SPDX-License-Identifier: AGPL-3.0-or-later¶

Commercial license available¶

© Concepts 1996–2026 Miroslav Šotek. All rights reserved.¶

© Code 2020–2026 Miroslav Šotek. All rights reserved.¶

ORCID: 0009-0009-3560-0851¶

Contact: www.anulum.li | protoscience@anulum.li¶

scpn-quantum-control — Theoretical Foundations¶

Theoretical Foundations¶

The Self-Consistent Phase Network (SCPN) and its quantum simulation.

The SCPN Framework¶

The Sentient-Consciousness Projection Network (Šotek, 2025) is a 15+1 layer architecture modelling coupled oscillatory dynamics across physical scales. Each layer represents a distinct ontological domain — from quantum biology (L1) through neural synchronisation (L4) to collective dynamics (L12) and meta-universal closure (L16). The mathematical backbone is the Unified Phase Dynamics Equation (UPDE), a generalised Kuramoto model with layer-specific couplings.

Source: "God of the Math — The SCPN Master Publications" (Šotek, 2025), DOI: 10.5281/zenodo.17419678

The 15+1 Layers¶

Domain	Layers	Physical Content
I: Biological Substrate	L1–L4	Quantum bio → neurochemical → genomic → cellular sync
II: Organismal & Planetary	L5–L8	Self → biosphere → symbolic → cosmic phase-locking
III–IV: Memory & Control	L9–L12	Memory → boundary control → noosphere → Gaian sync
V: Meta-Universal	L13–L15	Source-field → transdimensional → Consilium
VI: Cybernetic Closure	L16	The Anulum — recursive self-observation loop

Three Axioms¶

Primacy of Consciousness (Ψ): Consciousness is the primary, irreducible ground of being — not emergent from matter.
Language of Information Geometry: The native language is geometric. Meaning is encoded in the geometry of informational spaces (Fisher Information Metric).
Teleological Optimisation: The universe is guided by an inherent drive to maximise future possibilities (Causal Entropic Forces).

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 ontological 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 (v0.9.5):

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 \(p_{h_1} = 0.72\) (measured on the TCBO's coupling-weighted simplicial complex) quantifies how much of the SCPN's layer-coupling topology creates persistent 1-cycles — information loops that sustain coherent circuits through the hierarchical structure.

This is computed on the coupling-weighted filtration (not Vietoris–Rips on phase configurations), where the SCPN's specific sparse hierarchical structure creates topological features that dense random graphs cannot.

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.

The NAQT (noise-assisted quantum transport) mechanism, validated in photosynthetic complexes (Plenio & Huelga, 2008; Mohseni et al., 2008), shows that biology tunes noise to maximise quantum transport — from \(\sim 70\%\) to \(\sim 99\%\) efficiency. The SCPN's stochastic computing architecture (sc-neurocore) is the computational implementation of this biological principle.

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 and classical. New J. Phys. 10, 113019.
Mohseni, M. et al. (2008). Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys. 129, 174106.

Developed by ANULUM / Fortis Studio