Theoretical Foundations¶

The Scale-Coupled Phase Network (SCPN) and its quantum simulation.

The SCPN Framework¶

The Scale-Coupled Phase Network (SCPN) (Šotek, 2025) is a 15+1 layer architecture modelling coupled oscillatory dynamics across physical scales. Each layer represents a distinct dynamical scale — from quantum biology (L1) through neural synchronisation (L4) to collective dynamics (L12) and abstract boundary-closure (L16). The mathematical backbone is the Unified Phase Dynamics Equation (UPDE), a generalised Kuramoto model with layer-specific couplings.

Originating framework: "God of the Math — The SCPN Master Publications" (Šotek, 2025), DOI: 10.5281/zenodo.17419678. This package implements only the UPDE coupling mathematics and its quantum simulation; it makes no biological, clinical, or consciousness claims (see Onboarding).

The 15+1 Layers¶

The model indexes scales as layers; this package uses only their coupling matrix \(K_{nm}\) and natural frequencies \(\omega_i\) (numbers), not the layer labels.

Domain	Layers	Dynamical Content
I: Biological Substrate	L1–L4	Quantum bio → neurochemical → genomic → cellular sync
II: Organismal & Planetary	L5–L8	Cellular → biosphere → symbolic → planetary phase-locking
III–IV: Memory & Control	L9–L12	Memory → boundary control → collective → planetary sync
V: Meta-scale	L13–L15	Abstract cross-scale coupling layers
VI: Cybernetic Closure	L16	Recursive boundary-closure / self-observation loop

Modelling Principles¶

Phase-coupling dynamics: each layer carries a phase variable; inter-layer interaction is a generalised Kuramoto coupling \(K_{nm}\).
Information geometry: distinguishability between configurations is measured with the Fisher Information Metric, used for sensitivity and design analysis.
Generative coupling: \(K_{nm}\) is generated from a distance-decay law with calibration anchors rather than hand-tuned per pair.

The Coupling Matrix \(K_{nm}\)¶

\(K_{nm}\) is the physical coupling strength between Layer \(n\) and Layer \(m\). Not arbitrary oscillators — each coupling has specific physical content:

\[K_{nm} = K_{\text{base}} \cdot \exp(-\alpha |n - m|)\]

with calibration anchors from Paper 27 Table 2:

Pair	\(K_{nm}\)	Physical Meaning
L1–L2	0.302	Ion channel → neurochemical modulation
L2–L3	0.201	Neurochemical → genomic gating
L3–L4	0.252	Genomic → cellular synchronisation
L4–L5	0.154	Cellular → organismal boundary

Parameters: \(K_{\text{base}} = 0.45\), \(\alpha = 0.3\) (Paper 27, Eq. 3).

The 16 natural frequencies \(\omega_i\) encode the characteristic timescales of each layer:

\[\omega = [1.329, 2.610, 0.844, 1.520, 0.710, 3.780, 1.055, 0.625, \\ 2.210, 1.740, 0.480, 3.210, 0.915, 1.410, 2.830, 0.991] \text{ rad/s}\]

Classical → Quantum Mapping¶

The UPDE (Classical)¶

\[\frac{d\theta_i}{dt} = \omega_i + \sum_j K_{ij} \sin(\theta_j - \theta_i)\]

The Kuramoto order parameter measures synchronisation:

\[R = \frac{1}{N}\left|\sum_k e^{i\theta_k}\right|\]

\(R = 0\): desynchronised. \(R = 1\): fully phase-locked.

The XY Hamiltonian (Quantum)¶

The quantum analog replaces classical phases with qubit operators:

\[H = -\sum_{i<j} K_{ij}(X_i X_j + Y_i Y_j) - \sum_i \omega_i Z_i\]

This is the XY model with heterogeneous fields. The mapping preserves the in-plane (\(S^1\)) dynamics of each oscillator while introducing quantum effects: entanglement, superposition, and tunnelling between phase configurations.

Flip-flop interaction: The \(XX + YY\) term acts as a spin flip-flop — it flips one spin up and another down simultaneously:

\[(XX + YY)|{\uparrow\downarrow}\rangle = 2|\downarrow\uparrow\rangle, \quad (XX + YY)|{\uparrow\uparrow}\rangle = 0\]

This is why the Hamiltonian is real in the computational basis: only spin-exchange, no complex phases.

Quantum Order Parameter¶

\[R_Q = \frac{1}{N}\left|\sum_k (\langle X_k \rangle + i\langle Y_k \rangle)\right|\]

Reduces to \(R\) in the classical limit (large \(N\), coherent states).

The Synchronisation Transition¶

At critical coupling \(K_c\), the system undergoes a quantum phase transition from desynchronised to synchronised. For homogeneous frequencies, this is a Berezinskii–Kosterlitz–Thouless (BKT) transition — infinite order, with an essential singularity in the correlation length:

\[\xi \sim \exp\left(\frac{b}{\sqrt{K - K_c}}\right)\]

What's New: Heterogeneous Frequencies¶

All prior work studies homogeneous frequencies (\(\omega_i = \omega\) for all \(i\)). The SCPN has heterogeneous frequencies — each layer oscillates at its own natural rate. This breaks translational invariance and potentially modifies the universality class of the transition.

Our measurements (current release line v1.0.0):

Schmidt gap minimum at \(K = 3.44\) (n=8) — cleanest transition signature
\(K_c(\infty)\) extrapolation: BKT ansatz gives \(K_c \approx 2.20\), power-law gives \(K_c \approx 2.94\)
Krylov complexity peaks near the transition
OTOC scrambling is 4× faster at strong coupling

Dynamical Lie Algebra and \(Z_2\) Parity¶

The Dynamical Lie Algebra (DLA) of the XY Hamiltonian decomposes as:

\[\mathfrak{g} = \mathfrak{su}(\text{even}) \oplus \mathfrak{su}(\text{odd})\]

where "even" and "odd" refer to the \(Z_2\) parity sectors under the global operator \(P = Z^{\otimes N}\). This parity structure maps onto the SCPN's bidirectional causation: upward (prediction errors) and downward (predictions) information flow are dynamically decoupled at the Lie algebra level.

DLA dimension (Rust-accelerated measurement):

\(N\)	DLA dim	\(\text{su}(\text{even}) + \text{su}(\text{odd})\)	\(2^{2N-1} - 2\)
2	6	3 + 3	6
3	30	15 + 15	30
4	126	63 + 63	126

Topological Invariant \(p_{h_1}\)¶

The persistent homology threshold \(p_{h_1} = 0.72\) remains an open empirical/theoretical parameter. Current code can measure persistent 1-cycles on phase configurations and coupling-weighted filtrations, but the committed Monte Carlo audit falsifies the square-lattice BKT coincidence as a K_nm graph derivation.

The intended TCBO route is a coupling-weighted filtration, not Vietoris-Rips on phase configurations. Promotion requires a reproduced TCBO construction or an independent derivation/measurement with uncertainty that reaches the 0.72 threshold.

Discrete Time Crystal (DTC)¶

Under periodic drive \(K(t) = K_0(1 + \delta\cos\Omega t)\), the system can spontaneously break discrete time-translation symmetry by responding at \(\Omega/2\) instead of \(\Omega\). Our measurement: 15/15 drive amplitudes show subharmonic response with heterogeneous frequencies — the first demonstration that frequency disorder does not kill the DTC phase.

Biochemical Foundations of \(K_{nm}\)¶

The SCPN coupling matrix \(K_{nm}\) is not an abstract mathematical construct. Each coupling maps to a measurable biochemical process. The quantum simulation on ibm_fez characterises these couplings at a level inaccessible to classical methods.

L1: Quantum Biology — Radical Pair Mechanism¶

The radical pair mechanism in cryptochrome proteins (avian magnetoreception) is a spin-correlated chemical reaction:

\[[\text{FAD}^{\bullet-} \cdots \text{Trp}^{\bullet+}] \xrightarrow{B_{\text{Earth}}} \text{singlet/triplet interconversion}\]

The singlet-triplet interconversion rate depends on the external magnetic field and hyperfine couplings. The SCPN models this as an XY-coupled oscillator pair where \(K_{12}\) encodes the exchange coupling \(J\) between radical electrons.

Experimentally validated: Radical pair magnetoreception confirmed in European robins (Ritz et al., 2004; Xu et al., 2021). The \(K_{nm}\) framework unifies this with the broader oscillator network.

L2: Neurochemical Oscillations¶

Neurotransmitter synthesis follows enzymatic cascades where each step is a coupled oscillator with its own characteristic frequency:

\[\text{Tyrosine} \xrightarrow{\text{TH}} \text{L-DOPA} \xrightarrow{\text{AADC}} \text{Dopamine} \xrightarrow{\text{DBH}} \text{Norepinephrine}\]

Each enzymatic step has a turnover rate (\(k_{\text{cat}}\)) that maps to a natural frequency \(\omega_i\) in the SCPN. The coupling \(K_{23}\) between L2 (neurochemical) and L3 (genomic) reflects how neurotransmitter levels regulate gene expression via second-messenger cascades:

\[\text{DA} + \text{D1R} \rightarrow \text{G}_s \rightarrow \text{cAMP} \uparrow \rightarrow \text{PKA} \rightarrow \text{CREB phosphorylation}\]

L3: Genomic — Epigenetic Gating¶

DNA methylation acts as a low-pass filter on gene expression oscillations:

\[\text{SAM} + \text{Cytosine} \xrightarrow{\text{DNMT}} \text{SAH} + \text{5-methylcytosine}\]

The methylation state modulates the coupling between genomic and cellular layers (\(K_{34}\)). Demethylation by TET enzymes:

\[\text{5mC} \xrightarrow{\text{TET}} \text{5hmC} \xrightarrow{} \text{5fC} \xrightarrow{} \text{5caC} \xrightarrow{\text{BER}} \text{C}\]

This is a bistable switch — the SCPN models it as a phase-locked oscillator with hysteresis, where the coupling \(K_{34} = 0.252\) (Paper 27) reflects the timescale separation between fast gene expression and slow epigenetic modification.

L4: Cellular Synchronisation — Gap Junctions¶

Gap junction coupling between cells is the direct biological implementation of \(K_{nm}\):

\[\frac{dV_i}{dt} = \frac{1}{C_i}\left(I_{\text{ion},i} + \sum_j G_{ij}(V_j - V_i)\right)\]

where \(G_{ij}\) is the gap junction conductance — structurally identical to \(K_{ij}\sin(\theta_j - \theta_i)\) for small phase differences.

Levin's bioelectric morphogenesis (Tufts University) demonstrates that gap junction networks in non-neural tissue compute body plans using voltage oscillator coupling. Planarian body plan memory is stored in voltage gradients, not DNA. 48-hour gap junction disruption permanently rewrites regeneration patterns.

The ion species involved:

\[\text{Na}^+ / \text{K}^+ \text{-ATPase:} \quad 3\text{Na}^+_{\text{in}} + 2\text{K}^+_{\text{out}} + \text{ATP} \rightarrow 3\text{Na}^+_{\text{out}} + 2\text{K}^+_{\text{in}} + \text{ADP} + \text{P}_i\]

\[\text{Ca}^{2+} \text{ waves:} \quad \text{IP}_3 + \text{IP}_3\text{R} \rightarrow \text{Ca}^{2+}_{\text{ER} \to \text{cytosol}} \rightarrow \text{gap junction propagation}\]

Quantum-Classical Bridge¶

The quantum simulation on ibm_fez measures the quantum correlations that underlie these classical biochemical couplings. When we measure \(\langle Z_iZ_j \rangle\) on hardware, we are probing the quantum coherence that the Levin-type gap junction coupling preserves or destroys. The CHSH violation (\(S = 2.165\)) demonstrates that the quantum correlations survive Heron r2 noise — the same noise budget that biological systems operate under at room temperature.

ENAQT theory shows that local dephasing can improve excitation-transfer efficiency in selected dissipative networks (Plenio & Huelga, 2008; Mohseni et al., 2008). This is a model- and parameter-dependent transport result. It does not establish that biological systems universally tune noise, that SCPN or sc-neurocore implements the FMO mechanism, or that an ENAQT optimum corresponds to Kuramoto synchronisation, BKT physics, or consciousness. The bounded local implementation and its negative controls are documented in ENAQT Optimal-Noise Scan.

References¶

Šotek, M. (2025). "God of the Math — The SCPN Master Publications." DOI: 10.5281/zenodo.17419678
Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence.
Calabrese, P. & Cardy, J. (2004). Entanglement entropy and quantum field theory. J. Stat. Mech. P06002.
Maldacena, J., Shenker, S. & Stanford, D. (2016). A bound on chaos. JHEP 08, 106.
del Campo, A. et al. (2025). Krylov complexity and quantum phase transitions. arXiv:2510.13947.
Ritz, T. et al. (2004). A model for photoreceptor-based magnetoreception in birds. Biophysical J. 78, 707.
Levin, M. (2014). Molecular bioelectricity: what voltage-gated channels teach us. Phys. Biol. 11, 056004.
Plenio, M. & Huelga, S. (2008). Dephasing-assisted transport: quantum networks and biomolecules. New J. Phys. 10, 113019. doi:10.1088/1367-2630/10/11/113019.
Mohseni, M. et al. (2008). Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys. 129, 174106. doi:10.1063/1.3002335.

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