Experimental implementations of Zeckendorf-arithmetic operations and their effects on small physics and number-theoretic test problems. This is a collection of honest experimental notes, not a research claim. Results include passes, artifacts, and inconclusive outcomes — all documented as-is.
Zeckendorf (1972) proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. This project explores whether arithmetic built on that representation — and on the algebraic field Q(√5) — behaves differently from standard floating-point arithmetic when applied to physics ODEs. The companion Rust simulator (zeckendorf-verifier) is where this arithmetic will eventually run inside an ODE integration loop for direct comparison.
core/
zeckendorf.py Zeckendorf representation engine: to/from conversion,
round-trip verified for integers 1–100,000.
phi_field.py Q(√5) algebraic field: numbers as (a + bφ)/d with
exact arithmetic, conjugate, norm.
fibonacci_arithmetic.py Native φ-basis arithmetic: add, subtract, multiply,
divide, power — within the representation.
constants.py Physical constants in standard and φ-field form.
experiments/
exp01_decay.py Radioactive decay in φ-basis.
exp01b_prediction.py Out-of-sample isotope prediction.
exp02_entropy.py Entropy evolution without explicit time parameter.
exp02_resolution.py Entropy resolution sweep.
exp03_harmonic.py Simple harmonic oscillator in φ-basis.
exp03b_phi_recurrence.py Second-order recurrences: SHO vs Fibonacci vs φ-hybrid.
exp04_growth.py Logistic growth and phyllotaxis angle.
exp04b_oos_prediction.py Out-of-sample logistic growth.
exp05_wave.py Wave propagation on Fibonacci vs uniform grids.
exp05b_convergence.py Richardson extrapolation on wave speed.
exp06_gravity.py Two-body orbit with φ-modulated force.
exp06b_convergence.py Precession vs timestep (artifact test).
exp06c_controlled.py Precession across arithmetic backends.
exp06d_gr_comparison.py φ-modulated precession vs Schwarzschild GR.
exp07_emergence.py Free evolution: 32 agents, 10,000 steps.
analysis/
divergence_map.py Cross-experiment error metrics and heatmap.
time_detection.py Automated scan for monotonic (time-like) quantities.
constants_analysis.py Fine structure constant and mass ratios in the φ-field.
report_generator.py Synthesis report.
docs/
FINDINGS.md Living experiment log with raw numerical results.
CLAIMS_REFRAMED.md Each claim restated with evidence and limitations.
output/
exp01/ … exp07/ Per-experiment: divergence plots, metrics JSON.
analysis/ Divergence heatmap, time detection output.
exp01 / exp01b — Radioactive decay. Tested whether φ-field arithmetic reproduces the standard exponential decay law for C-14, U-238, and Po-210. The φ-basis decay formula was implemented and compared against the analytic exponential. Maximum relative error across all three isotopes: ~3.9×10⁻⁹. Correlation with standard values: ≈1.0000000. A blind out-of-sample prediction (exp01b) on three unseen isotopes — Rn-222, Cs-137, Co-60 — produced the same error level and passed the 0.1% threshold for all three. This shows the φ-field correctly encodes the exponential function to near floating-point precision; it does not imply special physical meaning.
exp02 — Entropy. Tested whether a fib-log entropy measure (indexed by the largest Fibonacci number ≤ the system size Ω) increases monotonically along simulation steps without an explicit time parameter. Result: not monotonic. The fib-log entropy correlates with Boltzmann entropy at 0.9997, but monotonicity fails — it is not a time surrogate on its own. A resolution sweep (exp02 resolution) confirmed the non-monotonicity is not a finite-size artifact at the grid sizes tested.
exp03 — Simple harmonic oscillator. Implemented the SHO via a discrete φ-basis recurrence and compared against the analytic cosine. RMS error: 9.4×10⁻¹⁴ (consistent with double-precision roundoff). A second test (exp03b) compared second-order recurrences using SHO coefficients, Fibonacci coefficients, and a φ-hybrid; the Fibonacci and φ-hybrid recurrences showed non-periodic growth rather than oscillation, as expected from the eigenstructure of those recurrences. The SHO result confirms the φ-field arithmetic is numerically clean; the recurrence comparison confirms it is not a drop-in substitute for standard SHO dynamics.
exp04 — Growth and phyllotaxis. Applied φ-scaled Euler integration to logistic growth and compared with the analytic logistic. Maximum |ΔN| ≈ 0.62; correlation ≈ 0.99998. Phyllotaxis divergence angle computed from φ: 137.508° (standard value: 137.508°). The phyllotaxis result is a consequence of φ's definition and is not a new observation; the logistic correlation confirms the arithmetic tracks the function without large instability.
exp05 / exp05b — Wave propagation. Compared 1D wave propagation on a Fibonacci-spaced grid against a uniform grid at matched physical time (T ≈ 0.782). The Fibonacci spacing creates a highly non-uniform grid (minimum Δx ≈ 1/ΣF), requiring ~2× as many timesteps to match elapsed time. The energy-centroid speed proxy differed: uniform ≈ 0.407, Fibonacci ≈ 0.380 (c = 1.0). A Richardson extrapolation study (exp05b) was run to test whether the gap closes as grid resolution increases; the two largest grid sizes timed out before completion. The evidence suggests the gap is a discretization artifact from grid non-uniformity rather than a structural difference in the arithmetic, but the extrapolation was not completed at fine enough resolution to confirm.
exp06 — Orbital precession. Added a φ-modulated angular perturbation to the Newtonian gravitational force on a two-body orbit and measured periapsis shift. The perturbation produced a shift of ~2.3×10⁻⁵°. A timestep convergence study (exp06b) showed the shift goes to zero as dt → 0, confirming it is a discretization artifact. A controlled arithmetic comparison (exp06c) ran the same setup with float64, Python Decimal, and a φ-approximation modulation — all three produced the same perturbation order, ruling out the φ-field as the cause. A GR ratio comparison (exp06d) found the φ-modulated precession is ~2.3× the Schwarzschild GR precession for an Earth-like orbit, but at a different force scale — the numbers are not comparable without matching the perturbation scale to physical GR corrections.
exp07 — Free emergence. Ran 32 agents under a ℤ[φ] re-quantization scheme with mean-field coupling and noise for 10,000 steps. Tested for any monotonically increasing (time-like) quantity. No monotonic candidates emerged. An unexpected result: agent variance collapsed to 0 — all agents locked to the same discrete value under the re-quantization coupling. This is a consequence of the discrete arithmetic (ℤ[φ] has finite resolution at each scale), not evidence of time-like ordering.
Constants analysis. Computed the fine structure constant α and the electron/proton mass ratio in the φ-field Q(√5). The nearest Z[φ] approximation to α has absolute error ~1.0×10⁻¹¹ at moderate term count. No special closed-form structure was found that would not appear in any algebraically dense field. Negative result.
Requirements: Python 3.11+, numpy, matplotlib, psutil.
pip install -e ".[dev]"
# Run core arithmetic tests first
python -m pytest tests/ -v
# Run a specific experiment
python -m experiments.exp01_decay
# Generate the synthesis report (run after all experiments)
python -m analysis.report_generatorEach experiment writes output to output/<exp_name>/. A per-experiment safety module enforces a 300-second deadline and a memory guard (>85% RAM triggers graceful early exit with partial results saved).
- Complete the Richardson extrapolation in exp05b at finer Fibonacci grid resolution to settle the wave speed convergence question
- Wire the φ-basis arithmetic into the
zv-engineintegration step inzeckendorf-verifierand compare per-step energy drift against the float64 baseline on the same solar system initial conditions - Test whether the variance collapse in exp07 is robust to different re-quantization thresholds or is specific to the ℤ[φ] step size used
- Run the constants analysis with higher Fibonacci term counts to determine whether the α fit error continues decreasing or plateaus
This repository uses a dual license:
- Python code (
core/,experiments/,analysis/,tests/): MIT. SeeLICENSE-CODE. - Research notes, derivations, and prose (
docs/,output/, all.mdfiles): CC BY-NC-SA 4.0. SeeLICENSE-NOTES.