Phase Transitions in Parameterized Families of Infinite Products over Even–Odd Integer Pairs
Dionisio Alberto Lopez III
Independent Researcher — Cranberry Township, Pennsylvania, USA
February 2026
This repository contains the white paper, presentation deck, simulation data, and supporting figures for The Pi Phase Hypothesis — a conjectural framework proposing that the appearance of π in the classical Wallis product is not an isolated identity but an instance of a broader phase phenomenon in parameterized families of infinite products.
The central object of study is the Bernoulli–Wallis product: at each index n, a biased coin (probability α) decides whether the Wallis factor W(n) = (2n)² / ((2n−1)(2n+1)) is applied or replaced by a neutral factor of 1. When α = 1 every factor is active and the product recovers π/2 exactly. When α = 0 the product is trivially 1.
The hypothesis conjectures a critical threshold α* ∈ (0, 1) at which the arithmetic nature of the limiting product undergoes a qualitative transition:
| Regime | Behavior |
|---|---|
| α > α* | Product converges almost surely to a value in ℚ · π |
| α < α* | Product converges to a limit not of the form qπ (q rational) |
| α = α* | Variance of the limiting distribution exhibits non-analytic behavior |
Simulations across 21 bias values (α = 0.00 to 1.00 in steps of 0.05), each with 30 independent trials of 20,000 factors, show:
- Monotone convergence of the mean product toward π/2 as α → 1
- Variance peaking at intermediate bias (α ≈ 0.4–0.6), consistent with the analytic formula Var[log P(α)] ∝ α(1−α)
- Distributional evolution from a right-skewed, π-agnostic shape (low α) to a tight, π-centered cluster (high α)
- Exact recovery of π/2 at α = 1.0 to floating-point precision
pi-phase-hypothesis/
├── README.md # This file
├── pi-phase-deck.md # 15-slide presentation deck with presenter notes
├── pi-phase-hypothesis.pdf # PDF export of the hypothesis
├── pi-phase-hypothesis.mp4 # Video presentation
├── .gitignore
└── white-paper/
├── pi_phase_hypothesis.md # Full white paper (Markdown)
├── pi_phase_hypothesis.tex # Full white paper (LaTeX source)
├── pi_phase_simulation_data.csv# Simulation data (21 bias values × 6 statistics)
├── fig1_mean_vs_bias.png # Mean product vs bias (±1 std dev shaded)
├── fig2_std_vs_bias.png # Standard deviation vs bias
├── fig3_ratio_vs_bias.png # Ratio to π/2 vs bias
└── fig4_histograms.png # Distribution of P(α) at six bias values
For bias parameter α ∈ [0, 1], let {B_n} be independent Bernoulli(α) random variables. The Bernoulli–Wallis product is:
P(α) = ∏ W(n)^{B_n}
where W(n) = (2n)² / ((2n−1)(2n+1)) is the classical Wallis factor.
The log-product log P(α) = ∑ B_n log W(n) converges almost surely by the Kolmogorov two-series theorem (since log W(n) ~ 1/(4n²)).
The deterministic interpolation P_det(α) = (π/2)^α is smooth with no phase transition. The stochastic model is fundamentally different: it explores the space of all sub-products of the Wallis product, weighted by α. Different random subsets of active factors yield different limits, and the hypothesis concerns whether there is a critical density above which these limits are arithmetically constrained to lie in ℚ·π.
The paper situates the hypothesis relative to:
- Viète (1593) — First infinite product for π
- Wallis (1655) — The classical product π/2 = ∏ (2n)²/((2n−1)(2n+1))
- Euler — Sine product factorization (setting z = 1/2 yields the Wallis product)
- Friedmann & Hagen (2015) — Quantum-mechanical derivation via hydrogen atom variational method
- Granville & Soundararajan (2003) — Random Euler products in probabilistic number theory
- Sanderson / 3Blue1Brown (2018) — Geometric lighthouse proof
- Farrell (2019) — Generalized Wallis products via gamma-function ratios
The white paper poses seven formal open problems:
- Existence of α* — Is α* interior to (0,1), or is it 0 or 1?
- Variance scaling — Does a refined "arithmetic residual" exhibit non-analytic behavior at α*?
- Other constants — Can the framework extend to products converging to e, log 2, or γ?
- Multidimensional phase diagram — Independent bias per prime index α_p: regions converging to different transcendentals?
- n-dependent bias — Characterize (α_n) sequences (e.g., α_n = min(1, c/n^β)) yielding ℚ·π limits
- Correlated factors — Markov-chain factor selection: universality classes?
- Lighthouse percolation — Geometric formulation via 3Blue1Brown's construction
| Figure | Description |
|---|---|
 |
Fig 1. Mean product converges monotonically toward π/2 as bias increases. Shaded region = ±1 std dev. |
 |
Fig 2. Variance peaks at intermediate bias and collapses at both endpoints, consistent with α(1−α) scaling. |
 |
Fig 3. Fraction of π/2 achieved approaches 1.0 as α → 1. |
 |
Fig 4. Distribution of P(α) across 100 trials at six bias values, showing evolution from π-agnostic to π-centered. |
The file white-paper/pi_phase_simulation_data.csv contains summary statistics for 21 bias values:
| Column | Description |
|---|---|
alpha |
Bias parameter (0.00 to 1.00 in steps of 0.05) |
mean |
Sample mean of P(α) across 30 trials |
std |
Sample standard deviation |
median |
Sample median |
min |
Minimum observed product value |
max |
Maximum observed product value |
ratio_to_pi_half |
Mean / (π/2), where 1.0 = exact Wallis limit |
If you reference this work:
Lopez III, D. A. (2026). The Pi Phase Hypothesis: Phase Transitions in
Parameterized Families of Infinite Products over Even–Odd Integer Pairs.
https://github.com/zfifteen/pi-phase-hypothesis
This project is released under the MIT License.
Dionisio Alberto Lopez III
Independent Researcher
Cranberry Township, Pennsylvania, USA
- GitHub: @zfifteen
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- Reddit: u/NewspaperNo4249