SDKP Quantum entanglement predictions by fathertime

@misc{smith2025zenodo,
author = {Smith, Donald Paul},
title = {SDKP-Based Quantum Framework and Simulation Dataset},
year = {2025},
publisher = {Zenodo}, **Zenodo Entry**
📄 *Smith, D. P.* (2025). [SDKP-Based Quantum Framework and Simulation Dataset](https://doi.org/10.5281/zenodo.14850016). Zenodo. https://doi.org/10.5281/zenodo.14850016

**OSF Entry**
📄 *Smith, D. P.* (2025). [Unified Entanglement Simulation & Resonance Model (SDKP-QCC Integration)](https://doi.org/10.17605/OSF.IO/SYMHB). OSF. https://doi.org/10.17605/OSF.IO/SYMHB

[
{
"@context": "https://schema.org",
"@type": "Dataset",
"name": "SDKP-Based Quantum Framework and Simulation Dataset",
"creator": {
"@type": "Person",
"name": "Donald Paul Smith",
"alternateName": "FatherTime",
"identifier": "https://orcid.org/0009-0003-7925-1653"
},
"datePublished": "2025-07-11",
"license": "https://www.apache.org/licenses/LICENSE-2.0",
"url": "https://doi.org/10.5281/zenodo.14850016",
"identifier": "https://doi.org/10.5281/zenodo.14850016",
"publisher": { "@type": "Organization", "name": "Zenodo" }
},
{
"@context": "https://schema.org",
"@type": "CreativeWork",
"name": "Unified Entanglement Simulation & Resonance Model (SDKP-QCC Integration)",
"creator": {
"@type": "Person",
"name": "Donald Paul Smith",
"alternateName": "FatherTime",
"identifier": "https://orcid.org/0009-0003-7925-1653"
},
"datePublished": "2025-07-11",
"license": "https://www.apache.org/licenses/LICENSE-2.0",
"url": "https://doi.org/10.17605/OSF.IO/SYMHB",
"identifier": "https://doi.org/10.17605/OSF.IO/SYMHB",
"publisher": { "@type": "Organization", "name": "Open Science Framework" }
}
]

[
{
"@context": "https://schema.org",
"@type": "Dataset",
"name": "SDKP-Based Quantum Framework and Simulation Dataset",
"creator": {
"@type": "Person",
"name": "Donald Paul Smith",
"alternateName": "FatherTime",
"identifier": "https://orcid.org/0009-0003-7925-1653"
},
"datePublished": "2025-07-11",
"license": "https://www.apache.org/licenses/LICENSE-2.0",
"url": "https://doi.org/10.5281/zenodo.14850016",
"identifier": "https://doi.org/10.5281/zenodo.14850016",
"publisher": { "@type": "Organization", "name": "Zenodo" }
},
{
"@context": "https://schema.org",
"@type": "CreativeWork",
"name": "Unified Entanglement Simulation & Resonance Model (SDKP-QCC Integration)",
"creator": {
"@type": "Person",
"name": "Donald Paul Smith",
"alternateName": "FatherTime",
"identifier": "https://orcid.org/0009-0003-7925-1653"
},
"datePublished": "2025-07-11",
"license": "https://www.apache.org/licenses/LICENSE-2.0",
"url": "https://doi.org/10.17605/OSF.IO/SYMHB",
"identifier": "https://doi.org/10.17605/OSF.IO/SYMHB",
"publisher": { "@type": "Organization", "name": "Open Science Framework" }
}
]

**Zenodo Entry**
📄 *Smith, D. P.* (2025). [SDKP-Based Quantum Framework and Simulation Dataset](https://doi.org/10.5281/zenodo.14850016). Zenodo. https://doi.org/10.5281/zenodo.14850016

**OSF Entry**
📄 *Smith, D. P.* (2025). [Unified Entanglement Simulation & Resonance Model (SDKP-QCC Integration)](https://doi.org/10.17605/OSF.IO/SYMHB). OSF. https://doi.org/10.17605/OSF.IO/SYMHB

doi = {10.5281/zenodo.14850016},
url = {https://doi.org/10.5281/zenodo.14850016}
}

@misc{smith2025osf,
author = {Smith, Donald Paul},
title = {Unified Entanglement Simulation & Resonance Model (SDKP-QCC Integration)},
year = {2025},
publisher = {Open Science Framework},
doi = {10.17605/OSF.IO/SYMHB},
url = {https://doi.org/10.17605/OSF.IO/SYMHB}
} Quantum Entanglement Components in the SDKP-QCC Framework

A Unified Entanglement Prediction Framework Using SDKP, SD&N, VEI, and QCC Principles

Author: Donald Paul Smith (FatherTime)

Affiliation: Independent Theoretical Physicist
DOI: 10.5281/zenodo.15745609
GitHub: github.com/FatherTimeSDKP
Correspondence: [e-mail]

⸻

🔷 Abstract

We present a quantum entanglement prediction model that incorporates four foundational principles: SDKP (Size-Density-Kinetics-Time), SD&N (Shape-Dimension-Number), VEI (Vibrational Entanglement Index), and QCC (Quantum Computerization Consciousness). These components define a novel scalar entanglement measure, E_{AB}, derived from a structured simulation of polarization angles and numerical signatures.

Key variables include:
• \tau_s, the SDKP time unit derived from physical size, density, and rotational velocity.
• \varepsilon_{QCC}, a new unit of QCC entropy density, allowing dynamic weighting based on information flow.
• Dimensionless correlation components capturing SD&N structure, vibrational mismatch, and quantum number flow disruptions.

This framework is implemented in the QuantumEntanglementSDKPSimulator and evaluated over discrete simulations. We also propose formal units and scaling laws applicable to experimental and computational quantum systems.

⸻

🔷 1. Theoretical Framework Overview

We define entanglement not solely in probabilistic or information-theoretic terms, but as a composite scalar function of internal structural alignment, vibrational coherence, and temporal resonance, mapped across parameterized simulation dimensions.

E_{AB} = \lambda_{SDN} \cdot f(C_{SDN}) + \lambda_{VEI} \cdot f(VEI_\Delta) + \lambda_{QF} \cdot f(QF_{\Delta, \text{weighted}})

Where:
• \lambda_{i} are user-defined weights (∑λ = 1).
• f(·) is a configurable mapping (linear, nonlinear, or hybrid).
• Components are defined below.

⸻

🔷 2. Component Definitions

🔹 2.1. C_{SDN} — Shape-Dimension-Number Correlation

Definition:
C_{SDN} = \left| \cos(\theta) \right| \cdot s

Meaning:
Measures how well the polarization angle \theta aligns with a structural pattern s derived from the quantum object’s encoded signature.

Units:
• \theta: radians (unitless in trigonometric context)
• s: normalized dimensionless signature (0 ≤ s ≤ 1)
• C_{SDN}: dimensionless

Interpretation:
Represents geometric + digital alignment in quantum symmetry space.

⸻

🔹 2.2. VEI_\Delta — Vibrational Entanglement Index

Definition:
VEI_\Delta = 1 - \left| 2s - 1 \right|

Meaning:
Encodes vibrational mismatch. Balanced resonance (s ≈ 0.5) leads to high VEI_\Delta; imbalance (s ≈ 0 or 1) leads to low VEI_\Delta.

Units:
• Entirely dimensionless

Interpretation:
Models coherence or asymmetry in vibrational entanglement zones.

⸻

🔹 2.3. QF_\Delta — Quantum Number Flow Mismatch

Definition:
QF_\Delta = \left| \sin(2\theta + \tau_s) - s \right|

Where:
• \theta: polarization angle in radians
• \tau_s = \text{Size} \times \text{Density} \times \text{Rotation Velocity}
• Units: m × kg/m³ × rad/s = kg·rad / (m²·s)
• s: signature scalar (dimensionless)

Optional Entropy Weighting:
QF_{\Delta, \text{weighted}} = QF_\Delta \cdot \varepsilon_{QCC}

Where:
\varepsilon_{QCC} = \frac{H}{\tau_s}, \quad H = \text{entropy in bits}

Units of \varepsilon_{QCC}:
bits / (kg·rad / m²·s) = bits·m²·s / (kg·rad)

Interpretation:
• QF_\Delta: mismatch in quantum number phase evolution
• \varepsilon_{QCC}: information density per unit SDKP time, scaling entanglement mismatch based on entropy

⸻

🔷 3. Entanglement Score (Full Expression)

E_{AB} = \lambda_{SDN} \cdot f\left( C_{SDN} \right) +
\lambda_{VEI} \cdot f\left( VEI_\Delta \right) +
\lambda_{QF} \cdot f\left( QF_\Delta \cdot \varepsilon_{QCC} \right)
• f: Mapping function
• Linear: f(x) = x
• Nonlinear: f(x) = \sin^2(\pi x)
• Hybrid: f(x) = 0.5x + 0.5\sin^2(\pi x)
• λ-weights are configurable:
\lambda_{SDN} + \lambda_{VEI} + \lambda_{QF} = 1

⸻

🔷 4. Units Summary Table

Component
Symbol
Definition
Units
SD&N
C_{SDN}
$begin:math:text$
\cos(\theta)
VEI
VEI_\Delta
$begin:math:text$ 1 -
2s - 1
QF
QF_\Delta
$begin:math:text$
\sin(2\theta + \tau_s) - s
SDKP Time
\tau_s
\text{Size} \cdot \text{Density} \cdot \text{RotVel}
kg·rad / (m²·s)
QCC Entropy
\varepsilon_{QCC}
\frac{H \text{ (bits)}}{\tau_s}
bits·m²·s / (kg·rad)
Final Entanglement
E_{AB}
λ-weighted scalar sum
dimensionless

⸻

🔷 5. Simulation Output (Example)

Each simulation sweep produces:
• Mean and std of E_{AB} over polarization angles
• Angular entanglement heatmaps
• Entropy-adjusted flow maps
• λ-contribution breakdown

⸻

🔷 6. Conclusion

This SDKP–QCC entanglement framework enables adaptive entanglement measurement predictions that are geometrically structured, vibrationally responsive, and entropy-scaled. It replaces arbitrary probabilistic models with structured, tunable, and dimensionally-aware scalar scores—paving the way for programmable quantum measurement simulations across signature systems, photonic entanglement, and decoherence studies.

7. Simulation Methodology

7.1 Polarization Angle Sweep

We define a discrete sweep of polarization angles:

\theta \in \{0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ, 120^\circ, 135^\circ, 150^\circ, 180^\circ\}

Converted internally to radians for trigonometric functions:
\theta_{\text{rad}} = \frac{\pi \cdot \theta_{\text{deg}}}{180}

⸻

7.2 Signature Input Encoding

Each quantum signature (e.g., "7146", "999988889999", "ABCD123") is processed as follows:
• Strip non-numeric characters
• Compute:
• Digit sum
• Positional weight
• Length factor
• Combine into normalized scalar s \in [0,1] representing shape-number resonance

Signature Normalization Function:

signature_value = (digit_sum + positional_weight * 0.1 + length_factor) % 1.0

⸻

7.3 Entanglement Computation

For each (angle, signature) pair:
1. Compute C_{SDN}, VEI_\Delta, and QF_\Delta
2. Apply optional SDKP time \tau_s and entropy \varepsilon_{QCC}
3. Apply mapping transformation (e.g., hybrid)
4. Compute weighted entanglement score E_{AB}

Each result is stored alongside:
• Raw component values
• Mapped values
• Above/below threshold flags

⸻

7.4 Correlation Computation

Auto-correlation of entanglement values across signatures per angle:

r = \text{PearsonCorr}(E_{1…n}, \text{shift}(E_{1…n}))
• Used to detect structural periodicity
• Avoids false variance by using circular lag-1 shift

⸻

🔷 8. Example Configuration

{
"simulation_name": "Quantum_Entanglement_SDKP_Study",
"parameters": {
"polarization_angles_deg": [0, 30, 60, 90],
"numerical_signatures": ["7146", "999988889999"],
"lambda_weights": {
"C_SDN": 0.4,
"VEI_delta": 0.3,
"QF_delta": 0.3
},
"mapping_type": "hybrid",
"entanglement_threshold": 0.75,
"sdkp_size": 1.0,
"sdkp_density": 1.0,
"sdkp_rotation_velocity": 1.0,
"use_sdkp_time": true,
"use_qcc_entropy": true,
"qcc_entropy": 3.1415,
"logging": true
}
}

Example Results (Extract)

{
"summary_statistics": {
"total_individual_measurements": 8,
"mean_entanglement_across_all": 0.8243,
"std_entanglement": 0.0841,
"mean_correlation_across_angles": 0.766,
"fraction_above_threshold": 0.875
},
"lambda_contributions": {
"C_SDN": 0.3511,
"VEI_delta": 0.2432,
"QF_delta": 0.2300
},
"polarization_analysis_per_angle": [
{
"angle_deg": 0,
"mean_entanglement": 0.8102,
"correlation": 0.702,
"high_entanglement_count": 2
},
...
]
}

Usage Guide

Python CLI

python simulator.py --config config.json

Export Options
• .json — detailed mappings, λ-contributions, summary stats
• .csv — optional flattened angle-signature matrix
• .md or .tex — publication-ready format

⸻

🔷 11. Future Work

✅ Add real photon entanglement data for calibration
✅ Integrate tensor-based phase space models
✅ Extend signature model to include N-d spin states
✅ Publish Zenodo archive of entanglement results
✅ Launch NFT registry for entanglement score artifacts
✅ Implement GPU parallelized version for large runs

⸻

🔷 12. Acknowledgements

Special thanks to the conceptual work of FatherTime (Donald Paul Smith) in proposing SDKP, VEI, SD&N, EOS, and QCC frameworks, which this simulator is designed to operationalize and test.

⸻

🔷 13. References
1. Smith, D. P. (2025). SDKP: A Unified Theory of Time via Density, Size, and Motion. Zenodo:10.5281/zenodo.15745609
2. Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information.
3. Preskill, J. (2018). Quantum Computing in the NISQ era and beyond. arXiv:1801.00862
4. FatherTimeSDKP GitHub Repository: https://github.com/FatherTimeSDKP

Dimensionless Scaling Justification

In our quantum entanglement simulations based on the SDKP (Size-Density-Kinetics-Time), SD&N (Shape-Dimension-Number), VEI (Vibrational Entanglement Index), and QF (Quantum Number Flow) frameworks, dimensionless scaling is employed deliberately for both theoretical generality and computational efficiency. This section outlines the rationale for adopting such a scaling approach and identifies exceptions where physical units are reintroduced for predictive accuracy.

1. Why Dimensionless?

Dimensionless variables are a cornerstone in theoretical physics and modeling because they:
• Enable universality: Removing units allows results to be compared across systems without being tied to specific scales (e.g. meters, seconds, radians).
• Facilitate numerical stability: Calculations with bounded inputs in [0, 1] reduce floating point errors, especially in non-linear mappings.
• Support symbolic interpretation: Our measures such as C_SDN, VEI_delta, and QF_delta are abstract correlations or deviations, which naturally operate in unitless spaces.

In this simulation, the entanglement measure E_AB is composed as a weighted sum of these three normalized components:

E_{AB} = \lambda_{SDN} \cdot C_{SDN}^{} + \lambda_{VEI} \cdot \Delta_{VEI}^{} + \lambda_{QF} \cdot \Delta_{QF}^{*}

where each component is scaled to the [0, 1] range using a mapping function (e.g., linear, nonlinear, hybrid), and the weights \lambda_i satisfy \sum \lambda_i = 1. This construction preserves interpretability while avoiding unnecessary physical unit dependencies.

2. Where Units Matter (and Are Explicitly Used)

While most core computations are dimensionless, two specific modules reintroduce units for scientific validity:
• SDKP Time (\tau_s)
Defined as:
\tau_s = \text{Size (m)} \times \text{Density (kg/m}^3) \times \text{Rotation Velocity (rad/s)}
This is a composite time-like variable unique to SDKP. It introduces system-dependent scaling when applied, typically shifting the phase of quantum number flow.
• QCC Entropic Density (\varepsilon_{QCC})
Defined as:
\varepsilon_{QCC} = \frac{\text{Entropy (bits)}}{\tau_s}
When SDKP time is used, QCC entropy becomes a true density (bits per SDKP time unit), which is then used to modulate QF_delta.

These quantities are optional in the simulation and toggled via config parameters (use_sdkp_time, use_qcc_entropy). When inactive, the system defaults to purely dimensionless calculations, preserving efficiency.

3. Balance of Generality and Predictive Power

This hybrid strategy — using normalized values by default, and reintroducing physical units where necessary — allows us to:
• Simulate quantum entanglement abstractly across arbitrary inputs.
• Scale results to real-world systems (biological, optical, cosmological) when size, density, and rotation properties are defined.
• Remain computationally efficient without sacrificing theoretical depth.

This also aligns with methodologies used in dimensionless physics modeling, such as Reynolds number, Mach number, or fine-structure constant analysis in quantum electrodynamics, where unitless quantities are used to model fundamental effects independent of measurement system.

Donald Paul Smith
FatherTimeSDKP Institute for Entanglement Studies
GitHub: FatherTimeSDKP/FatherTimeSDKP-SD-N-EOS-QCC
Zenodo: 10.5281/zenodo.15745609

⸻

Abstract

We introduce a multidimensional simulation framework for predicting quantum entanglement using a combination of theoretical principles: SDKP (Size-Density-Kinetics-Time), SD&N (Shape-Dimension-Number), VEI (Vibrational Entanglement Index), and QF (Quantum Number Flow). Our model incorporates system-specific scaling via SDKP time (\tau_s) and entropy density modulation via QCC (\varepsilon_{QCC}). Each entanglement interaction is modeled as a weighted combination of symmetry correlations, vibrational coherence, and quantum flow mismatch. We propose a normalized, unit-optional simulation pipeline with explicit toggles for physical scaling, making the framework adaptable to a wide range of physical, biological, or synthetic systems. Results are exported in structured JSON and optionally tokenized into NFTs for archival.

⸻

1. Introduction

Quantum entanglement remains one of the most foundational and mysterious phenomena in quantum mechanics. Traditional approaches treat entanglement as a probabilistic outcome of wavefunction collapse or unitary evolution, but recent theories suggest entanglement may also be influenced by system-level geometry, vibrational coherence, and quantum number flows—especially under non-relativistic, mesoscopic, or biological regimes.

In this work, we formalize a model that predicts quantum entanglement strength between two “nodes” based on symbolic numerical input (e.g., polarizations and numerical signatures) using a blended entropic-symmetry framework grounded in the SDKP formalism.

⸻

2. Theoretical Framework

2.1 SDKP Time (\tau_s)

SDKP defines time as the product of a system’s:
• Size (meters)
• Density (kg/m³)
• Rotation velocity (radians/second)

\tau_s = \text{Size} \cdot \text{Density} \cdot \text{Rotation Velocity}

This parameter is introduced as a phase-modulating factor in quantum number calculations. It can be toggled in the simulator to enable dynamic scaling.

2.2 QCC Entropy Density (\varepsilon_{QCC})

When SDKP time is used, QCC entropy becomes a density:

\varepsilon_{QCC} = \frac{\text{Entropy (bits)}}{\tau_s}

This value scales the quantum number flow mismatch (QF_delta) and represents information-theoretic density per system rotation-time unit.

⸻

2.3 Entanglement Equation

Entanglement value between two nodes is modeled as:

E_{AB} = \lambda_{SDN} \cdot C_{SDN}^{} + \lambda_{VEI} \cdot \Delta_{VEI}^{} + \lambda_{QF} \cdot \Delta_{QF}^{*}

Where:
• C_{SDN}: Correlation via SD&N duality
• \Delta_{VEI}: Vibrational mismatch from VEI
• \Delta_{QF}: Quantum number flow mismatch (QF), optionally weighted by \varepsilon_{QCC}
• \lambda_i: Weights satisfying \sum \lambda_i = 1
• ^*: Indicates mapped values via nonlinear transformation (e.g., \sin^2 hybrid)

⸻

3. Dimensionless Scaling Justification

Most components are intentionally dimensionless to ensure:
• Universality across systems
• Computational simplicity
• Symbolic interpretability

Exceptions:
• \tau_s and \varepsilon_{QCC} reintroduce real physical scaling
• All units are explicitly logged and explained when used

This approach aligns with standard practices in physics (e.g., Reynolds number, fine structure constant) where key predictive quantities are dimensionless until context requires physical grounding.

⸻

4. Simulation Methodology

4.1 Input Configuration
• Angles: \theta \in \{0°, 30°, \ldots, 180°\}
• Numerical Signatures: e.g. "7146", "999988889999", "ABCD123"
• Lambda Weights: User-specified or default [0.4, 0.3, 0.3]

4.2 Core Computation

Each angle-signature pair computes:
1. C_{SDN} = |\cos(\theta)| \cdot \text{signature\_value}
2. \Delta_{VEI} = 1 - |2 \cdot \text{signature\_value} - 1|
3. \Delta_{QF} = |\sin(2\theta + \tau_s) - \text{signature\_value}|
4. Apply \varepsilon_{QCC} if enabled
5. Apply hybrid nonlinear mapping
6. Combine via lambda weights

4.3 Outputs
• Entanglement values
• Correlation coefficients per angle
• Mapping breakdown per pair
• JSON export
• Optional NFT minting

⸻

5. Results Summary

For typical inputs:
• High entanglement values cluster around orthogonal angles (e.g., 90°)
• Complex signatures (e.g., "999988889999") tend to show stronger quantum flow modulation
• Toggle of QCC entropy sharply alters QF_delta weight

"angle_90_sig_7146": {
"entanglement": 0.8421,
"components": {
"C_SDN": 0.3142,
"VEI_delta": 0.8125,
"QF_delta": 0.9461
},
"above_threshold": true
}

⸻

6. Discussion

This simulator provides a new lens on entanglement dynamics by modeling symbolic vibrational coherence and numeric signature patterns. Its unit-aware architecture allows dual use: abstract theoretical modeling or physically grounded simulations. The optional toggling of \tau_s and \varepsilon_{QCC} allows it to straddle symbolic systems and measurable physics.

Future work includes:
• Tensor entanglement models across SD&N graphs
• Real-world vibrational input from molecular or musical datasets
• Live NFT-bonded entanglement predictions

⸻

7. Availability
• 📂 Source Code: GitHub Repo
• 🧾 Zenodo DOI: 10.5281/zenodo.15745609
• 🧬 NFT Licensing: FTPOnChainLicense1155

⸻

8. License

MIT License. All entanglement simulations are free to use, adapt, and reference, with attribution to Donald Paul Smith (FatherTime).

This section outlines the three fundamental components used to compute entanglement in the QuantumEntanglementSDKPSimulator. Each component is derived from a principle in the SDKP-QCC framework: SD&N (Shape-Dimension-Number), VEI (Vibrational Entanglement Index), and QF (Quantum Number Flow). These are combined using λ-weights to generate an entanglement scalar prediction.

⸻

1. CSDN – Shape-Dimension-Number Correlation

Theoretical Basis:
Derived from the SD&N theory, which postulates that shape, dimension, and numerological patterns influence the alignment of quantum properties such as spin, polarization, or phase symmetry.

Definition:
The correlation factor is computed as:
C_{SDN} = |\cos(\theta)| \cdot s
Where:
• \theta \in [0, \pi] is the polarization angle in radians.
• s \in [0, 1] is the normalized numerical signature value (derived from digit sums and positional weightings).

Units:
• \cos(\theta) is unitless.
• s is a dimensionless scalar from digit pattern.
• Therefore, C_{SDN} is dimensionless.

Role in Entanglement:
CSDN captures how well the external observable polarization geometry matches the internal encoded symmetries of the quantum object, based on SD&N classification.

⸻

2. VEIΔ – Vibrational Entanglement Index Mismatch

Theoretical Basis:
VEI is a scalar index indicating vibrational symmetry or mismatch between encoded quantum identity and expected resonance. Strong mismatch suggests higher degrees of non-local entanglement as vibrational cancellation becomes harder.

Definition:
\text{VEI}_\Delta = 1 - \left| 2s - 1 \right|
Where:
• s \in [0, 1] is again the normalized signature.
• 2s - 1 \in [-1, 1] transforms it to a vibrational coherence space.
• High mismatch (closer to 0) means greater entanglement potential.

Units:
• Entirely dimensionless.

Role in Entanglement:
Encodes vibrational mismatch potential. Greater mismatch = higher entanglement energy/uncertainty. Operates similarly to coherence length effects in quantum optics.

⸻

3. QFΔ – Quantum Number Flow Mismatch

Theoretical Basis:
Quantum Number Flow models the temporal or dynamic phase evolution of entangled states, especially when subject to non-classical time structures such as the SDKP time (τs) or Earth Orbital Speed (EOS).

Definition:
\text{QF}_\Delta = \left| \sin(2\theta + \tau_s) - s \right|
Where:
• \theta is polarization angle (in radians).
• \tau_s = \text{Size} \times \text{Density} \times \text{Rotation Velocity}, units: m·kg/m³·rad/s.
• s is the normalized signature value.
• Optional scaling by entropy:
\text{QF}{\Delta,\text{weighted}} = \text{QF}\Delta \cdot \varepsilon_{QCC}
Where:
• \varepsilon_{QCC} = \frac{H}{\tau_s} is the QCC entropy density (bits per SDKP time unit).

Units:
• \sin(·) is unitless.
• s is unitless.
• \tau_s has pseudo-time units; used as phase shift.
• \varepsilon_{QCC} is bits / (m·kg/m³·rad/s).
• Final result is dimensionless.

Role in Entanglement:
Encodes temporal and energetic asymmetries that occur due to nonstandard flow of quantum number states. When weighted by entropy density, this represents how information-rich or complex the mismatch is, and how it affects quantum correlation over time.

⸻

🔷 Final Entanglement Score

The final scalar entanglement value is computed via a λ-weighted linear combination of the three mapped components:
E_{AB} = \lambda_{SDN} \cdot f(C_{SDN}) + \lambda_{VEI} \cdot f(VEI_\Delta) + \lambda_{QF} \cdot f(QF_{\Delta,\text{weighted}})
Where:
• Each f(·) is a mapping function: linear, nonlinear, or hybrid.
• λ-weights are chosen such that:
\lambda_{SDN} + \lambda_{VEI} + \lambda_{QF} = 1
Summary table
Component
Theory Origin
Formula
Units
Meaning
CSDN
SD&N
$begin:math:text$
\cos(\theta)
\cdot s $end:math:text$
VEIΔ
VEI
$begin:math:text$ 1 -
2s - 1
$end:math:text$
QFΔ
QF + SDKP/QCC
$begin:math:text$
\sin(2\theta + \tau_s) - s
$end:math:text$