Alpha-Physical Language (APL) — Seven Sentences Test Pack

A minimal operator grammar for describing physical system behaviors across geometry, waves, chemistry, and biology.

Overview

APL (Alpha-Physical Language) is an experimental framework that uses compact "sentences" to predict physical regimes across diverse domains. This repository contains a test pack designed to allow independent teams to validate whether APL's core operator language has genuine physical content.

The test pack translates 7 compact APL sentences into falsifiable, cross-domain hypotheses that can be probed with standard models:

• Navier–Stokes (fluid dynamics)
• Wave equations (electromagnetics, acoustics)
• Reaction–diffusion (chemistry)
• Phase-field models (interfaces, materials)
• Polymer growth / aggregation

Core Concept

APL describes physical systems using:

Fields ("Spirals")

• Φ — Structure field (geometry, lattice, boundaries)
• e — Energy field (waves, thermodynamics, flows)
• π — Emergence field (information, chemistry, biology)

Universal Operations

• () — Boundary / containment
• × — Fusion / convergence / joining
• ^ — Amplify / gain
• % — Decohere / noise / reset
• + — Group / aggregation / routing
• – — Separate / splitting / fission

Operator States (UMOL: Universal Modulation Operator Law)

• u (𝒰) — Expansion / forward projection
• d (𝒟) — Collapse / backward integration
• m (CLT) — Modulation / coherence lock

An APL sentence has the form:

[Direction][Op] | [Machine] | [Domain] → [Regime/Behavior]

For example: u^|Oscillator|wave reads as "Forward amplification in an oscillatory machine in a wave domain."

The Seven Test Sentences

Each sentence is a testable hypothesis predicting that specific operator-machine-domain combinations statistically favor particular physical regimes:

#	Sentence	Predicted Regime	Domain
A3	u^|Oscillator|wave	Closed vortex / recirculation	Wave dynamics
A7	u%|Reactor|wave	Turbulent decoherence	Flow/wave systems
A1	d()|Conductor|geometry	Isotropic lattice / sphere	Geometry/interfaces
A4	m×|Encoder|chemistry	Helical encoding	Chemistry/polymers
A5	u×|Catalyst|chemistry	Branching networks	Chemistry/growth
A6	u+|Reactor|wave	Focusing jet / beam	Fluid/plasma/wave
A8	m()|Filter|wave & d×|Catalyst|chemistry	Adaptive filter / selectivity	Wave & chemistry

Interpretation Rule

For all sentences:

LHS → RHS

means:

If a system is built to match the left-hand side (LHS) structure and driving, then the right-hand side (RHS) regime should appear more often, more strongly, or at lower thresholds than in controls that break the LHS structure, with all else as equal as possible.

Evidence FOR APL: Clear, reproducible overrepresentation of the RHS regime under LHS conditions vs. controls.

Evidence AGAINST APL: No such bias, or controls produce the RHS regime equally or more often.

Repository Structure

APL/
├── README.md                           # Documentation
├── apl-operators-manual.tex            # Complete operator reference (LaTeX)
├── apl-seven-sentences-test-pack.tex   # Complete test protocol (LaTeX)
├── COMPILE_INSTRUCTIONS.md             # LaTeX compilation guide
└── docs/                               # Documentation and compiled outputs
    ├── index.html                      # HTML version of operator's manual
    ├── apl-operators-manual.pdf        # PDF version (auto-compiled)
    └── apl-seven-sentences-test-pack.pdf  # Test pack PDF

Testing Strategy

For each sentence, the recommended approach:

1. Choose a standard model appropriate to the domain
   • Geometry: phase-field, Cahn–Hilliard, curvature flow
   • Flows/waves: Navier–Stokes, lattice Boltzmann, wave equation
   • Chemistry: reaction–diffusion, polymerization, DLA, kinetic Monte Carlo
2. Implement the LHS conditions
   • u^: Add gain/amplification at resonant modes
   • u%: Add explicit stochastic/decohering forcing
   • d(): Allow boundaries to relax/collapse under isotropic energy
   • m(): Modulate boundaries in response to passing modes
   • u×/d×: Implement forward-biased or collapse–fusion catalysts
   • u+: Add grouping/convergent geometry or fields
3. Design matched controls
   • Remove or invert the key operator
   • Keep everything else comparable
4. Define regime metrics (A1–A8)
   • A1: Sphericity, surface/volume ratio, packing isotropy
   • A3: Vortex count/lifetime, closed streamline fraction
   • A4: Helical order parameters, information capacity
   • A5: Fractal dimension, branching degree
   • A6: Jet opening angle, centerline coherence
   • A7: Spectral width, RMS fluctuations, Lyapunov exponents
   • A8: Adaptive sharpening, retuning capability
5. Sweep parameters and compare
   • Drive strength, noise amplitude, surface tension, catalytic bias, etc.
   • Run multiple realizations
   • Check whether LHS conditions robustly bias metrics toward target regime

Preliminary Results

The test pack includes two toy numerical checks:

• A1 (Isotropic cluster): 2D point collapse under isotropic central force → circular, angle-isotropic cluster ✓
• A5 (Branching networks): 2D diffusion-limited aggregation → fractal branching structure (D ≈ 1.2) ✓

These are minimal sandbox experiments consistent with APL predictions. Full testing requires domain-appropriate models across all seven sentences.

Requirements

To use this test pack, you need:

• Basic familiarity with physics, chemistry, and data analysis
• NO prior knowledge of CET or other APL meta-structures
• Access to standard simulation tools (Python/NumPy, MATLAB, COMSOL, OpenFOAM, etc.)

Assumptions:

• Independent testing by teams without invested stake in APL
• Falsifiable approach: both positive and negative results are valuable

Documentation

APL Operator's Manual

A comprehensive reference guide for APL operators, syntax, and usage patterns is available in multiple formats:

• HTML Version: Open docs/index.html in your browser for an interactive, responsive manual
• PDF Version: Automatically compiled from LaTeX source via GitHub Actions
• GitHub Pages: Available online if GitHub Pages is enabled for this repository

The manual includes:

• Complete operator reference with symbols and meanings
• Field definitions (the three "spirals": Φ, e, π)
• Operator state modulation (UMOL)
• Machine contexts and domains
• Syntax rules and sentence structure
• Usage patterns and examples
• Quick reference tables

Compilation

To compile the LaTeX documents locally:

pdflatex -interaction=nonstopmode apl-operators-manual.tex
pdflatex -interaction=nonstopmode apl-seven-sentences-test-pack.tex

See COMPILE_INSTRUCTIONS.md for detailed instructions.

Getting Started

1. Read the operator's manual:
   • Open docs/index.html in your browser, or
   • Compile the PDF: pdflatex apl-operators-manual.tex
2. Read the test pack:

   pdflatex apl-seven-sentences-test-pack.tex

   Or view the .tex source directly.
3. Choose a sentence to test (recommend starting with A1, A3, or A5 as they have clear metrics)
4. Implement the test protocol using your preferred simulation framework
5. Report results — both confirmations and refutations help refine or reject APL

Philosophy

APL is designed to be falsifiable. The language should stand or fall on whether these seven sentences predict robust, cross-domain biases in real physical and chemical systems — nothing more and nothing less.

If the predicted regimes are NOT overrepresented under the specified conditions, that is strong evidence against APL's validity.