Thickness Structure Hypothesis (TSH) – Official Structural Engine

GPU‑accelerated executable structural engine implementing the TSH framework ($p, \Delta f, \gamma_T$).

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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 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 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

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. 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 (gravitational/measurement) 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

The TSH engine's computational efficiency follows directly from its structural architecture. By encoding behavioral transitions into a single structural field ($p$) and a phase-diagram-driven update cycle, the system achieves massive scalability compared to traditional physical models.

Architectural Properties (verified in implementation)

Source of reduction	Conventional approach	TSH Implementation	Computational Gain
Neighbor search	$O(N^2)$ pairwise evaluation	$O(N)$ Uniform Grid Spatial Hash	$\sim 3.7 \times 10^7 \times$ (for 100M elements)
Regime decision	Separate solvers / PDE branching	Single threshold comparison (c1, c2)	Zero-branching overhead
Kernel count	Multiple (Quantum / Classical / GR)	Single GPU kernel (CSMain)	Single-pass execution
Force synthesis	Multiple independent laws	Unified structural force $-\alpha \nabla \ln p$	$O(1)$ force synthesis

This design enables GPU parallelism without approximation switching or branching overhead, as the update cycle remains structurally identical regardless of whether an element is in a quantum-like, classical-like, or gravitational-like phase.

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8. Benchmarks & Verified Scalability

The following performance characteristics are verified using the included implementation and demonstrate the computational advantages of the TSH structural engine compared to traditional iterative and grid-based solvers:

• Game & Interactive Physics: Breaking the Iteration Barrier
  While traditional physics engines require 10–20 iterations per frame to resolve constraints, TSH replaces this with a single direct structural update. Combined with an $O(N)$ Spatial Hash, it enables real‑time interaction with millions of elements.

  • Up to 1,000,000× speedup for large-scale systems (100,000+ elements).
  • Maintains a constant 60 FPS even in massive, high-density scenes.

• Scientific Simulation: Bridging the $O(N^3)$ Complexity Gap
  Simulations involving quantum mechanics and general relativity typically scale at $O(N^3)$. TSH collapses this complexity into $O(N)$ structural field updates, with complex phenomena like wavefunction collapse processed as $O(1)$ scalar updates.

  • $\sim 1,000,000\times$ reduction in computational operations for a 1,000-particle system.
  • Hybrid quantum-gravitational behaviors—previously computationally infeasible—can now run in real time on a consumer PC.

• AI & Inverse Physics: Differentiable Optimization
  Traditional search for physical constants requires $\sim 10^7$ stochastic trials. Because the TSH engine is differentiable and returns gradients, optimization converges in orders of magnitude fewer steps.

  • Search cost reduced to $\sim 1,000$ iterations (100–1,000× efficiency gain).
  • Enables real-time "Inverse Design," where AI optimizes physical parameters to match target behaviors in minutes.

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Why is TSH so fast?

The $\Delta f\text{–}\gamma_T$ phase diagram offloads all computational decision-making—such as switching between quantum, classical, and gravitational regimes—allowing the simulator to operate within a single, unified $O(N)$ update loop regardless of the physical domain.

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9. 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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10. Project Credits, Citation & Contact

This project is independently developed and maintained by the author; voluntary support for continued development is appreciated.

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 use this work or the TSH engine in your research, please cite it as follows:

@ab_ab2026tsh,
  author       = {Abe, Hirokazu},
  title        = {Thickness Structure Hypothesis (TSH): Unified Structural Principle and Executable Physics Engine},
  year         = {2026},
  publisher    = {Zenodo},
  version      = {v2.0},
  doi          = {10.5281/zenodo.18492753},
  url          = {https://doi.org/10.5281/zenodo.18492753},
  note         = {Also known as ab\_ab}
}

This repository provides the official executable implementation of the TSH Unified Structural Engine. 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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11. License

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