Thickness Structure Hypothesis (TSH)

Unified Structural Principle + Executable Structural Engine

Author: Hirokazu Abe (2026)
Zenodo DOI: https://zenodo.org/records/20078368
(For detailed academic derivations, complete mathematical formulations, and theoretical proofs, please refer to the full manuscript available via the Zenodo DOI above.)

Thickness Structure Hypothesis (TSH) is a structural model for unified physics, describing quantum, classical, and gravitational behavior within a single Quantum Gravity–class framework based on three internal variables ($p, \Delta f, \gamma_{T}$).

This repository contains the executable Structural Engine and conceptual overview. The full mathematical details are available in the Zenodo manuscript.

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1. Overview — Minimal 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 intensity: 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 minimal structural principle.

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2. Unified Structural Dynamics (Conceptual Summary)

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 phase: quantum-like behavior
• Composite phase: classical-like behavior
• Core phase: gravitational / observational behavior

The system computes the following loop as a continuous function:

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 continuously iterating this loop, the three structural phases (Stable / Composite / Core) deform smoothly, and quantum-like, classical-like, and gravitational-like 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 computed directly from this single equation of motion alone.

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4. Interaction Slots — Open Integration Architecture

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 rules 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-Driven Computation Reduction

Another major feature of TSH is that the $\Delta f\text{–}\gamma_{T}$ phase diagram is structured to reduce the computational cost itself.

In conventional physics models, separate equations, separate approximations, and separate branching logic 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 returns which structural force to apply
• The phase diagram directly provides the update rule for the next step

As a result, all computation is completed within a single update loop:

• Zero branching
• Zero approximation switching
• No need to evaluate multiple physical laws
• Runs at $O(N)$ on GPU

This yields a structure that is nearly impossible to achieve in conventional physics simulation.

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6. TSH Execution Stack — Structural Engine & AI Structural Engine

TSH is not only a theoretical framework; it is an executable structural environment that directly runs the structural dynamics defined by $p(x), \Delta f, \gamma_{T}$.

6.1 TSH Structural Engine — Unified Structural Engine

A GPU-accelerated execution stack (Unity ECS + HLSL compute + Python) that implements the TSH structural dynamics in real time.

Core Implementation

• Structural field $p(x)$ computed as a Gaussian-weighted sum over neighboring structural elements (p_total)
• $\Delta f$ and $\gamma_{T}$ updated per step from field gradients and accumulated tension
• Phase determined from $p(x)$ against material-defined thresholds (strong_threshold, core_threshold); irreversible lock into Core phase enforced
• 4 abstract interaction channels (charges.xyzw: EM / Strong / Weak / Custom) — interaction domain switchable via materials.json
• Relativistic extension: 4-velocity $u^{\mu}$, Lorentz factor $\gamma$, and proper time $\tau$ per structural element
• $O(N)$ neighbor search via Spatial Hash (supports 100M+ elements)

3D Volumetric Visualization (3 HLSL kernels)

• Phase Map (_BaseFieldTex): R = phase state, G = $\Delta f$ (interference), B = $\gamma_{T}$ (collapse intensity)
• Channel Map (_ChannelFieldTex): q1–q4 interaction channels rendered as hue-coded volume
• Boundary Map (_BoundaryTex): Procedural contour lines at phase-transition boundaries

Implementation Files TSHUnifiedForce.compute / TSHCore.cs / TSHFieldCompiler.cs / TSHPositionUpdateSystem.cs / TSH_Core.py

6.2 TSH AI Structural Engine — Structural Exploration Interface

A Python API (tsh_ai_api.py) that allows AI systems to interact with the TSH structural simulation through a standard Observe → Infer → Apply → Verify loop.

• Observe — get_observables() retrieves structural quantities ($m_\text{eff}$, $E_\text{total}$, $\Phi_\text{struct}$, $\Delta f$, $\gamma_{T}$, phase distance) per structural element. export_observables() saves them as .npy arrays for use with PyTorch / TensorFlow.
• Evaluate — evaluate_phase_topology() scores core density, strong-phase coverage, and structural entropy from the $p(x)$ field. evaluate_irreversibility() measures collapse efficiency and resistance to phase reversal.
• Apply — edit_material() rewrites physical constants ($\alpha$, $\beta$, $k_\text{tension}$, collapse_rate) in materials.json. The simulator reloads this file and the structural behavior changes in real time.
• Compile — export_compiler_results() writes phase-boundary thresholds to compiler_out.json for downstream use.

This loop enables AI-driven exploration of the $\Delta f\text{–}\gamma_{T}$ phase space and optimization of structural behavior — without modifying the TSH structural laws themselves.

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7. Computational Performance — Structural Design Characteristics

The TSH engine's computational efficiency follows directly from its structural architecture. All three behavioral domains (quantum-like, classical-like, gravitational-like) are handled by a single GPU kernel (CSMain) with no separate code paths per domain.

Architectural Properties (verified in implementation)

• Single kernel dispatch — CSMain computes $p(x)$, $\Delta f$, $\gamma_{T}$, phase, force, and position update in one pass
• Phase determined by threshold comparison — no iterative solver or regime-specific evaluation; p_total is compared against c1 / c2 directly
• No inter-domain branching — the same kernel runs identically regardless of whether an element is in Stable, Composite, or Core phase
• $O(N)$ neighbor search — Uniform Grid Spatial Hash; designed to support 100M+ structural elements
• AI Inverse Physics Solver — backpropagation-based optimizer (tsh_ai_api.py) finds material constants from target phase topologies, reducing parameter search cost

Design Consequence

Because the $\Delta f\text{–}\gamma_{T}$ phase diagram encodes all behavioral transitions as a lookup rather than as separate physical laws, the update cycle remains structurally identical across all phases — enabling GPU parallelism without approximation switching or branching overhead.

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8. 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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9. License

• Code and Scripts: MIT License.
• Theoretical Content: The TSH paper (PDF/HTML), theoretical content in this README, and figures are © 2026 Hirokazu Abe. Unauthorized redistribution is prohibited.