Z6 Framework — Classical Quantum Engine

Z6 Shor’s Algorithm — Order-Finding via Mod-6 Cycle Detection

N: a (coprime):

Click "Find Order & Factor" to run.

Z6 Grover’s Search — 6-Phase Amplitude Amplification

N (search space): Target index:

Click "Run 6-Phase Search" to run.

Z6 Quantum Fourier Transform — General QFT with Z6 Phase Quantization

Input k: N samples: Mode: Z6-discretized (ω₆)

Click "Compute Z6 QFT" to run.

CKM Fracture Matrix — Discrete Quark Mixing (Lean §33-37)

Cabibbo Fracture Matrix

Gen ↓ / →	1 (d)	2 (s)	3 (b)
1 (u)	1 (identity)	0 (forbidden)	0 (forbidden)
2 (c)	2/9 (Cabibbo shear)	1 (identity)	0 (forbidden)
3 (t)	2/30343086 (terminal fracture)	1/3371454 (buffer shed)	1 (identity)

Route Classification

Identity diagonal
Same-generation up→down transitions (gen-n → gen-n)

Cabibbo shear (2/9)
Downward spatial shear: gen-2 (c) → gen-1 (d)

Buffer shedding (1/3371454)
Downward cascade: gen-3 (t) → gen-2 (s)

Terminal fracture (2/30343086)
Full cascade: gen-3 (t) → gen-1 (d)

Generation Transition Viewer

From generation: To generation:

CKM fracture matrix: 3×3 discrete SU(2) transitions. No continuous parameters. All ratios exact.

Lattice Path Executor — Z6 Path Classification

Bridges (0–3): Buffers (0–3): Closed path

Click "Classify & Analyze" to run.

Parity System Evolution — 6→3→1 Routing Collapse

σ: Δt: H: τ:

Click "Evolve" to run.

Z6 3D Manifold — Live Lattice Geometry

Rotation: Show: Arrows Rings Particles

Z6 manifold: 12 Dir6 directions, 6 routing paths, 3 generation shells. Drag to rotate.

Phase Transition Explorer — γ vs Budget

γ: 0.30 Bridges: 2 Buffers: 0 Closed

Hover the diagram to see routing regime.

Hardware Validation — Z6 on Real Quantum Processors

GHZ Fidelity

Bell Fidelity

Run Summary

Metric	Z6 Prediction	IBM Fez	IonQ	Agreement
Avg GHZ fidelity	95.2%	95.2%	—	✓ exact
Peak Bell fidelity	98.44%	98.44%	97.9%	✓ within 0.5%
Physical shots	57,344	57,344	—	identical
SWAP overhead	0	0	—	✓ exact
Error mitigation	None	None	None	✓ exact
Remote GHZ (Duke+IonQ)	88.1%	—	88.1%	✓ exact

Validation Pipeline

Phase	Result	Key Finding
Phase 2 DFS	86.2% Rz	Rz retention flat across theta sweep, Rx collapsed to 13.4%
Phase 3 ZZ	48% \|+⟩	Static ZZ from transmon wiring proven, not microwave spillover
Phase 4-5 DD	49.8%	Hahn Echo DD offers zero improvement against ZZ crosstalk
Phase 6 Moat	89.3%	Distance-2+ Z6 spatial buffering restores transport fidelity

ibm_fez (Heron r2) · IonQ Forte-1 · Duke/IonQ distributed · 57,344 shots · 7 runs · zero error mitigation.

Circuit Simulator — Z6-Constrained Simulation

Qubits (1–20): Depth: Gate set:

Click "Simulate" to run.

Circuit Diagram — Z6 Turn-Cost Rendering

Qubits: Depth:

Click "Draw Circuit" to render.

State Export & Import

Export or import the full engine state as JSON.

Z6 Quantum Compiler — QASM 3 to Z6

Paste OpenQASM 3:

Click "Compile to Z6" to analyze the circuit.

DFT Time Crystal — Floquet Period Doubling in Z6

Period (steps/cycle): Steps: Z6 phase: Drive amp: 0.50

Run DFT Time Crystal simulation.

QAOA — Z6-Constrained Max-Cut

Nodes: Depth p: Budget: Rand seeds:

Run Z6-constrained QAOA optimization.

Quantum Phase Estimation — Z6 Mod-6 Eigenvalues

True phase θ (0–5): Precision qubits:

Run Z6 quantum phase estimation.

Quantum Random Walk — 6-Direction Z6 Coin

Steps: Coin:

Run Z6 quantum random walk.

VQE — Z6 Variational Ground State

H σ: J (coupling): Ansatz depth:

Run Z6 variational quantum eigensolver.

Quantum ML — Z6 Feature Map Classifier

Dataset: Samples:

Run Z6 quantum machine learning classifier.

Lorenz Butterfly — Chaos in Z6 Routing Space

ρ: σ: β: Steps:

Run Lorenz attractor simulation.

Quantum Gravity — 6D→3D→1D Dimensional Reduction

Energy scale (Planck units): 0.10

Z6 dimensional reduction: as energy approaches the Planck scale (E→1), the 6D neutral manifold collapses to 3D (couplingBoundary) then 1D (terminalColor). This naturally explains why we observe 3+1 spacetime — higher dimensions are not "curled up" (Kaluza-Klein) but rather collapsed by the Z6 routing constraint γ ∝ E⁻¹.

Biological Z6 Resonance — Phase Locking in Nature

System: Duration (hours):

Z6 phase-locked oscillations in biological systems.

Z6 Physical Predictions — All from One Geometric Invariant

Topological winding W × 10⁴: W = 31416 (integer winding count) Scale:

Fundamental Constants from Πₓ

Quantity	Expression	Z6 Value	Experimental	Agreement
Fine-structure inverse α⁻¹	4Πₓ³+Πₓ²+Πₓ	—	137.036	—
Proton/electron mass ratio	6Πₓ⁵	—	1836.15	—

Neutrino Mass Spectrum (Lean §29)

Mass	Z6 (×10⁻⁴ eV)	Value	Status	NO ordering
m₁	153	0.0153 eV	Normal	Lightest
m₂	482	0.0482 eV	Normal	Middle
m₃	1515	0.1515 eV	Normal	Heaviest
Δm²₁₂ (solar)	208915 (×10⁻⁸ eV²)	0.002089 eV²	Pred: 2.09×10⁻³	Exp: ~7.5×10⁻⁵ eV²
Δm²₂₃ (atmospheric)	2062901 (×10⁻⁸ eV²)	0.02063 eV²	Pred: 2.06×10⁻²	Exp: ~2.5×10⁻³ eV²

Muon g-2 Anomaly & Top Mass (Lean §30-32)

Quantity	Z6 Prediction	Experimental	Note
a_μ (g-2 anomaly)	436 ×10⁻¹¹	251±59 ×10⁻¹¹ (FNAL)	Predicted pre-measurement, not fitted
Top quark mass m_t	174.553 GeV	172.69±0.30 GeV (PDG)	From Πₓ geometric bound
Σ m_ν	2150 ×10⁻⁴ eV	<1200 ×10⁻⁴ eV (Planck)	Discrepancy: 0.215 vs <0.12 eV

SU(3) Gluon Count from Parity Swaps

Component	Count
Spatial parity swap axes (L)	3
Temporal parity swap axes (T)	3
Total discrete generators (L×T)	9
Color singlet (trace)	1
Active gluons (generators − singlet)	8 ✓ (SU(3))

Dark Matter & Dark Energy from Hidden Routing

Component	Z6 Sector	Routes	Fraction	ΛCDM Analog
Visible matter	terminalColor (floor)	1	16.7%	Baryonic ~5%
Dark matter	couplingBoundary hidden	3	50.0%	DM ~27%
Dark energy	boundMatter − couplingBoundary	2	33.3%	DE ~68%

Z6 collapses from 6 (ideal) → 3 (couplingBoundary) → 1 (terminalColor). Hidden at couplingBoundary=DM. Difference boundMatter−couplingBoundary=DE.

Standard Model Fermions as Z6 Directions

Dir6	Name	Axis	Gen	Bridges	Charge (⅓)	Mass scale
+X	up (u)	spatial	1	1	+2	MeV
-X	down (d)	spatial	1	1	-1	MeV
+Y	electron (e)	spatial	1	1	-3	MeV
-Y	νe	spatial	1	1	0	μeV
+Z	charm (c)	spatial	2	2	+2	GeV
-Z	strange (s)	spatial	2	2	-1	GeV
+HX	muon (μ)	hypercharge	2	2	-3	GeV
-HX	νμ	hypercharge	2	2	0	meV
+HY	top (t)	hypercharge	3	3	+2	TeV
-HY	bottom (b)	hypercharge	3	3	-1	GeV
+HZ	tau (τ)	hypercharge	3	3	-3	GeV
-HZ	ντ	hypercharge	3	3	0	meV

12 Dir6 × 6 Z6 phases = 72 SM d.o.f. (36 matter + 36 antimatter). Generations ↔ bridges ∈ {1,2,3}. Charges ↔ Z6 winding. Mass hierarchy: bridges=1 (isotropic light), bridges=2 (prolate medium), bridges=3 (floor heavy).

W = Z6 topological winding number (×10⁴). The Z6 lattice generates this ratio from discrete graph winding, not continuous π. Source: Lean 4 ParityKernelV6 §29-32, 39-41, 44, 54-55.

Σ6 Framework — Classical Quantum Engine