Foundations: The Triangle
From the raw arithmetic substrate to the polynomial that encodes the universe. This path covers everything you need before branching into particles, forces, or cosmology.
What you'll understand after this path
What the UGP state space is, why the survivor space is the valid subset, and how the Chinese Remainder Theorem encodes a tape in one number.
How p(L,C,R) = C+R−CR−LCR updates a tape,
why it equals Rule 110 on binary inputs, and why that is a theorem not a design choice.
How the UGP arithmetic, the GTE orbit, and the polynomial are three descriptions of the same object — and what that means for physics.
Lessons
The Problem of Physics
Why does the Standard Model have 25 free parameters? What would it mean to derive them all from one rule?
The Principle: PSC and MDL
Perfect Self-Containment and Minimum Description Length — the two ideas that force a unique physical substrate.
The GTE Polynomial Step by Step
What the formula p(L,C,R) = C+R−CR−LCR does, computed by hand with actual tapes in both binary and 7-state systems.
The UGP State Space and Survivor Space
What a cell state is (a prime residue), what the survivor space filters out, and why the UWCA is native — not imposed.
How the UWCA Works
The four-pass sweep (P1–P4), the penalty function, and how one sweep implements exactly one step of Rule 110.
GTE Orbits: From Arithmetic to Particle Masses
The canonical orbit (1,73,823)→(9,42,1023)→(5,275,65535), the ridge, and how three numbers predict particle masses.
The Triangle: One Substrate, Three Views
How UGP arithmetic, the GTE orbit, and the polynomial p are three descriptions of the same algebraic object — machine-certified.
No prerequisites for this path. The lessons build from scratch — residues are explained, primes are explained, cellular automata are explained. If you can do arithmetic, you can follow Path 1.