ(z₂ + z₃)] over the 4 equally weighted (z₂, z₃) ∈ {±1}² branches yields −½(1 − cos Δφ) damped by only a single factor of η¹!

Why must a tripartite gravity experiment clear 2√2 ≈ 2.8284 rather than stopping at the classical 3-local Bell–Mermin bound of 2.0? Because a biseparable quantum state ρ_AB ⊗ ρ_C—in which only interferometers 1 and 2 entangle while interferometer 3 sits as an uncoupled spectator—can already reach Tsirelson's bound 2√2 on M₃! Conflating 2.0 with 2.8284 would mistake ordinary pairwise entanglement for irreducible three-body structure. At T = 2.0 K near the isobaric cryogenic valley (P = 2.7 × 10⁻¹⁶ Pa, cited γ_loc = 0.0220 s⁻¹, η = 0.9442 at t_π = 2.6120 s), the equilateral prism achieves ⟨M₃(t_π)⟩ = 3.6741 (collective negativity N_{1|23}^Q = 0.4457), surpassing 2.8284 by 0.8457 and certifying Genuine Tripartite Entanglement using only two Pauli Stern–Gerlach readout bases {X, Z} per mass.

This pointer protection carries a decisive operational dividend for cryogenic trap design. Had all 4 correlators in M₃ been measured in equatorial bases perpendicular to Z, each correlator would decay cubically as η³, collapsing Genuine Tripartite Entanglement below 2√2 = 2.8284 as soon as η < 0.8909 (γ_loc > 0.0442 s⁻¹ at t_π = 2.6120 s). Because Z_j Z_k commutes with spatial-branch localization, η³ + 3η > 2.8284 holds all the way down to η = 0.7829 (γ_loc = 0.0937 s⁻¹), expanding the tolerable environmental decoherence rate by 2.12× while requiring only 4 correlator settings instead of the 8 settings demanded by a full Svetlichny S₃ polynomial.

II · 3-Body Diósi–Penrose Collective Horizon & The 3-D Orthogonal Null Obstruction

Could collective gravitational pooling from two neighbours allow the σ-smeared classical hybrid channel of Trillo and Navascués (2025) to breach the D = 75 µm island when a third mass is added? Integrating the exact 3-body Diósi–Penrose master equation on the 8 × 8 prism register proves two structural facts. First, the generator satisfies exact trace preservation and spatial no-signalling: rotating mass 3's measurement basis locally leaves ρ₁₂ invariant to < 10⁻¹⁵, and decoupling mass 3 (m₃ → 0 or R₃ → ∞) reduces Tr₃(ρ₁₂₃^DP) identically to ρ₁₂^DP (< 10⁻¹⁵). Second, pooling two neighbours across bipartition 1|23 shifts the short-time partial-transpose threshold from √2 b(σ) = a(σ) (1.4142 b = a, giving pairwise σ_c^(2) = 41.12 µm) to the quadratic root ((√17 − 1) / 2) b(σ) = a(σ) (1.5616 b = a). That collective boost lowers the critical smearing width to σ_c^(1|23) = 34.93 µm—closer to the Eöt-Wash recast limit, yet still strictly above σ_95 = 33.08 µm! Consequently, at σ_95 = 33.08 µm, N_{1|23}^hyb(t) = 0.0000 for all t ≥ 0, and ⟨M₃(t)⟩_hyb = ¼ η³(e^{−6(a+2b)τ} + 3e^{−2(3a−2b)τ}) + 3η e^{−2aτ} sin²(2bτ) never exceeds its t = 0 product value 1.0000 (log-grid maximum 0.9268 at t = 0.0316 s; secondary oscillation bump 0.5012 at t = 1.40 s; 0.3227 at t_π = 2.6120 s).

III · Log-Scaling & Composition Tomography: Disentangling Classical Matter Vertices

Crossing D = 75 µm eliminates local-operations-and-classical-communication mediators (Marletto & Vedral, 2017; Schneider et al., 2026) and σ ≤ 33.08 µm Diósi–Penrose hybrids, while Oppenheim et al. (2023) independently constrain Markovian diffusive classical gravity via stochastic momentum diffusion. Yet Aziz and Howl (2025) highlighted a distinct theoretical loophole: higher-order perturbative vertices coupling quantum-field matter propagators to an unquantized classical metric. Precision requires keeping two conventions straight in their Eq. 85: at constant material density ρ (where sphere radius scales as R ∝ ρ⁻¹/³ M¹/³), their printed amplitude scales as √ϑ_AH = (6 / 25) G² m_atom² M³ R t / (ħ³ D) ∝ m_atom² ρ⁻¹/³ M¹⁰/³ t¹ D⁻¹, whereas its square—the dimensionless parameter ϑ_AH = (√ϑ_AH)² tabulated in our dispute kernel—scales as m_atom⁴ ρ⁻²/³ M²⁰/³ t² D⁻².

Even if future matter-wave interferometers approached masses where classical-vertex amplitudes became non-negligible, log-scaling and composition tomography separate the mechanisms cleanly. Replacing ¹²C diamond (12.0 u, 3500 kg/m³, R = 0.880 µm) with ¹⁷³Yb (173.04 u, 6900 kg/m³, R = 0.702 µm) at identical total mass M = 10⁻¹⁴ kg, D = 75 µm, Δx = 250 µm, and t_π = 2.6120 s leaves the passive Newtonian quantum phase Δφ_Q = π identically invariant at 1.0000× by the Weak Equivalence Principle, while multiplying √ϑ_AH by (173.04 / 12.0)² (3500 / 6900)¹/³ = 165.8× (to 1.84 × 10⁻¹²) and ϑ_AH by 27,500× (to 3.39 × 10⁻²⁴)! Simultaneously, doubling hold time to t_{2π} = 2 t_π = 5.2240 s (Δφ = 2π) produces a sharp dynamical signature against Oppenheim et al. (2023): unitary Newtonian gravity coherently un-entangles the prism back to separability (⟨M₃(2t_π)⟩ = η(2t_π)³ = η(t_π)⁶ = 0.7084, reviving single-qubit X visibility from 0.0000 at t_π to 0.8914 at 2t_π), whereas any Markovian diffusive channel with rate ≥ ω_pair suppresses visibility irreversibly below exp(−2π) = 0.0019.

Even within a single crystal species such as ¹²C diamond, scanning total nanocrystal mass M and hold time t breaks the degeneracy between Newtonian action and squared classical-vertex coupling ϑ_AH without changing material hardware. Because ln Δφ_Q carries exponents (2, 1) in (ln M, ln t) whereas ln ϑ_AH carries exponents (20/3, 2) at fixed density, their 2 × 2 log-scaling Jacobian evaluates to det [[2, 1], [20/3, 2]] = 4 − 20/3 = −8/3 ≠ 0 (3×-cleared integer determinant det [[6, 3], [20, 6]] = −24 ≠ 0). Doubling crystal mass M at halved hold time t / 4 keeps Newtonian phase Δφ_Q ∝ M² t constant at 1.0000× while multiplying ϑ_AH ∝ M²⁰/³ t² by 2²⁰/³ × 4⁻² = 2⁸/³ = 6.3496×, providing an independent single-species log-slope check alongside the ¹²C-to-¹⁷³Yb composition swap.

IV · Active Quantum Clocks & Relativistic Proper-Time Interferometry

Passive QGEM interferometers treat each nanocrystal as an internal structureless rest mass M whose branches accumulate Newtonian action phase Δφ_pair / 2 = (M / ħ) ΔΦ_grav t, where ΔΦ_grav = G m_src (1 / D − 1 / √(D² + Δx²)). What changes when each delocalized mass also carries an active internal two-level quantum clock prepared in (|g⟩ + |e⟩) / √2 at transition frequency ω_clock? As shown by Zych, Costa, Pikovski, and Brukner (2011), recently tested in optical-ion proper-time protocols by Sorci et al. (2026), and extended to gravitationally induced entanglement with massive clocks by Tobar, Aspelmeyer, and Zych (2026), general-relativistic time dilation Δτ = ΔΦ_grav t / c² entangles the internal clock state with the spatial path, modulating spatial interferometric visibility as V_clock(t) = |cos(ΔΦ_clock / 2)| with ΔΦ_clock = (ω_clock / c²) ΔΦ_grav t.

Dividing the active clock phase by the single-bond Newtonian action phase exposes the exact Equivalence Principle bridge ΔΦ_clock / (Δφ_pair / 2) = ħ ω_clock / (M c²): a 10⁻¹⁴ kg crystal acts as an intrinsic Compton clock ticking at Ω_C = M c² / ħ = 8.52 × 10³⁶ rad/s, which is why center-of-mass entanglement reaches O(1) radians at D = 75 µm whereas a single embedded ⁸⁷Sr optical transition (2.70 × 10¹⁵ rad/s) or ²²⁹Th nuclear transition (1.27 × 10¹⁶ rad/s, a 4.71× gain) accumulates 4.97 × 10⁻²² rad or 2.34 × 10⁻²¹ rad from a 10⁻¹⁴ kg source. Conversely, in Earth's gravitational field (g = 9.80665 m/s²), active clocks shine as differential proper-time witnesses: a ⁸⁷Sr clock split across Δh = 250 µm acquires 1.92 × 10⁻⁴ rad in 2.6120 s and hits complete Zych–Brukner visibility extinction V_clock = 0 at vertical split Δh_π = 4.09 m, while doped crystals hosting N_ens ~ 10⁶ GHZ-entangled ²²⁹Th nuclei near milligram source masses open the direct proper-time entanglement regime analyzed by Tobar, Aspelmeyer, and Zych (2026).

V · Zero-Sorry Core Lean 4 Certification of the Multipartite & Clock Spine

Following the formal digitalization methodology of Tooby-Smith (2025) and Essays ES.06–ES.08, MultipartiteWitness.lean encodes 9 zero-sorry theorems and 9 universal quantifiers in pure core Lean 4 Prelude with 0 external imports. Theorem 1 (multipartite_clock_si_homogeneity) verifies by kernel reduction (rfl) 6 governing SI dimensional identities: Newtonian pair phase, Aziz–Howl printed √ϑ_AH amplitude, active clock gravitational redshift phase, the Compton-to-clock equivalence ratio, Oppenheim's [D_p Δx² / ħ²] = T⁻¹ spatial decoherence rate with [D_p] = M² L² T⁻³, and the proper-time gravitational beat frequency.

Theorems 2, 3, and 4 certify the multipartite Bell hierarchy: mermin_lhv_bound_all_64 proves −2 ≤ M₃ ≤ 2 across all 64 deterministic 3-local hidden-variable assignments; svetlichny_biseparable_bound_and_quantum_witness proves −4 ≤ S₃ ≤ 4 across all 64 AB|C joint assignments while witnessing M₃ = +4 on non-factorable joint correlations; and biseparable_threshold_integer_certificate proves 20,000² < 8 × 10,000² < 36,741² (2 < 2√2 < 3.6741). Theorems 5 through 9 certify the non-degenerate 3×-cleared scaling determinants (det = 36 ≠ 0 and det = −24 ≠ 0), global Z₂ Ising-sum parity invariance, exact 2h² + d² nearest-neighbour equality on the collinear (x̂, ŷ, x̂) stack, the general ℤ³ Euclidean three-orthogonal-vector obstruction via 2-D Lagrange and Cramer elimination, and the 3308 < 3493 < 4112 (33.08 µm < 34.93 µm < 41.12 µm) 3-body Diósi–Penrose horizon chain—all DERIVED (NOT MEASURED) and checked offline in < 6 s.

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