PDT-AutoMath: Automated Mathematical Discovery

Derive — Discover — Name

Pisot Dimensional Theory | Stephanie Alexander · 2026

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This repository applies the Omega Institute's AutoMath methodology — Derive, Discover, Name — to the polynomial pair at the unique Pisot boundary of the family xⁿ = x + 1:

P3: x³ − x − 1 = 0   →   ρ = 1.32472  (plastic constant, 3D classical sector)
P4: x⁴ − x − 1 = 0   →   Q = 1.22074  (quartic root,     4D quantum sector)
Lock:  N(ρQ) = −1     →   algebraic unit norm identity (proved)

95 theorems across 10 branches of mathematics, all derived from these three axioms.

Comparison with AutoMath

	AutoMath (Omega Institute)	PDT-AutoMath
Seed polynomial	x² = x + 1 (n=2)	x³=x+1 AND x⁴=x+1 (n=3,4)
Roots	φ = 1.61803 (golden ratio)	ρ = 1.32472, Q = 1.22074
Additional axiom	none	N(ρQ) = −1 (proved)
Compositum degree	2	12
Unit group rank	1	5 (Dirichlet)
Theorems	~2,350 (Lean 4)	95 (Python, 95/95 verified)
Physics predictions	none	89 at 0.75% mean error

The n=2 seed (AutoMath's x²=x+1) is the special case just below the Pisot boundary.
PDT uses the two members straddling the boundary — where classical physics meets quantum physics.

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The Ten Branches

Branch	Name	Theorems
I	Core Algebra	12
II	Norm Theory — The Lock N(ρQ)=−1	12
III	Integer Sequences (Padovan, Perrin)	8
IV	Convergence and Spectral Theory	11
V	Compositum Structure and Unit Group	6
VI	Elliptic Curves and L-functions	9
VII	Dynamics and Substitution Systems	6
VIII	Arithmetic Geometry	7
IX	Physics Staircase (coupling constants)	14
X	Golden Ratio and Fibonacci Connections	10
Total		95

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Usage

Run in Google Colab — no installation needed:

# Install dependencies
pip install sympy numpy

# Run all 10 branches
python PDT_AutoMath.py

# Run a single branch — set BRANCH = 2 in the CONFIG block at the top
python PDT_AutoMath.py

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Key Results

N(ρ)  = +1   (ρ is a positive-norm unit — classical, Pisot, self-dual)
N(Q)  = −1   (Q is a negative-norm unit — quantum, non-Pisot, anti-self-dual)
N(ρQ) = −1   (the algebraic lock — proved in degree-12 compositum)

The unit norm identity N(ρQ) = −1 is not a numerical observation. It is a proved theorem, verifiable in 30 seconds via PARI/GP (see ratio-unit-norm repository).

Selected Theorems

• I.4: resultant(P3, P4) = 1 — the polynomials share no root in any extension
• II.3: N(ρQ) = −1 — the algebraic lock [THE CENTRAL RESULT]
• III.3: Per(19) = 209 = gravity staircase floor (15²−15−1=209)
• III.4: Perrin primality test verified for all primes up to 31
• IV.7: Newton power sums p₁=0, p₂=2, p₃=3 for P3-roots
• IV.8: Koide ratio p₂/p₃ = 2/3 (lepton mass ratio from polynomial)
• IX.1: α⁻¹ = (ρQ)¹⁵/π² = 137.063 (measured 137.036, error 0.020%)
• IX.11: γ_BI = λ₄ρ = 0.2396 (LQG Barbero-Immirzi value 0.2375, error 0.86%)
• IX.14: G = 6.6741×10⁻¹¹ from floor 224 + screening correction (error 0.003%)

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Physical Predictions

The same two polynomials, with the electron mass as the one dimensional scale, give:

Quantity	Formula	Predicted	Measured	Error
α⁻¹	(ρQ)¹⁵/π²	137.063	137.036	0.020%
sin²θ_W	λ₄/χ³	0.2311	0.2312	0.04%
Y_p	λ₃	0.2451	0.2449	0.08%
γ_BI	λ₄ρ	0.2396	0.2375	0.86%
G	complete formula	6.6741×10⁻¹¹	6.6743×10⁻¹¹	0.003%

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Next Step: Lean 4 Formal Verification

The Omega Institute's AutoMath infrastructure can formally verify these branches in Lean 4.
The three axioms to add:

-- P3: the cubic polynomial
def P3 (x : ℝ) : ℝ := x^3 - x - 1

-- P4: the quartic polynomial  
def P4 (x : ℝ) : ℝ := x^4 - x - 1

-- norm_lock: the algebraic unit identity (proved in PARI/GP)
axiom norm_lock : norm (rho * Q) = -1

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Related Repositories

• golden-ratio-unit-norm — PARI/GP proof of N(ρQ)=−1
• dimensional-origin-Newton — derivation of Newton's constant from the Pisot boundary

Citation

@software{alexander2026pdt_automath,
  author    = {Alexander, Stephanie},
  title     = {PDT-AutoMath: Automated Mathematical Discovery from the Pisot Boundary},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.XXXXXXX},
  url       = {https://github.com/stalex444/pdt-automath}
}

License

MIT License — free to use, modify, and distribute with attribution.

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"Everything solved itself along the way."
The question was why space has three dimensions. Newton's constant was in the way.