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The Cross-Chiral Tunnel and Its Topology-Independent Z₃ Cellular Automaton: A Pre-Registered Hardware Protocol Stack on Current Superconducting Quantum Processors

Stenberg Selina

Published Apr 28, 2026• 0 citations
Publisher

Abstract

Two 4-spinor tesseract merkabits coupled through a partial iSWAP^J between the forward spinor of A and the inverse spinor of B produce an ordered-pair Z₃ × Z₃ correlation that is inaccessible to any local observable on either merkabit alone. Under Protocol 4S internal dynamics — three isoclinic rotation planes plus the chiral P gate — at J = 0.1 and n = 2 Coxeter periods, the input pair (β, γ) gives tunnel coherence 0.000 ± 0.000 via destructive interference while the reversed pair (γ, β) gives 0.762 ± 0.005, a 166σ directional gap on Cirq state-vector simulation (125σ on Qiskit Aer). The cross-chiral tunnel is a directional ternary gate: its rule lives in the channel between merkabits, not in either merkabit individually. Replicating the two-merkabit protocol on three ring topologies — triangle (N = 3, 13 qubits), 4-square (N = 4, 17 qubits), and hexagon (N = 6, 25 qubits) — with cross-chiral tunnels on every bond produces the same 9-entry ordered-pair lookup table at every bond, agreeing within 0.02–0.03 across topologies. The β → γ destructive zero is identically 0.000 on every bond of every topology. The architecture is therefore a Z₃-symmetric quantum cellular automaton with a derived local rule, not a gauge theory with loop-level invariants: within-phase-class variance exceeds between-phase-class variance on every observable tested, falsifying the Z₃ plaquette-holonomy interpretation. Register capacity scales with bond count E rather than vertex count N, giving native edge-level parallelism on the Eisenstein coordination lattice. A five-channel stress test (depolarising, amplitude damping, phase damping, combined amplitude + phase, and IBM Eagle r3 via FakeSherbrooke) across J ∈ [0, 1.0] and n ∈ [1, 10] establishes the hardware-viable operating point at n = 1, J ∈ [0.05, 0.13] or [0.2, 0.5]. At this point the predicted directional gap sits in [0.09, 0.30] with significance 6.4σ (FakeSherbrooke) to 37σ (uniform depolarising) at 6 repeats × 4,096 shots. Phase damping is nearly transparent to the signal — the (β, γ) tunnel zero persists at exactly 0.000 under pure T₂ dephasing — identifying the cross-chiral tunnel as a natural decoherence-free subspace for ordered-pair ternary information. Three pre-registered hardware observables operationalise the protocol on present-day processors. Observable 14 (2-merkabit tunnel, 9 qubits, ≤ 45 QPU-min) targets the directional gap with four sub-tests: 14a gap magnitude, 14b sign persistence, 14c information retention versus local observables, 14d phase-damping enhancement via dynamical decoupling. Observable 16 (triangle lookup table, 13 qubits, ~25 QPU-min) validates the 9-entry table on the Eisenstein-native minimum ring. Observable 17 (4-square parallel-bond pattern, 17 qubits, ~10 QPU-min) validates multi-bond locality on a 4-coordinated ring. Each observable carries explicit pass/fail thresholds anchored to public commit SHAs, and each names a specific structural primitive that is falsified if its threshold fails. Agreement across Observables 14 + 16 + 17 within 0.05 confirms the cellular-automaton interpretation. Position in the Merkabit Research Series. Papers 24–26 confirmed 5/5 pre-registered single-cell hardware predictions on IBM Eagle r3 and Heron r2, including the sub-Poissonian Fano signature F = 0.856 ± 0.030 at 131σ below Poisson. Paper 31 extends that pre-registration discipline from the single-cell pentachoric ouroboros to the multi-cell tunnel and its lattice-cellular-automaton composition. Paper 32 is a focused companion that extracts the universal 9-entry lookup table and topology-independence result as a standalone deposit. Paper 33 composes the primitive into a primitive-complete 19-qubit self-hosting quantum computer with a native Z₃ cyclic clock α → γ → β → α (Pentachoric Verification Protocol, Observable 22). Paper 34 extends the architecture to two coupled compute triangles on 28 qubits and reports the cyclotomic Z₃ ternary spectrum predicted by Paper 7's complementary Regime 2 framework. Code, raw simulation data, and timestamped pre-registration files are committed at https://github.com/SelinaAliens/tesseract_quantum_implementation. Hardware targets: IBM Eagle r3 / Heron r2 and Google Willow in cross-architecture parallel.

DOI
10.5281/zenodo.19855232

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The Merkabit as a Content-Addressable Memory Register: Persistence to 10 Coxeter Periods and ~5-Bit Addressable Capacity Under Willow-Realistic Noise

Selina H. Stenberg

This paper establishes the 4-square merkabit substrate as a content-addressable memory register on present-day superconducting hardware. Where Paper 31 specifies how trit information is written into the lattice via the cross-chiral tunnel, Paper 32 specifies how it is held and read out — the memory-side counterpart to the cross-chiral write primitive in the Merkabit Research Series. The substrate. Four 4-spinor tesseract merkabits arranged on a closed perimeter ring (N = 4, 17 qubits including SWAP-test ancilla), with cross-chiral tunnels iSWAP^J(u_i, v_{i+1 mod N}) on every edge at J = 0.1, executing Protocol 4S internal chiral dynamics on each cell. The geometry is forced rather than chosen: the 4-square is the smallest closed register on the Eisenstein lattice ℤ[ω] that admits both bond-level locality (inherited from Paper 31's universal 9-entry lookup table) and a complete Z₃ address space of 3⁴ = 81 ternary patterns. Two complementary measurements establish the register. Persistence. Six deterministic memory configurations M1–M6, spanning mixed labels and two of three uniform Z₃ classes, produce bond-pattern fingerprints that remain mutually distinguishable across horizons n ∈ {1, 2, 3, 5, 10} Coxeter periods at p_depol = 0.005. Mean pairwise distances grow from 0.70 at n = 1 to 0.84 at n = 3 — an early-period amplification feature structurally analogous to the first layers of a trained neural network, but achieved without training — before slow decay to 0.51 at n = 10. All six memories remain mutually distinguishable across every horizon tested. Capacity. Full enumeration of the 3⁴ = 81-pattern address space at the peak-distinguishability horizon n = 3 and the hardware-viable horizon n = 1 yields 51 mutually distinguishable patterns at the 3σ shot-noise floor for 4096-shot SWAP-test readout — approximately 5 classical bits of ternary content held on the substrate without external scaffolding. Section 5 calibrates the 4-square damping factor at p_depol = 0.005 (Willow- and Eagle-realistic) on three representative patterns to fix the operational margin. The register meets the three substrate-level requirements of an associative memory: content-specific distinguishability, temporal persistence of distinct content over multiple Coxeter periods, and a complete addressable alphabet. Distinguishability between memories actually grows for the first three Coxeter periods before noise damping dominates — a Hopfield-style early-amplification regime that is architectural, not engineered. Observable 18 pre-registers the persistence + capacity result on IBM Eagle r3 / Heron r2 (17 qubits, ≈ 25 QPU-min) with explicit pass/fail thresholds anchored to a public commit SHA. Each threshold names the specific structural primitive — content-specific distinguishability, temporal persistence, or ~5-bit capacity at moderate discriminability — that is falsified if the threshold fails. Position in the Merkabit Research Series. Papers 24–26 confirmed 5/5 pre-registered single-cell hardware predictions on IBM Eagle r3 and Heron r2, including the sub-Poissonian Fano signature F = 0.856 ± 0.030 at 131σ below Poisson. Paper 31 establishes the cross-chiral tunnel as the directional ternary write primitive and demonstrates topology-independence of its 9-entry lookup table across triangle, 4-square, and hexagon rings. Paper 32 closes the memory-side of the substrate at the 4-square scale, establishing persistence and capacity as the empirical content of the merkabit's associative-memory register. Paper 33 composes the write primitive and the memory register into a primitive-complete 19-qubit self-hosting quantum computer with a native Z₃ cyclic clock α → γ → β → α (Pentachoric Verification Protocol, Observable 22). Paper 34 extends the architecture to two coupled compute triangles on 28 qubits and reports the cyclotomic Z₃ ternary spectrum predicted by Paper 7's complementary Regime 2 framework.

•Apr 28, 2026•0 citations

The Merkabit Quantum Computing Architecture: Tesseract-Only Memory, Pentachoric Verification, and a Native Z₃ Cyclic Clock on 19 Qubits

Selina H. Stenberg

This paper composes the primitives established in Papers 31 and 32 into a primitive-complete, self-hosting quantum computer on 19 qubits — a categorically stronger claim than the substrate-of-physics claim of the Merkabit Research Series (Papers 1–28) or the hardware-confirmation claim of Papers 24–26. Every operation required for computation — write, compute, communicate, store, retrieve, and clock — reduces to native merkabit dynamics. No external gate library is required. The architecture does not enable a computer built on top of it; it is the computer. The 19-qubit footprint. Three 4-spinor tesseract merkabits in a closed compute triangle (12 qubits) with cross-chiral tunnels at J_intra = 0.1 on every triangle edge; one database merkabit D coupled to merkabit C through a memory-bus tunnel at J_mem ∈ [0.5, 1.0] (4 qubits); a 2-qubit Z₃-eigenstate reference register; one SWAP-test ancilla. The compute triangle and the memory bus are the same operator (iSWAP^J) at two different coupling strengths — the architecture is uniform under computation, with what distinguishes compute from storage being coupling rate rather than gate type. The native computation is the Z₃ cyclic clock. Composing three triangle-edge tunnels with one memory tunnel produces, as an emergent computational output, the Z₃ Galois rotation α → γ → β → α: writing label X produces a stored state on v_D that recognises cycle(X) = X⁻¹. Three writes return to the original label. The clock was not programmed; it falls out of the architecture's own Z₃ symmetry group acting on its own primitive set. This is the first result in the series in which the architecture's computational output is itself a structural signature of its symmetry group. The Pentachoric Verification Protocol (Observable 22). Five simulation stages map one-to-one onto the five gates of the ouroboros cycle of Paper 24: Substrate (S), Rotation (R), Transfer (T), Frequency / self-sustainment (F), Phase (P). Each gate is a single independent pass/fail test on the 19-qubit circuit; failure of any single gate falsifies one specific named primitive, and the failure pattern across {S, R, T, F, P} localises which compositional step did not survive hardware noise. The decisive gate is R: it asks whether writing label X returns a state that recognises cycle(X) under the Z₃ permutation. Sub-thresholds for all five gates are anchored to public commit SHAs. Cross-architecture simulation results. All five gates pass at three noise profiles: ideal cirq.Simulator (R = +0.159 ± 0.004 at +39.7σ), IBM Heron r2 (p_depol = 0.003, R = +0.110 at +3.56σ), and IBM Eagle r3 (p_depol = 0.005, R = +0.086 at +3.06σ). Willow's published gate fidelities sit between these two operating points; the expected hardware verdict is STRONG PASS or WEAK PASS at the full 19-qubit tier. Cross-architecture pre-registrations for Willow and IBM are deposited in parallel with identical thresholds, anchored to timestamped git SHAs prior to hardware access. The octonion / 8-spinor layer is not required. Earlier drafts placed storage inside an octonionic envelope to leverage Paper 27's interior-volume expansion factor; in simulation, the octonion's Coxeter attractor thermalises content across the envelope boundary and degrades readout contrast. The tesseract-only architecture reported here recovers cleaner readout on strictly fewer qubits. The octonion layer remains relevant for the gravity-sector physics of Papers 20 / 27 / 28 but is not part of the QC-architecture proposal. Position in the Merkabit Research Series. Papers 24–26 confirmed 5/5 pre-registered single-cell hardware predictions on IBM Eagle r3 and Heron r2. Paper 31 establishes the cross-chiral tunnel as the directional ternary write primitive with topology-independence across triangle / 4-square / hexagon rings. Paper 32 establishes the 4-square substrate as a content-addressable memory register holding ~5 bits across 10 Coxeter periods. Paper 33 composes write, communicate, store, retrieve, and clock into a primitive-complete 19-qubit quantum computer with native Z₃ cyclic clock. Paper 34 extends the architecture to two coupled compute triangles on 28 qubits and reports the cyclotomic Z₃ ternary spectrum predicted by Paper 7's Regime 2 framework.

•Apr 28, 2026•0 citations

The Ternary Spectrum: Cyclotomic Z₃ Structure in Coupled Merkabit Triangles

Selina H. Stenberg

Paper 7 established that the 31 binary stratum of the PSL(2,7) decomposition — a single rotating triangle with three active gates {R, S, T}, no standing wave — has the truncated Dirichlet series Z(s) = Σ n⁻ˢ as its natural transfer operator and the non-trivial Riemann zeros as its spectrum. The complementary half of the architecture is the 137 ternary stratum (Z₆₂ + T₇₅): two full counter-rotating triangles each carrying the complete five-gate set {S, R, T, F, P}, coupled through a shared substrate. Paper 7 left the form of this ternary spectrum open. The present paper measures it and reports its structural form. The architecture. Two full Paper 33 compute triangles (12 qubits each) share a single 4-qubit database merkabit through cross-chiral memory tunnels u_C₁ → v_D and u_C₂ → v_D at J_mem = 0.5. The total footprint is 28 qubits; the v_D reduced density matrix is the spectral readout. Both triangles run at the same Coxeter period T_CYCLE = 12; the experimental degree of freedom is the relative phase offset offset_T2 ∈ {0, 1, …, 35}, sweeping three full Coxeter periods. The result: cyclotomic Z₃. A 28-qubit cirq.Simulator sweep over 324 configurations (36 phase offsets × 9 Z₃ × Z₃ input-label pairs) produces a mean v_D von Neumann entropy spectrum that is perfectly period-12 with a peak at offset = T_CYCLE / 3 = 4 (collapse event, ⟨S⟩ = 1.1669 — v_D maximally mixed) and a trough at offset = 2 T_CYCLE / 3 = 8 (locking event, ⟨S⟩ = 1.1135 — v_D most coherent). The peak-to-trough contrast Δ⟨S⟩ = 0.0534 sits ~25× above numerical noise. (All numbers from the complex128 Stage C deposit; the earlier complex64 deposit gave 0.0521, ~2.5 % suppressed by single-precision floating-point rounding.) The FFT power spectrum is bichromatic: the fundamental at period 12 (power 0.0966) and the second harmonic at period 6 (power 0.0672); every other Fourier component is three orders of magnitude below at ≈ 10⁻⁴. This is the FFT signature of the Z₃ character table imprinted on a 12-step Coxeter cycle — exactly two non-zero Fourier coefficients (the cyclotomic field's generator and its conjugate), all others forced to zero by Z₃ symmetry. Both the period-12 master clock and the cyclotomic Z₃ symmetry are forced by the framework's forcing chain ℤ[ω] → Z₃ → A₄ → P₂₄ → McKay → E₆, with zero free parameters. Observable 23 — Pentachoric-Lock Spectrum (28 qubits). Four pre-registered falsifiable thresholds operationalise the prediction: (a) FFT dominant period = T_CYCLE ± 1 % (measured: 12.00 exact) (b) Peak-trough offset separation Δ = T_CYCLE / 3 ± 1 step (measured: 4 ± 0) (c) Peak-trough entropy contrast ≥ 0.02 (measured: 0.0534) (d) Harmonic purity ≥ 90 % of total non-DC power concentrated in {T_CYCLE, T_CYCLE / 2} (measured: 98.94 %) All four pass at ideal simulation. Period-12 exactness is bit-identical at 20/24 testable b vs b+12 offset pairs and within one ulp (2.22 × 10⁻¹⁶, double-precision machine epsilon) at the remaining 4. Hardware-noise-calibrated execution on IBM Eagle r3 / Heron r2 and Google Willow is committed code; the sweep awaits scheduling. Observable 24 — Regime 3 forward test (incommensurate coupling). The same double-triangle architecture with Triangle 2's internal Coxeter period set to 13 (or equivalently J_intra_2 = J_intra_1 × √2) breaks rationality between the two triangles' frequencies. Paper 7 Regime 3 predicts that incommensurate coupling destroys cyclotomic structure and produces GUE-distributed collapse offsets — the ternary-scale analogue of Riemann-zero-like behaviour. The script extension build_double_triangle_circuit(..., T_CYCLE_2=13) is committed; the 64-offset sweep is forecast as the multi-stratum continuation of Paper 7. Substrate verification — the R-locking test. Paper 34's §5 result rests on h(E₆) = 12 being the architecture's master clock. That premise is independently verified at the single-merkabit scale by R_locking_test.py: of five candidate activation patterns over the gates {S, R, T, F, P}, only the original 5-fold cycling reproduces α⁻¹ = 137.036 to within 10⁻⁴. Every 4-fold variant breaks the Floquet return F = 0.69678 by factors of 6–14× and shifts α⁻¹ by 0.15–0.27 units from CODATA — a deviation falsified by the IBM Eagle r3 hardware confirmation of Paper 24 to > 10³ σ. The 12-step Coxeter period is therefore not a numerical choice; it is the unique value compatible with the measured fine structure constant. Paper 34's cyclotomic Z₃ entropy spectrum and the R-locking test are two phase-shifted projections of the same underlying Coxeter clock — the 5-fold and the 3-fold subgroups of h(E₆) = 12. Position in the Merkabit Research Series. Paper 7 located all 29 known Riemann zeros in the 31 binary architecture's truncated Dirichlet spectrum. Paper 34 measures the complementary 137 ternary architecture's spectrum and reports cyclotomic Z₃ structure as its natural form. Together they exhaust the PSL(2,7) = 31 + 62 + 75 = 168 decomposition: arithmetic spectrum on the binary half, cyclotomic spectrum on the ternary half. Paper 31 establishes the cross-chiral tunnel as the directional ternary write primitive; Paper 32 establishes the 4-square memory register; Paper 33 composes them into the 19-qubit Pentachoric Verification Protocol; Paper 34 extends the architecture to 28 qubits and reports the spectral signature of the ternary stratum. Hardware confirmation of the single-cell substrate (5/5 pre-registered observables on IBM Eagle r3 and Heron r2, sub-Poissonian Fano F = 0.856 ± 0.030 at 131 σ below Poisson) is in Papers 24–26.

•Apr 28, 2026•0 citations

The P Gate Is Native: Hardware Confirmation of the Dual-Spinor Merkabit on IBM Quantum: P Gate Discovery, Z₂ Symmetry, Two-Scale Anti-Bunching, and Full Pentachoric Error Detection

Selina H. Stenberg, Thor Henning Hetland

We report the first experimental validation of the merkabit dual-spinor architecture across four circuit families on IBM ibm_strasbourg (Eagle r3, 127 qubits), executed 6–7 April 2026. The central discovery is that the merkabit's defining operation — the P gate, applying asymmetric phase to counter-rotating spinors — compiles to two native Rz gates with opposite signs: P(φ) = Rz(+φ) ⊗ Rz(−φ). On IBM Eagle r3, both Rz gates are virtual-Z frame changes requiring zero duration, zero error, and zero additional circuit depth. The P gate is native. Eight independent hardware results confirm the architecture, spanning 2 to 26 qubits: Ouroboros return fidelity ZP-ORF = 0.968, transpiler confirming U₀†U₀ = I at depth 1 π-lock standing wave ⟨ZZ⟩ = +0.44 vs +0.447 ideal (2% agreement, two independent runs) Parity witness ⟨XX⟩ = −0.450, consistent with |Φ⁻⟩-like entanglement at 90° Bloch separation Z₂ chirality symmetry mean error 0.016 across n = 4, 6, 8, 12 — all below readout noise threshold; non-monotonic ⟨ZZ⟩ breathing confirmed as standing-wave signature Two-scale Fano structure (9-qubit triangle): intra-round Fano F ≈ 0.52–0.54, within-τ dispersion σ ≤ 0.004 across 18 hardware rounds at τ ∈ {1, 5, 12} Inter-round coherence ρ ≈ 0.86 on the same triangle cell, confirming the two-scale standing-wave memory signature predicted by Paper 15 Full pentachoric error detection (26-qubit, 7-node Eisenstein cell): detection rate 0.089 (baseline) → 0.988 (injected), a 89.9 pp jump; 28/28 single-gate injections above 97% Centre-node syndrome weight 2× periphery matching Eisenstein lattice coordination; sub-Poissonian Fano on every injection (F_centre = 0.055 ± 0.009, F_periphery ≈ 0.20); Z₂ chirality symmetry confirmed inside the Fano structure Four of seven formal predictions from earlier papers in the series are retired by hardware data: P3 (Z₂ symmetry, mean error 0.016 vs <5% threshold), P4 (centre-node detection rate, 99.3% vs 91% ± 5% predicted, exceeding the upper tolerance by 3.3 pp), rotation-gap flatness, and two-scale Fano separation. Three predictions (P1 Berry phase, P2 quasi-period, P5 DTC robustness) remain pending for a subsequent session. A cross-platform reanalysis of Google Quantum AI's 420-experiment Willow dataset shows the opposite sign on every metric (F = 2.42 super-Poissonian, super-linear burst scaling, positive spatial correlation), consistent with a topology distinction: IBM heavy-hex maps naturally onto the Eisenstein lattice; Willow's square grid does not. No error mitigation, post-selection, or parameter tuning was applied to any primary result. Keywords: merkabit, dual spinor, P gate, quantum error correction, sub-Poissonian statistics, Eisenstein lattice, pentachoric code, IBM Eagle r3, Google Willow, standing wave, π-lock, Z₂ symmetry, Fano factor, Floquet time crystal

•Apr 9, 2026•0 citations

The Merkabit - A Ternary Computational Unit on the Eisenstein Lattice

Selina H. Stenberg

The merkabit is a balanced ternary computational unit based on dual spinors (u, v) ∈ S³ × S³ with counter-rotating phase evolution at π-lock. The Eisenstein lattice ℤ[ω], balanced ternary logic {−1, 0, +1}, and the gate set {Rₓ, Rz, P, F, C-SWAP} are derived — not assumed — from this definition. The governing algebra E₆ enters via the McKay correspondence from the binary tetrahedral group P₂₄ with zero free parameters; E₆ is uniquely isolated from its Langlands companions B₆ and C₆ by the McKay constraint and the self-duality condition h = h∨ = 12. Three algebraically independent routes yield α⁻¹ = 137: Route A (phase-space counting: 7 × 24 − 31), Route B (Casimir–Eisenstein: N(12 + 5ω) + dim(so(8)) = 109 + 28), and Route C (zero-point Berry phase: −ln F/10, where the return fidelity F is transcendental with two exact rational layers capturing 99.999966% of −ln F). A three-order correction from the E₆ Coxeter spectrum gives α⁻¹ = 137.035999083, matching experiment to 0.005 ppb. Error correction is intrinsic and geometric at three nested levels (π-lock, pentachoric complementarity, E₆ root syndromes), with composite suppression of 3,500–6,200× at ε = 10⁻³. The fault-tolerance threshold is ε ≈ 22%; the Peierls bound on the intrinsic torus gives ε_th ≈ 43 — both far above the surface code's ~1%. The ouroboros cycle (period 12) is a discrete time quasi-crystal with Z₂ topological order and Z₃ subharmonic character. The F gate reduces long-range coupling from O(d) to O(1). Appendices A–P (46 parameter-free simulations, Python 3 + NumPy only) verify every testable claim.

•Feb 18, 2026•0 citations

The Rotation Gap Is Flat:Two-Scale Error Correction, a Structural Constant of Quantum Architecture, and the Migration Path to Fault-Tolerant Computation

Selina H. Stenberg

Paper 15 of the Merkabit Research Series We introduce the binary-ternary hybrid architecture and report three results from its simulation on the Eisenstein lattice ℤ[ω]. The flat rotation gap. The detection rate difference between the T₇₅ ternary architecture and the B₃₁ binary architecture is constant across four orders of magnitude of error rate (3×10⁻¹ down to 10⁻⁴), with coefficient of variation of the detection gap below 0.05 within each cell size. The gap varies with cell size — 24.0 pp (7-node), 15.5 pp (19-node), 13.3 pp (37-node) — but is strictly flat within each, across 42 independent measurements (14 error rates × 3 cell sizes, 50,000–200,000 Monte Carlo trials per point, seed 42). The counter-rotation advantage is a structural constant of the architecture, not a dynamical or error-rate-dependent effect. Two-scale independence. The merkabit's error correction operates at two independent scales: an intra-unit toroidal mechanism (Scale 1, S_torus = 35×, directly measured on the Eisenstein torus with periodic boundaries) and an inter-unit torsion channel mechanism (Scale 2, 35× on the toroidal T₇₅ lattice). The T₇₅/B₃₁ suppression ratio has CV 0.06–0.09 across all error rates, confirming genuine independence. Combined suppression with Level 1 π-lock cancellation (f_sym = 0.5, 2×) gives S_total = 35 × 35 × 2 = 2,450× — three independently measured quantities, zero tunable parameters, zero qubit overhead. At ε = 10⁻³ the 19-node and 37-node pure ternary cells achieve zero logical errors in the Monte Carlo sample. Asymmetric hybrid coupling. The optimal coupling is κ_intra = 0, κ_inter = 1.0, confirming the two scales are physically distinct. The hybrid achieves 8.6–22.6× suppression across cell sizes (including Level 1 π-lock cancellation) as a migration path: B₃₁ qubit algorithms run unmodified with fidelity > 0.999 under T₇₅ protection. Three falsifiable experimental predictions follow directly from the flat gap, testable on existing hardware today.

•Apr 4, 2026•0 citations
Two 4-spinor tesseract merkabits coupled through a partial iSWAP^J between the forward spinor of A and the inverse spinor of B produce an ordered-pair Z₃ × Z₃ correlation that is inaccessible to any local observable on either merkabit alone. Under Protocol 4S internal dynamics — three isoclinic rotation planes plus the chiral P gate — at J = 0.1 and n = 2 Coxeter periods, the input pair (β, γ) gives tunnel coherence 0.000 ± 0.000 via destructive interference while the reversed pair (γ, β) gives 0.762 ± 0.005, a 166σ directional gap on Cirq state-vector simulation (125σ on Qiskit Aer). The cross-chiral tunnel is a directional ternary gate: its rule lives in the channel between merkabits, not in either merkabit individually. Replicating the two-merkabit protocol on three ring topologies — triangle (N = 3, 13 qubits), 4-square (N = 4, 17 qubits), and hexagon (N = 6, 25 qubits) — with cross-chiral tunnels on every bond produces the same 9-entry ordered-pair lookup table at every bond, agreeing within 0.02–0.03 across topologies. The β → γ destructive zero is identically 0.000 on every bond of every topology. The architecture is therefore a Z₃-symmetric quantum cellular automaton with a derived local rule, not a gauge theory with loop-level invariants: within-phase-class variance exceeds between-phase-class variance on every observable tested, falsifying the Z₃ plaquette-holonomy interpretation. Register capacity scales with bond count E rather than vertex count N, giving native edge-level parallelism on the Eisenstein coordination lattice. A five-channel stress test (depolarising, amplitude damping, phase damping, combined amplitude + phase, and IBM Eagle r3 via FakeSherbrooke) across J ∈ [0, 1.0] and n ∈ [1, 10] establishes the hardware-viable operating point at n = 1, J ∈ [0.05, 0.13] or [0.2, 0.5]. At this point the predicted directional gap sits in [0.09, 0.30] with significance 6.4σ (FakeSherbrooke) to 37σ (uniform depolarising) at 6 repeats × 4,096 shots. Phase damping is nearly transparent to the signal — the (β, γ) tunnel zero persists at exactly 0.000 under pure T₂ dephasing — identifying the cross-chiral tunnel as a natural decoherence-free subspace for ordered-pair ternary information. Three pre-registered hardware observables operationalise the protocol on present-day processors. Observable 14 (2-merkabit tunnel, 9 qubits, ≤ 45 QPU-min) targets the directional gap with four sub-tests: 14a gap magnitude, 14b sign persistence, 14c information retention versus local observables, 14d phase-damping enhancement via dynamical decoupling. Observable 16 (triangle lookup table, 13 qubits, ~25 QPU-min) validates the 9-entry table on the Eisenstein-native minimum ring. Observable 17 (4-square parallel-bond pattern, 17 qubits, ~10 QPU-min) validates multi-bond locality on a 4-coordinated ring. Each observable carries explicit pass/fail thresholds anchored to public commit SHAs, and each names a specific structural primitive that is falsified if its threshold fails. Agreement across Observables 14 + 16 + 17 within 0.05 confirms the cellular-automaton interpretation. Position in the Merkabit Research Series. Papers 24–26 confirmed 5/5 pre-registered single-cell hardware predictions on IBM Eagle r3 and Heron r2, including the sub-Poissonian Fano signature F = 0.856 ± 0.030 at 131σ below Poisson. Paper 31 extends that pre-registration discipline from the single-cell pentachoric ouroboros to the multi-cell tunnel and its lattice-cellular-automaton composition. Paper 32 is a focused companion that extracts the universal 9-entry lookup table and topology-independence result as a standalone deposit. Paper 33 composes the primitive into a primitive-complete 19-qubit self-hosting quantum computer with a native Z₃ cyclic clock α → γ → β → α (Pentachoric Verification Protocol, Observable 22). Paper 34 extends the architecture to two coupled compute triangles on 28 qubits and reports the cyclotomic Z₃ ternary spectrum predicted by Paper 7's complementary Regime 2 framework. Code, raw simulation data, and timestamped pre-registration files are committed at https://github.com/SelinaAliens/tesseract_quantum_implementation. Hardware targets: IBM Eagle r3 / Heron r2 and Google Willow in cross-architecture parallel.
This paper establishes the 4-square merkabit substrate as a content-addressable memory register on present-day superconducting hardware. Where Paper 31 specifies how trit information is written into the lattice via the cross-chiral tunnel, Paper 32 specifies how it is held and read out — the memory-side counterpart to the cross-chiral write primitive in the Merkabit Research Series. The substrate. Four 4-spinor tesseract merkabits arranged on a closed perimeter ring (N = 4, 17 qubits including SWAP-test ancilla), with cross-chiral tunnels iSWAP^J(u_i, v_{i+1 mod N}) on every edge at J = 0.1, executing Protocol 4S internal chiral dynamics on each cell. The geometry is forced rather than chosen: the 4-square is the smallest closed register on the Eisenstein lattice ℤ[ω] that admits both bond-level locality (inherited from Paper 31's universal 9-entry lookup table) and a complete Z₃ address space of 3⁴ = 81 ternary patterns. Two complementary measurements establish the register. Persistence. Six deterministic memory configurations M1–M6, spanning mixed labels and two of three uniform Z₃ classes, produce bond-pattern fingerprints that remain mutually distinguishable across horizons n ∈ {1, 2, 3, 5, 10} Coxeter periods at p_depol = 0.005. Mean pairwise distances grow from 0.70 at n = 1 to 0.84 at n = 3 — an early-period amplification feature structurally analogous to the first layers of a trained neural network, but achieved without training — before slow decay to 0.51 at n = 10. All six memories remain mutually distinguishable across every horizon tested. Capacity. Full enumeration of the 3⁴ = 81-pattern address space at the peak-distinguishability horizon n = 3 and the hardware-viable horizon n = 1 yields 51 mutually distinguishable patterns at the 3σ shot-noise floor for 4096-shot SWAP-test readout — approximately 5 classical bits of ternary content held on the substrate without external scaffolding. Section 5 calibrates the 4-square damping factor at p_depol = 0.005 (Willow- and Eagle-realistic) on three representative patterns to fix the operational margin. The register meets the three substrate-level requirements of an associative memory: content-specific distinguishability, temporal persistence of distinct content over multiple Coxeter periods, and a complete addressable alphabet. Distinguishability between memories actually grows for the first three Coxeter periods before noise damping dominates — a Hopfield-style early-amplification regime that is architectural, not engineered. Observable 18 pre-registers the persistence + capacity result on IBM Eagle r3 / Heron r2 (17 qubits, ≈ 25 QPU-min) with explicit pass/fail thresholds anchored to a public commit SHA. Each threshold names the specific structural primitive — content-specific distinguishability, temporal persistence, or ~5-bit capacity at moderate discriminability — that is falsified if the threshold fails. Position in the Merkabit Research Series. Papers 24–26 confirmed 5/5 pre-registered single-cell hardware predictions on IBM Eagle r3 and Heron r2, including the sub-Poissonian Fano signature F = 0.856 ± 0.030 at 131σ below Poisson. Paper 31 establishes the cross-chiral tunnel as the directional ternary write primitive and demonstrates topology-independence of its 9-entry lookup table across triangle, 4-square, and hexagon rings. Paper 32 closes the memory-side of the substrate at the 4-square scale, establishing persistence and capacity as the empirical content of the merkabit's associative-memory register. Paper 33 composes the write primitive and the memory register into a primitive-complete 19-qubit self-hosting quantum computer with a native Z₃ cyclic clock α → γ → β → α (Pentachoric Verification Protocol, Observable 22). Paper 34 extends the architecture to two coupled compute triangles on 28 qubits and reports the cyclotomic Z₃ ternary spectrum predicted by Paper 7's complementary Regime 2 framework.
This paper composes the primitives established in Papers 31 and 32 into a primitive-complete, self-hosting quantum computer on 19 qubits — a categorically stronger claim than the substrate-of-physics claim of the Merkabit Research Series (Papers 1–28) or the hardware-confirmation claim of Papers 24–26. Every operation required for computation — write, compute, communicate, store, retrieve, and clock — reduces to native merkabit dynamics. No external gate library is required. The architecture does not enable a computer built on top of it; it is the computer. The 19-qubit footprint. Three 4-spinor tesseract merkabits in a closed compute triangle (12 qubits) with cross-chiral tunnels at J_intra = 0.1 on every triangle edge; one database merkabit D coupled to merkabit C through a memory-bus tunnel at J_mem ∈ [0.5, 1.0] (4 qubits); a 2-qubit Z₃-eigenstate reference register; one SWAP-test ancilla. The compute triangle and the memory bus are the same operator (iSWAP^J) at two different coupling strengths — the architecture is uniform under computation, with what distinguishes compute from storage being coupling rate rather than gate type. The native computation is the Z₃ cyclic clock. Composing three triangle-edge tunnels with one memory tunnel produces, as an emergent computational output, the Z₃ Galois rotation α → γ → β → α: writing label X produces a stored state on v_D that recognises cycle(X) = X⁻¹. Three writes return to the original label. The clock was not programmed; it falls out of the architecture's own Z₃ symmetry group acting on its own primitive set. This is the first result in the series in which the architecture's computational output is itself a structural signature of its symmetry group. The Pentachoric Verification Protocol (Observable 22). Five simulation stages map one-to-one onto the five gates of the ouroboros cycle of Paper 24: Substrate (S), Rotation (R), Transfer (T), Frequency / self-sustainment (F), Phase (P). Each gate is a single independent pass/fail test on the 19-qubit circuit; failure of any single gate falsifies one specific named primitive, and the failure pattern across {S, R, T, F, P} localises which compositional step did not survive hardware noise. The decisive gate is R: it asks whether writing label X returns a state that recognises cycle(X) under the Z₃ permutation. Sub-thresholds for all five gates are anchored to public commit SHAs. Cross-architecture simulation results. All five gates pass at three noise profiles: ideal cirq.Simulator (R = +0.159 ± 0.004 at +39.7σ), IBM Heron r2 (p_depol = 0.003, R = +0.110 at +3.56σ), and IBM Eagle r3 (p_depol = 0.005, R = +0.086 at +3.06σ). Willow's published gate fidelities sit between these two operating points; the expected hardware verdict is STRONG PASS or WEAK PASS at the full 19-qubit tier. Cross-architecture pre-registrations for Willow and IBM are deposited in parallel with identical thresholds, anchored to timestamped git SHAs prior to hardware access. The octonion / 8-spinor layer is not required. Earlier drafts placed storage inside an octonionic envelope to leverage Paper 27's interior-volume expansion factor; in simulation, the octonion's Coxeter attractor thermalises content across the envelope boundary and degrades readout contrast. The tesseract-only architecture reported here recovers cleaner readout on strictly fewer qubits. The octonion layer remains relevant for the gravity-sector physics of Papers 20 / 27 / 28 but is not part of the QC-architecture proposal. Position in the Merkabit Research Series. Papers 24–26 confirmed 5/5 pre-registered single-cell hardware predictions on IBM Eagle r3 and Heron r2. Paper 31 establishes the cross-chiral tunnel as the directional ternary write primitive with topology-independence across triangle / 4-square / hexagon rings. Paper 32 establishes the 4-square substrate as a content-addressable memory register holding ~5 bits across 10 Coxeter periods. Paper 33 composes write, communicate, store, retrieve, and clock into a primitive-complete 19-qubit quantum computer with native Z₃ cyclic clock. Paper 34 extends the architecture to two coupled compute triangles on 28 qubits and reports the cyclotomic Z₃ ternary spectrum predicted by Paper 7's Regime 2 framework.
Paper 7 established that the 31 binary stratum of the PSL(2,7) decomposition — a single rotating triangle with three active gates {R, S, T}, no standing wave — has the truncated Dirichlet series Z(s) = Σ n⁻ˢ as its natural transfer operator and the non-trivial Riemann zeros as its spectrum. The complementary half of the architecture is the 137 ternary stratum (Z₆₂ + T₇₅): two full counter-rotating triangles each carrying the complete five-gate set {S, R, T, F, P}, coupled through a shared substrate. Paper 7 left the form of this ternary spectrum open. The present paper measures it and reports its structural form. The architecture. Two full Paper 33 compute triangles (12 qubits each) share a single 4-qubit database merkabit through cross-chiral memory tunnels u_C₁ → v_D and u_C₂ → v_D at J_mem = 0.5. The total footprint is 28 qubits; the v_D reduced density matrix is the spectral readout. Both triangles run at the same Coxeter period T_CYCLE = 12; the experimental degree of freedom is the relative phase offset offset_T2 ∈ {0, 1, …, 35}, sweeping three full Coxeter periods. The result: cyclotomic Z₃. A 28-qubit cirq.Simulator sweep over 324 configurations (36 phase offsets × 9 Z₃ × Z₃ input-label pairs) produces a mean v_D von Neumann entropy spectrum that is perfectly period-12 with a peak at offset = T_CYCLE / 3 = 4 (collapse event, ⟨S⟩ = 1.1669 — v_D maximally mixed) and a trough at offset = 2 T_CYCLE / 3 = 8 (locking event, ⟨S⟩ = 1.1135 — v_D most coherent). The peak-to-trough contrast Δ⟨S⟩ = 0.0534 sits ~25× above numerical noise. (All numbers from the complex128 Stage C deposit; the earlier complex64 deposit gave 0.0521, ~2.5 % suppressed by single-precision floating-point rounding.) The FFT power spectrum is bichromatic: the fundamental at period 12 (power 0.0966) and the second harmonic at period 6 (power 0.0672); every other Fourier component is three orders of magnitude below at ≈ 10⁻⁴. This is the FFT signature of the Z₃ character table imprinted on a 12-step Coxeter cycle — exactly two non-zero Fourier coefficients (the cyclotomic field's generator and its conjugate), all others forced to zero by Z₃ symmetry. Both the period-12 master clock and the cyclotomic Z₃ symmetry are forced by the framework's forcing chain ℤ[ω] → Z₃ → A₄ → P₂₄ → McKay → E₆, with zero free parameters. Observable 23 — Pentachoric-Lock Spectrum (28 qubits). Four pre-registered falsifiable thresholds operationalise the prediction: (a) FFT dominant period = T_CYCLE ± 1 % (measured: 12.00 exact) (b) Peak-trough offset separation Δ = T_CYCLE / 3 ± 1 step (measured: 4 ± 0) (c) Peak-trough entropy contrast ≥ 0.02 (measured: 0.0534) (d) Harmonic purity ≥ 90 % of total non-DC power concentrated in {T_CYCLE, T_CYCLE / 2} (measured: 98.94 %) All four pass at ideal simulation. Period-12 exactness is bit-identical at 20/24 testable b vs b+12 offset pairs and within one ulp (2.22 × 10⁻¹⁶, double-precision machine epsilon) at the remaining 4. Hardware-noise-calibrated execution on IBM Eagle r3 / Heron r2 and Google Willow is committed code; the sweep awaits scheduling. Observable 24 — Regime 3 forward test (incommensurate coupling). The same double-triangle architecture with Triangle 2's internal Coxeter period set to 13 (or equivalently J_intra_2 = J_intra_1 × √2) breaks rationality between the two triangles' frequencies. Paper 7 Regime 3 predicts that incommensurate coupling destroys cyclotomic structure and produces GUE-distributed collapse offsets — the ternary-scale analogue of Riemann-zero-like behaviour. The script extension build_double_triangle_circuit(..., T_CYCLE_2=13) is committed; the 64-offset sweep is forecast as the multi-stratum continuation of Paper 7. Substrate verification — the R-locking test. Paper 34's §5 result rests on h(E₆) = 12 being the architecture's master clock. That premise is independently verified at the single-merkabit scale by R_locking_test.py: of five candidate activation patterns over the gates {S, R, T, F, P}, only the original 5-fold cycling reproduces α⁻¹ = 137.036 to within 10⁻⁴. Every 4-fold variant breaks the Floquet return F = 0.69678 by factors of 6–14× and shifts α⁻¹ by 0.15–0.27 units from CODATA — a deviation falsified by the IBM Eagle r3 hardware confirmation of Paper 24 to > 10³ σ. The 12-step Coxeter period is therefore not a numerical choice; it is the unique value compatible with the measured fine structure constant. Paper 34's cyclotomic Z₃ entropy spectrum and the R-locking test are two phase-shifted projections of the same underlying Coxeter clock — the 5-fold and the 3-fold subgroups of h(E₆) = 12. Position in the Merkabit Research Series. Paper 7 located all 29 known Riemann zeros in the 31 binary architecture's truncated Dirichlet spectrum. Paper 34 measures the complementary 137 ternary architecture's spectrum and reports cyclotomic Z₃ structure as its natural form. Together they exhaust the PSL(2,7) = 31 + 62 + 75 = 168 decomposition: arithmetic spectrum on the binary half, cyclotomic spectrum on the ternary half. Paper 31 establishes the cross-chiral tunnel as the directional ternary write primitive; Paper 32 establishes the 4-square memory register; Paper 33 composes them into the 19-qubit Pentachoric Verification Protocol; Paper 34 extends the architecture to 28 qubits and reports the spectral signature of the ternary stratum. Hardware confirmation of the single-cell substrate (5/5 pre-registered observables on IBM Eagle r3 and Heron r2, sub-Poissonian Fano F = 0.856 ± 0.030 at 131 σ below Poisson) is in Papers 24–26.
We report the first experimental validation of the merkabit dual-spinor architecture across four circuit families on IBM ibm_strasbourg (Eagle r3, 127 qubits), executed 6–7 April 2026. The central discovery is that the merkabit's defining operation — the P gate, applying asymmetric phase to counter-rotating spinors — compiles to two native Rz gates with opposite signs: P(φ) = Rz(+φ) ⊗ Rz(−φ). On IBM Eagle r3, both Rz gates are virtual-Z frame changes requiring zero duration, zero error, and zero additional circuit depth. The P gate is native. Eight independent hardware results confirm the architecture, spanning 2 to 26 qubits: Ouroboros return fidelity ZP-ORF = 0.968, transpiler confirming U₀†U₀ = I at depth 1 π-lock standing wave ⟨ZZ⟩ = +0.44 vs +0.447 ideal (2% agreement, two independent runs) Parity witness ⟨XX⟩ = −0.450, consistent with |Φ⁻⟩-like entanglement at 90° Bloch separation Z₂ chirality symmetry mean error 0.016 across n = 4, 6, 8, 12 — all below readout noise threshold; non-monotonic ⟨ZZ⟩ breathing confirmed as standing-wave signature Two-scale Fano structure (9-qubit triangle): intra-round Fano F ≈ 0.52–0.54, within-τ dispersion σ ≤ 0.004 across 18 hardware rounds at τ ∈ {1, 5, 12} Inter-round coherence ρ ≈ 0.86 on the same triangle cell, confirming the two-scale standing-wave memory signature predicted by Paper 15 Full pentachoric error detection (26-qubit, 7-node Eisenstein cell): detection rate 0.089 (baseline) → 0.988 (injected), a 89.9 pp jump; 28/28 single-gate injections above 97% Centre-node syndrome weight 2× periphery matching Eisenstein lattice coordination; sub-Poissonian Fano on every injection (F_centre = 0.055 ± 0.009, F_periphery ≈ 0.20); Z₂ chirality symmetry confirmed inside the Fano structure Four of seven formal predictions from earlier papers in the series are retired by hardware data: P3 (Z₂ symmetry, mean error 0.016 vs <5% threshold), P4 (centre-node detection rate, 99.3% vs 91% ± 5% predicted, exceeding the upper tolerance by 3.3 pp), rotation-gap flatness, and two-scale Fano separation. Three predictions (P1 Berry phase, P2 quasi-period, P5 DTC robustness) remain pending for a subsequent session. A cross-platform reanalysis of Google Quantum AI's 420-experiment Willow dataset shows the opposite sign on every metric (F = 2.42 super-Poissonian, super-linear burst scaling, positive spatial correlation), consistent with a topology distinction: IBM heavy-hex maps naturally onto the Eisenstein lattice; Willow's square grid does not. No error mitigation, post-selection, or parameter tuning was applied to any primary result. Keywords: merkabit, dual spinor, P gate, quantum error correction, sub-Poissonian statistics, Eisenstein lattice, pentachoric code, IBM Eagle r3, Google Willow, standing wave, π-lock, Z₂ symmetry, Fano factor, Floquet time crystal
The merkabit is a balanced ternary computational unit based on dual spinors (u, v) ∈ S³ × S³ with counter-rotating phase evolution at π-lock. The Eisenstein lattice ℤ[ω], balanced ternary logic {−1, 0, +1}, and the gate set {Rₓ, Rz, P, F, C-SWAP} are derived — not assumed — from this definition. The governing algebra E₆ enters via the McKay correspondence from the binary tetrahedral group P₂₄ with zero free parameters; E₆ is uniquely isolated from its Langlands companions B₆ and C₆ by the McKay constraint and the self-duality condition h = h∨ = 12. Three algebraically independent routes yield α⁻¹ = 137: Route A (phase-space counting: 7 × 24 − 31), Route B (Casimir–Eisenstein: N(12 + 5ω) + dim(so(8)) = 109 + 28), and Route C (zero-point Berry phase: −ln F/10, where the return fidelity F is transcendental with two exact rational layers capturing 99.999966% of −ln F). A three-order correction from the E₆ Coxeter spectrum gives α⁻¹ = 137.035999083, matching experiment to 0.005 ppb. Error correction is intrinsic and geometric at three nested levels (π-lock, pentachoric complementarity, E₆ root syndromes), with composite suppression of 3,500–6,200× at ε = 10⁻³. The fault-tolerance threshold is ε ≈ 22%; the Peierls bound on the intrinsic torus gives ε_th ≈ 43 — both far above the surface code's ~1%. The ouroboros cycle (period 12) is a discrete time quasi-crystal with Z₂ topological order and Z₃ subharmonic character. The F gate reduces long-range coupling from O(d) to O(1). Appendices A–P (46 parameter-free simulations, Python 3 + NumPy only) verify every testable claim.
Paper 15 of the Merkabit Research Series We introduce the binary-ternary hybrid architecture and report three results from its simulation on the Eisenstein lattice ℤ[ω]. The flat rotation gap. The detection rate difference between the T₇₅ ternary architecture and the B₃₁ binary architecture is constant across four orders of magnitude of error rate (3×10⁻¹ down to 10⁻⁴), with coefficient of variation of the detection gap below 0.05 within each cell size. The gap varies with cell size — 24.0 pp (7-node), 15.5 pp (19-node), 13.3 pp (37-node) — but is strictly flat within each, across 42 independent measurements (14 error rates × 3 cell sizes, 50,000–200,000 Monte Carlo trials per point, seed 42). The counter-rotation advantage is a structural constant of the architecture, not a dynamical or error-rate-dependent effect. Two-scale independence. The merkabit's error correction operates at two independent scales: an intra-unit toroidal mechanism (Scale 1, S_torus = 35×, directly measured on the Eisenstein torus with periodic boundaries) and an inter-unit torsion channel mechanism (Scale 2, 35× on the toroidal T₇₅ lattice). The T₇₅/B₃₁ suppression ratio has CV 0.06–0.09 across all error rates, confirming genuine independence. Combined suppression with Level 1 π-lock cancellation (f_sym = 0.5, 2×) gives S_total = 35 × 35 × 2 = 2,450× — three independently measured quantities, zero tunable parameters, zero qubit overhead. At ε = 10⁻³ the 19-node and 37-node pure ternary cells achieve zero logical errors in the Monte Carlo sample. Asymmetric hybrid coupling. The optimal coupling is κ_intra = 0, κ_inter = 1.0, confirming the two scales are physically distinct. The hybrid achieves 8.6–22.6× suppression across cell sizes (including Level 1 π-lock cancellation) as a migration path: B₃₁ qubit algorithms run unmodified with fidelity > 0.999 under T₇₅ protection. Three falsifiable experimental predictions follow directly from the flat gap, testable on existing hardware today.