SPPC — Single-Parameter Phase Coordinates

A coordinate system that parameterizes n-dimensional nested rotational spaces using a single parameter s:

theta_i(s) = 2 * pi * s * phi^(i-1)

where phi = (1 + sqrt(5)) / 2 is the golden ratio.

Instead of n independent coordinates, one value traces a quasi-periodic space-filling curve through nested rotational dimensions. The golden ratio's status as the "most irrational" number (its continued fraction is all 1s) creates exploitable structure — shortcuts cluster at Fibonacci distances, gaps can be traversed with Lucas numbers, and log-polar transforms amplify the underlying pattern.

Origin

Three ideas collided:

1. The oscilloscope: An electron beam tracing a phosphor screen creates perceived form from motion — a single moving point producing the illusion of structure. What if position itself is just a phase pattern from something moving?

2. The notch on a rod: A single mark on a stick can encode a number. But if the stick is wound through nested loops at irrational ratios, that one notch encodes a position in arbitrarily many dimensions. One mark, infinite address space.

3. The Library of Babel: The libraryofbabel.info website contains every possible page of text — every book ever written or ever will be — already existing in its algorithm, addressable by index. The data is all there, but it's useless without navigation. The key question was: can geometric constants and structurally elegant formulas help us find the hotspots of actual information in that space? And if the structure doesn't naturally cluster meaning, can we exploit it by purposely storing data at addresses that sit near those navigable landmarks?

This led to the core question: can we define a coordinate system where the "thread" — a single path through space — is fundamental, and where the geometry itself provides the map?

Structure

sppc/                    # Reusable Python module
  core.py                # s_to_phases(), s_to_position(), constants
  shortcuts.py           # Shortcut detection and topology
  navigation.py          # Fibonacci + Lucas navigation algorithms
  transforms.py          # Log-polar and lens transformations
  dual_encoding.py       # Polar twin encoding (s and s+0.5)
  figures*/              # Generated paper figures (v1-v3)
  paper*/                # LaTeX paper source (v1-v3)

paper/                   # Research paper PDFs and generators
  thread_coordinates_paper.tex
  single_parameter_phase_coordinates.tex
  *.pdf                  # Paper versions v1-v4

critical_analysis/       # Honest self-assessment
  01_initial_critique.md # Identifies circular reasoning and misleading metrics
  02_salvageable_ideas.md# What's actually worth exploring
  03-08_*.py             # Benchmark scripts (Hilbert curves, bronze mean, etc.)
  07_investigation_summary.md  # Results: what held up, what didn't

output/                  # Validation visualizations and result data
*.py                     # Standalone validation and exploration scripts

Key Documents

File	What it covers
FINDINGS_SUMMARY.md	Quantified results across all four navigation layers
THE_GOLDEN_THREAD.md	Conceptual framing as a thought experiment
thread_coordinate_theory.md	Mathematical foundation and connections to fiber bundles, Hopf fibration, ergodic theory
critical_analysis/	What's novel vs. established (Weyl 1916), what metrics are misleading, and what research directions survive scrutiny

Critical Analysis

The critical_analysis/ directory is the most important part of this repo. It identifies:

• What's not novel: Quasi-periodic space-filling on tori is Weyl's equidistribution theorem (1916). Fibonacci clustering is definitional when using phi. "Shortcuts" exist on any space-filling curve.
• What's misleading: Efficiency ratios that divide by small numbers, self-confirming test design, speculative physics language.
• What survives: The inverse problem (position to parameter) connects to lattice algorithms. Dual encoding with structured offsets is non-trivial. The bronze mean anomaly is unexplained. Benchmarking against Hilbert curves for spatial indexing is a real question.

Quick Start

pip install numpy matplotlib scipy

import sppc

# Encode a position with single parameter
phases = sppc.s_to_phases(0.42, n_dims=3)
pos = sppc.s_to_position(0.42, n_dims=3)

# Find shortcuts (points far on thread, close in space)
shortcuts = sppc.find_shortcuts(n_points=1000, threshold=0.1)

# Navigate using Fibonacci landmarks + Lucas gap traversal
path = sppc.two_stage_navigate(start=0.1, target=0.9, n_dims=3)

# Polar twin encoding
twin = sppc.get_polar_twin(0.42)  # returns 0.92

Run all validations:

python run_all_validations.py

Requirements

• Python 3
• NumPy >= 1.20
• Matplotlib >= 3.4
• SciPy >= 1.7

License

MIT