Verification code for "Arithmetic Geometry at the Pisot Boundary" — five arithmetic theorems and the Dimensional Norm-Hodge Theorem for the PDT polynomials

Companion repository for:

"Arithmetic Geometry at the Pisot Boundary: Galois Groups, Class Fields, and Implications for Physical Geometry and Loop Quantum Gravity"
Stephanie Alexander, April 2026
Zenodo DOI: https://doi.org/10.5281/zenodo.19561374

The script proves five theorems in algebraic number theory and one arithmetic-geometric theorem for the two Pisot polynomials:

x³ - x - 1 = 0    root ρ ≈ 1.32472    (ρ-sector polynomial, degree 3)
x⁴ - x - 1 = 0    root Q ≈ 1.22074    (Q-sector polynomial, degree 4)

Physical corollaries:

Click the Open in Colab badge above. Run all cells. Takes under 60 seconds.

pip install sympy
python3 pdt_arithmetic_geometry.py

See VERIFIED_OUTPUT.txt for the complete output from a verified Colab run.

Verified environments:

• Google Colab: Python 3.12.13, SymPy 1.14.0
• Local: Python 3.12.3, SymPy 1.14.0

No dependencies beyond sympy.

====================================================================
  THEOREM 4: SPLITTING FIELD OF x^3-x-1 = HILBERT CLASS FIELD OF Q(sqrt(-23))
====================================================================
  h(Q(sqrt(-23))) = 3

  *** THEOREM 4 VERIFIED ***
  The splitting field of x^3-x-1 = Hilbert class field of Q(sqrt(-23)).
  rho generates the class field of Q(sqrt(-23)) (class number 3).
  rho is arithmetically equivalent to a special value of the j-function.

====================================================================
  PHYSICAL COROLLARY A: BARBERO-IMMIRZI PARAMETER OF LQG
====================================================================
  gamma_BI = lambda_4 * rho = 0.2395454187

  Published values from black hole entropy matching:
    Meissner (2004):            gamma = 0.2375
    Domagala-Lewandowski (2004):gamma = 0.2427
    PDT derivation:             gamma = 0.2395

  Match to Meissner:  99.14%
  Match to DL:        98.70%

Pisot Dimensional Theory derives fundamental physical constants from the algebraic boundary between x³=x+1 (3D) and x⁴=x+1 (4D). The arithmetic properties established here — prime discriminants, maximal Galois groups, unit norms ±1, Hilbert class field structure, and equal class numbers — provide the number-theoretic foundation for why these two polynomials organize physical reality.

Key connections:

• N(Q) = −1 = ★² on 2-forms in 4D Lorentzian spacetime → arithmetic origin of Ashtekar's self-dual/anti-self-dual decomposition in LQG
• N(ρ) = +1 = ★² in 3D Riemannian space → arithmetic origin of the spatial sector
• γ_BI = λ₄ρ → Barbero–Immirzi parameter derived with zero free parameters
• Splitting field of x³−x−1 = Hilbert class field of ℚ(√−23) → ρ is a CM value of the j-function
• Level 23 newform → PDT enters the Langlands program at the prime |disc(x³−x−1)|

For the full PDT framework see:

• GRF Essay 2026: "The Dimensional Origin of Newton's Constant" — 2026 Gravity Research Foundation Essay Contest

@misc{alexander2026arithmetic,
  author    = {Alexander, Stephanie},
  title     = {Arithmetic Geometry at the Pisot Boundary: Galois Groups, 
               Class Fields, and Implications for Physical Geometry 
               and Loop Quantum Gravity},
  year      = 2026,
  publisher = Zenodo,
  doi       = https://doi.org/10.5281/zenodo.19561374
  url       = https://zenodo.org/records/19582084
}

MIT License — free to use, verify, and build on with attribution.