Thickness Structure Hypothesis (TSH) – Unified Theoretical Framework

Theoretical formulation of the Thickness Structure Hypothesis ($p, \Delta f, \gamma_T$), a unified structural framework connecting quantum phenomena and gravitational localization.

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1. Unified Structural Principle

Quantum theory and gravity have long been described using fundamentally different assumptions: one probabilistic, one geometric. TSH proposes that both can be understood as different structural states of a single underlying principle defined by three minimal degrees of freedom:

• $p(x)$ — existence thickness: a scalar field with the structural property of "existence thickness."
  Both the state observed as quantum-like spreading and the state observed as gravitational localization are described on a unified basis as differences in the structural states taken by $p(x)$, $\Delta f$, and $\gamma_{T}$.

• $\Delta f$ — Internal degree of freedom in the "spreading direction" of the thickness structure.

• $\gamma_{T}$ — Internal degree of freedom in the "contracting direction" of the thickness structure.

These three quantities cannot be further reduced, cannot be replaced by any other physical quantity, and must carry physical content — making them the Unified Structural Principle.

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2. Unified Dynamical Equation

The motion of TSH is described by the following single covariant equation:

$$\frac{Du^{\mu}}{D\tau} = -\nabla^{\mu} \ln p + F^{\mu}(\Delta f, \gamma_{T})$$

This equation integrates three contributions:

• Left-hand side: The geometric covariant acceleration in general relativity ($\frac{Du^{\mu}}{D\tau}$)
• Middle term: The "spreading tendency" generated by the shape of the thickness profile $p(x)$ ($-\nabla^{\mu} \ln p$)
• Right-hand side: The structural force ($F^{\mu}$) arising from the competition between the expanding degree of freedom $\Delta f$ and the contracting degree of freedom $\gamma_{T}$

This is why it holds as an equation of motion:

Next-step trajectory
= quantum spreading tendency determined by the thickness profile $p(x)$
+ structural force $F^{\mu}$ generated by the competition between expansion $\Delta f$ and contraction $\gamma_{T}$

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3. Structural Phases and Continuous Transitions

The internal state $(p, \Delta f, \gamma_{T})$ is organized into three structural phases:

• Stable (quantum): quantum behavior
• Composite (classical): classical behavior
• Core (gravitational/measurement): gravitational / observational behavior

The system describes the following loop as a continuous relation:

Phase Diagram → Structural Force → Motion → Updated Variables → Phase Diagram

$$ (p, \Delta f, \gamma_{T})_{t} \implies F^{\mu} \implies u^{\mu}(t+\delta t) \implies (p, \Delta f, \gamma_{T})_{t+\delta t} $$

By iterating this loop, the three structural phases (Stable / Composite / Core) deform smoothly, and quantum-like, classical-like, and gravitational behaviors transition continuously — as structural states — within a single covariant dynamics.

In other words, TSH enables the three domains of quantum, classical, and gravitational behavior to be formulated directly from this single equation of motion alone.

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4. Interaction Slots

The structural action of TSH is defined by a minimal principle that depends solely on $p(x)$, $\Delta f$, and $\gamma_{T}$. Because of this, even when external interactions (gauge fields, matter fields, etc.) are added:

• The structural dynamics of TSH do not change
• The update relations for the three internal degrees of freedom do not change
• The phase diagram (Stable / Composite / Core) does not change

This means that the internal structure of TSH is completely independent of external interactions — and any external interaction can be integrated simply by appending it to the right-hand side of the tensor equation.

Integrations Made Possible

The TSH tensor equation provides a hierarchical set of interaction slots into which external interactions can be freely inserted:

• Standard Model (SM)
• GUTs (SO(10), etc.)
• Effective field theories from string theory
• General matter fields: fluid, Higgs, Yang–Mills, Dirac, etc.

Furthermore, because the slots have a parallel structure:

• Multiple matter fields can be stacked without contradiction
• Multiple gauge fields can be stacked without contradiction
• Weak, strong, and electromagnetic interactions can be placed side by side without contradiction
• Multiple instances of the same type of interaction can be accumulated without contradiction
• Different types of interactions can be added simultaneously without contradiction

In short, TSH means:

"Whether matter, gauge field, or force — singly or in combination — any mix can be integrated."

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5. Phase-Diagram Formulation

A major theoretical feature of TSH is that the $\Delta f\text{–}\gamma_{T}$ phase diagram unifies structural regimes without requiring separate fundamental laws.

In conventional physics models, separate equations and separate assumptions are required for:

• The quantum domain
• The classical domain
• The gravitational domain

In TSH, however:

• The phase diagram uniquely determines which phase the system is in
• The phase diagram directly provides the structural force relation
• The phase diagram directly provides the continuous transition across regimes

This yields a unified theoretical architecture that seamlessly bridges microscopic spreading and macroscopic localization.

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6. Executable Structural Model

The Ultimate TSH Simulator provides a fully runnable implementation of the structural dynamics. It computes:

• $\Delta f - \gamma_{T}$ phase deformation
• Mass‑dependent boundary scaling
• Irreversible phase transitions
• Evolving thickness distribution $p(x)$

This allows real‑time simulation of structural behavior across the three phases.

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7. Project Credits, Citation & Contact

This project is independently developed and maintained by the author.

Author: Hirokazu Abe (ab_ab, 2026)
Zenodo DOI (Concept DOI): https://doi.org/10.5281/zenodo.18492753
GitHub: https://github.com/ababphysics
X (Twitter): https://x.com/abab162535

Citation (BibTeX)

If you refer to the TSH theoretical framework in your research, please cite it as follows:

@ab_ab2026tsh,
  author       = {Abe, Hirokazu},
  title        = {Thickness Structure Hypothesis (TSH): Unified Structural Principle},
  year         = {2026},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.18492753},
  url          = {https://doi.org/10.5281/zenodo.18492753},
  note         = {Also known as ab\_ab}
}

For the full theoretical derivation, mathematical formulation, and proofs, please refer to the Zenodo DOI:
https://doi.org/10.5281/zenodo.18492753

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8. Copyright & License

All theoretical formulations, concepts, texts, and mathematical derivations described in this repository and associated papers are © 2026 Hirokazu Abe. All rights reserved.