Foundational Research
Constants. Structures. Systems.
"We got a lot of flack from a lot of people, saying 'Oh it's crazy to say something like that.'"
— Professor David N. Schramm, University of Chicago (On his prediction of three particle families—later confirmed) This research is dedicated to him in memoriam
Baryonix Corp. conducts foundational research exploring the structures underlying physical and biological reality, and their applications to complex systems. Our AI-driven work spans theoretical physics, applied mathematics, signaling biology, and computational intelligence.
We investigate, for example, how fundamental constants and geometric relationships propagate through dimensional hierarchies, and how these insights can inform practical advances in artificial intelligence and beyond.
We exist to ask foundational questions—and to pursue answers that help move humanity forward.
Mathematical Foundations
Polynomial structures generating fundamental constants. Dimensional hierarchy theory. Relationships between the plastic constant, golden ratio, and physical phenomena.
Theoretical Physics
Quantum measurement and geometric projection. Emergent dimensional structures. Connections between algebraic constants and cosmological parameters.
Applied Intelligence
Novel approaches to neural architecture informed by fundamental mathematical relationships. Compression methods derived from structural constants.
Biological Structures
How signaling hubs coordinate competing inputs. Resolving paradoxes in immune regulation through structural and functional analysis.
Pisot Dimensional Theory: Proof Map
Reference document. Every claim in the framework with its logical dependencies, current status, and proof source — from Peano axioms to cosmological predictions. 43 claims established by theorem, 19 computationally verified, 6 open problems.
Quantum Mechanics as the Arithmetic of a Complex Place
Derives the kinematic formalism of a single quantum system — Hilbert space, the Born rule, unitarity, wave-function collapse — from one named identification about the field of x⁴ = x + 1. The central identities are kernel-checked in Lean 4 + Mathlib; the Mahler-extremality ground is proved outright, with the sweep deposited alongside the paper.
A Fibonacci–Lucas Spectral Law for the Trinomials xⁿ = x + 1
Establishes the unit-norm identity N(ρQ) = −1 and the Fibonacci–Lucas trace structure across the trinomial family.
Elliptic Curves Associated to the Pisot-Boundary Polynomials x³ − x − 1 and x⁴ − x − 1
Defines the elliptic curves E₃ and E₄ from the Pisot polynomials and catalogs their arithmetic invariants — conductors, discriminants, ranks, L-functions, modular forms.
Arithmetic Geometry at the Pisot Boundary: Galois Groups, Class Fields, and Implications for Physical Geometry and Loop Quantum Gravity
Six theorems: Galois groups S₃ and S₄, prime discriminants −23 and −283, and the Hilbert class field of ℚ(√−23). Derives the Barbero–Immirzi parameter of loop quantum gravity algebraically.
The Strong Coupling Constant as an Elliptic Period: αₛ(MЗ) from the Arithmetic Geometry of the Pisot Boundary
Identifies αₛ(MЗ) as the real period of the elliptic curve 1132b1 divided by 23. Matches the FLAG lattice average to 0.003% with zero free parameters.
The Ringing Universe at the Pisot Boundary: Why x³ = x + 1 Requires Oscillation
Trichotomy theorem: the plastic constant is the unique dimension in the trinomial family where integer convergence is oscillatory rather than monotonic.
Cosmological Ringing from the Pisot Boundary: A Zero-Parameter Prediction of the Ringermacher–Mead Oscillation
Tests the algebraic oscillation prediction against the 7.15 HHz Ringermacher–Mead signal in Pantheon+ supernova data. Matches to 0.16% with zero free parameters.
The Complete Guide to Pisot Dimensional Theory: From Zero to Deriving All Physical Constants
Self-contained tutorial. Derives the full predictions table from first principles with no prerequisites beyond basic mathematics.
The theory’s exposure, kept in public: named forks with their numbers and their kill conditions, scored against measurement as it arrives — hits and misses alike.
The Hubble tension is a seam, not an error
On the theory’s reading, the late-universe distance ladder and the early-universe CMB inference measure two different projections of one universe, offset by the fixed, parameter-free ratio ρ/Q = 1.0852. Planck’s 67.36 × 1.0852 = 73.10 km/s/Mpc. Current scoring: SH0ES 2022 (73.04 ± 1.04) −0.06σ; SH0ES 2025 (73.49 ± 0.93) +0.42σ; H0 Distance Network (73.5 ± 0.81) +0.50σ. Kill condition, stated in advance: as measurements sharpen, the ratio must converge to ρ/Q, not to one — if the tension melts away, or the CCHP value near 70 prevails (currently −1.63σ), this reading is wrong.
Cosmological ringing at the Pisot boundary
The trichotomy theorem makes integer convergence at x³ = x + 1 oscillatory rather than monotonic; the deposited zero-parameter prediction matches the 7.15 HHz Ringermacher–Mead oscillation in Pantheon+ supernova data to 0.16% (paper above). Kill condition: the oscillation fails to appear in larger supernova compilations.
Every quantitative claim above is reproducible. The computational repositories confirm their results in under a minute each; the two Lean repositories carry full kernel proofs — clone, lake build, and the machine checks every step (axiom audits included).
The kernel-verified core: single-system quantum kinematics, the two-rulers dagger identity, the symplectic form, trace-form signature certificates, and the settle/spiral boundary — 256 declarations in Lean 4 + Mathlib, no sorry, axiom-audited. DOI: 10.5281/zenodo.21210683
The arithmetic of time, kernel-checked: the arrow of time as a theorem (the settled history, read backwards, is provably a different object), an exact per-tick irreversibility rate, a tick proved aperiodic but never random, and the Pisot boundary. CI reruns the full build and axiom audit on every push.
Automated mathematical discovery pipeline. Derives and verifies 95 theorems across 10 branches of mathematics from three axioms.
Proof of the unit-norm identity N(ρQ) = −1 in the degree-12 compositum ℚ(ρ, Q).
Verification code for the algebraic derivation of Newton’s gravitational constant G to 0.003% of the CODATA value.
Verifies all six theorems of the arithmetic geometry paper and computes the Barbero–Immirzi parameter γᵢᵢ = λ₄ρ.
Computes the real period of the elliptic curve 1132b1, divides by 23, and reproduces αₛ(MЗ) = 0.11792 in under 30 seconds.
Stephanie Alexander
Independent Researcher & Author · University of Chicago (dual alumna)
Exploring fundamental mathematics, physics, and biology through AI-driven inquiry—guided by the conviction that foundational questions matter.
For research inquiries, collaboration proposals, or general correspondence.