8. Mass Function in SDKP: Shape, Number, Scale, and Density

SDKP posits that particle mass arises as a composite function of fundamental shape, dimension, and number parameters combined with local scale and density fields.

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8.1 Definitions

• Number ( N ): Integer count of fundamental entities (e.g., number of knots or topological components).
• Shape ( S ): Vector or tensor representing geometric/topological configuration (e.g., knot type, trefoil, unknotted loop).
• Scale ( s ): Local granular scale.
• Density ( \rho ): Local information/spatial density.

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8.2 General Mass Function

The mass ( m ) is given by a scale-density modulated function:

[ m = f(N, S) \cdot s^\alpha \cdot \rho^\beta ]

• ( f(N, S) ): Base mass function depending on discrete shape/number properties.
• ( \alpha, \beta ): Scaling exponents derived empirically or via fitting to known particle masses.

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8.3 Example: Electron and Proton Mass

• Electron:
  [ m_e = f(1, S_e) \cdot s^{\alpha_e} \cdot \rho^{\beta_e} ]

• Proton:
  [ m_p = f(3, S_p) \cdot s^{\alpha_p} \cdot \rho^{\beta_p} ]

where ( f(1, S_e) ) and ( f(3, S_p) ) reflect the knotting/topological number encoding each particle’s internal structure.

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8.4 Physical Interpretation

• ( s^{\alpha} ) modulates mass by the local granular scale (smaller scale → higher mass density).
• ( \rho^{\beta} ) modulates mass by density of underlying informational or spatial content.
• ( f(N, S) ) encodes the discrete quantum topological information intrinsic to the particle.

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9. Extensions: Quark and Composite Particle Modeling

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9.1 Incorporating Quark Masses and Confinement

• Quarks are modeled as composite configurations with specific ( N ), ( S ) combinations, embedded in local scale-density fields.
• Confinement arises from strong coupling potentials in ( V(\rho, s, \phi) ) tuned for quark interactions.
• Mass hierarchy and flavor differences emerge from variations in ( f(N, S) ) and local ( s, \rho ) environments.

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9.2 Higher-Dimensional Shape Encoding

• Advanced shape encodings use knot polynomials, homology classes, or tensorial invariants beyond simple trefoil/un-knot.
• These encode chirality, handedness, and other particle properties in ( S ).

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9.3 Composite Mass Calculation

• Composite particle mass is a nonlinear superposition of constituent mass functions, including interaction energy corrections from the potential ( V ).
• Dynamical scaling exponents (\alpha, \beta) may depend on particle environment and energy scale.

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10. Summary and Next Steps

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10.1 Summary

• SDKP provides a unified field-theoretic framework modeling scale, density, kinematics, and phase fields coupled via a Lagrangian density.
• The mass function integrates discrete quantum shape/number data with continuous scale-density modulation.
• Phase dynamics and compressive kinematic coupling underpin the emergence of granular time flow (Chronon Wake).
• Extensions to composite particles and quarks are natural within this framework using topological and shape encoding.

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10.2 Next Steps for Implementation

• Analytic Solutions: Explore special cases of the Euler-Lagrange equations, e.g., stationary states, soliton-like phase structures.
• Numerical Simulations: Develop PDE solvers to simulate coupled field dynamics, phase locking, and chronon wake generation.
• Mass Function Calibration: Fit scaling exponents (\alpha, \beta) and base function ( f(N,S) ) to known particle masses.
• Topology Encoding: Implement algorithms to represent shape ( S ) as computational knot invariants or tensor fields.
• Physical Testing: Propose experimental tests or simulation validations linking SDKP predictions with observed particle properties or granular time effects.

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