Abstract

Pattern Field Theory (PFT) predicts that stable loops emerge from three interacting forces—permission, tension, and resolution—represented by a trivergent structure. We test a concrete implication: if π encodes a universal instruction set, then small addressable windows of π digits should be sufficient to configure controllers that reliably achieve coherence in noisy, adversarial conditions. Using the BBP formula to seek digits without preloading, we construct windows (12–32 hex digits), score them by triad balance, and map them into parameter schedules for three families of machines: a strict triad controller, coupled phase oscillators (Kuramoto and Kuramoto–Sakaguchi), and digital phase-locked loops (PLL) including a dual-loop (DPLL) variant with cycle-slip stress. In progressively harder tests—higher detuning, frustration (phase lags), 1/f-like noise, quantization, saturation, and forced slips—π windows act as effective instruction blocks. In the harshest PLL/DPLL regimes, π outperforms random and is competitive with or better than shuffled-π on time-to-lock, recovery, and jitter.

Background

In PFT, coherence emerges when three forces close a loop. The integer part of π (=3) mirrors this triadic necessity; the fractional expansion is treated here as an instruction set—a compressed tape encoding reusable patterns. The Bailey–Borwein–Plouffe (BBP) formula enables direct addressing of π’s base-16 digits, allowing “read-head” experiments without loading the full constant.

Methods

Digit access & window selection

• Access: BBP hex addressing (no preload) to the first 4–8k digits.
• Windows: 12, 24, or 32 hex digits.
• Scoring: Triad balance — split windows into three equal parts (e.g., 8/8/8 for 24 digits); score = sum of absolute pairwise differences of part sums. Lower is more balanced.
• Comparators: (i) π window (as read), (ii) shuffled-π (same multiset, randomized order), (iii) random window (i.i.d. hex).

Machines & mappings

1. Triad controller: Three-phase alignment to 0°, 120°, 240° with time-varying gain matrices derived from windows; metrics include time-to-lock, recovery, final cost.
2. Kuramoto / Kuramoto–Sakaguchi: Coupled oscillators with detuning and (in K–S) phase lags (frustration) mapped from digits; strict lock thresholds; noise injections.
3. PLL & DPLL: Digital PLL tracking a drifting, noisy reference; window-driven PI gains (Kp/Ki) with anti-windup; extreme variants add 1/f-like noise, quantization, VCO saturation, bursts, and forced cycle slip. The DPLL adds a slow outer drift loop + a fast inner phase loop.

Metrics & statistics

• Lock metrics: time-to-lock; reacquisition after disturbance or slip.
• Quality metrics: RMS error, final jitter (tail RMS), overshoot, final composite cost.
• Failures: No-lock/recovery are imputed (horizon+1) to avoid bias.
• Inference: Nonparametric bootstrap on mean differences (π vs shuffled; π vs random); reports include effect direction and p-values in the accompanying CSVs.

Implementation details and parameter values are documented in the CSVs and code listings saved with each run.

Results

R1 — Triad controller

π windows reliably configure the controller to lock and recover after shocks. Against controls, π is consistently usable; differences vs shuffled-π are small at this difficulty, while π tends to outperform purely random windows.

R2 — Kuramoto & Kuramoto–Sakaguchi (frustration)

Under higher detuning and phase-lag frustration, π windows maintain competitive or better time-to-lock and final cost compared with controls. With stricter thresholds (≤ 0.02 rad) some comparators fail to lock; π maintains lock rates and recovery comparable to or better than controls.

R3 — PLL (strict) and PLL EXTREME

With stronger drift, 1/f-like noise, and larger bursts, π shows lower final jitter and faster recovery than random, and is competitive with or slightly better than shuffled-π. See boxplots and stats in the datasets below.

R4 — Dual-Loop PLL (hardest)

In the cascaded DPLL with quantization, saturation, and a forced cycle slip, π again delivers usable schedules. Medians favor π on RMS error and final jitter; time-to-lock and reacquisition are competitive with shuffled-π and better than random’s tails. Cycle-slip counts are dominated by the forced event (thus similar across groups).

All raw stats and effect estimates are available below.

Data & Artifacts

• Large triad/PLL/Kuramoto runs (CSV bundles):
  • pft_bakeoff_results.csv · pft_bakeoff_summary.csv
  • kuramoto_bakeoff_results.csv
  • kuramoto_nextlevel_results_compact.csv
  • kuramoto_32win_results.csv
  • ks_frustrated_results.csv
  • pll_results.csv · pll_stats.csv
  • pll_extreme_results.csv · pll_extreme_stats.csv
  • dpll_hard_results.csv · dpll_hard_stats.csv
• Candidate windows (by triad balance):
  • pft_bakeoff_candidates.csv
  • kuramoto_32win_candidates.csv
  • ks_frustrated_candidates.csv
  • pll_candidates.csv · pll_extreme_candidates.csv
  • dpll_hard_candidates.csv
• Figures: open from the inline images above, or download the PNGs in the sandbox paths named plot_*.

Discussion

These experiments do not claim that π is mystical. They show that short, addressable π windows behave as useful configuration payloads for real dynamical systems, even under severe stress. This supports the PFT proposition that universality emerges from triadic closure and that π’s expansion can serve as a compact instruction source.

Shuffled-π performs competitively in many regimes (as expected, since it preserves digit histograms). The added structure of π’s native order appears to help in the harshest conditions, notably when stability and recovery margins are razor-thin.

Limitations & Next Steps

• Effect sizes vary by machine and parameterization; some regimes show parity with shuffled-π.
• We’ve probed a finite digit range and window sizes; broader address sweeps and cross-validation would refine estimates.
• Future: spectral matching of windows to machine eigenstructures; analytic bounds connecting triad balance to control margins; hardware-in-loop tests.

Conclusion

Within Pattern Field Theory, π functions as a practical instruction set. Using nothing more than seekable π windows, we can configure controllers that lock, ride out shocks, and reacquire after slips. That is evidence of compressed, reusable structure in π’s expansion—and a concrete bridge from PFT’s triadic axioms to working machines.