The Ratio

This is Artist’s Proof 03 of The 420 Code. It opens Notebook II — Spacetime. The paper asks a question prior to physics: how do the fundamental constants that define the arena arise?

Specifically, are c and G independent constants, or are they conjugate expressions of a single symmetry-breaking event?

Eight sections carry the work. Section 1 names the two speed limits and proposes they are siblings. Section 2 shows that at the moment of symmetry breaking, only ε is available as a scale, and both c and G must inherit from it. Section 3 proves the minimal break produces exactly two sectors via the ℤ₂ representation theorem. Section 4 derives c from a minimal pre-geometric action and lands the conjugacy of c and G. Section 5 identifies the cosmological origin and the black hole event horizon as the same marginal surface ε = 0, traversed in opposite directions. Section 6 connects the small ε to the observed dark-to-visible energy ratio. Sections 7 and 8 carry the formal mathematics: the action principle and the ℤ₂ representation theorem.

AP03 is logically independent of AP01 and AP40. Its two postulates (E0, E1) are self-contained. Its derivations require only standard representation theory and scalar field theory. The paper can be read on its own; within The 420 Code, AP20 adds the structural upgrade that turns the conjugacy of c and G from conjecture to theorem.

A note on status. AP03 was originally published as a conjecture: c was derived from the pre-geometric action; G was conjectured to be determined by ε but not derived. The central open problem was the derivation of G. That problem is structurally resolved. AP20’s forcing proof — Axiom B forces exactly one parameter — makes the conjecture E-G a theorem consequence: G cannot be independent of ε. The explicit scaling G = κ/(εΛ²), equivalently c/G ∝ ε (correcting AP03’s original c/G ∝ ε²), was derived in subsequent work from the structural condensate density of section 7 and verified numerically. The conditional claims (E4, E-Identity, E-Hawking) inherit the upgrade. A note on level: AP03’s G = κ/(εΛ²) is a substrate-level scaling — it fixes how G responds to ε with the loop coefficient κ and the cutoff Λ free, anchored at order of magnitude (the dimensionless hierarchy α/α_G(e) ≈ 10⁴³). It is superseded as the statement of G’s value by AP28 (Notebook IV), which derives G = α_em²¹(1 + 1/π)ℏc/m_e² parameter-free (G ∝ ε²¹, 0.69% from measurement) and recovers the same hierarchy as α_em⁻²⁰/(1 + 1/π). The two operate at different architectural levels (substrate condensate vs. Mode-0 channel count) and are reconciled, not in conflict. G ∝ ε²¹ is canonical for the value; AP03’s ε⁻¹ scaling is retained as the substrate-level derivation AP28’s count refines.

Six kill switches are published with this paper. Every load-bearing claim attaches to a structural condition under which the claim would fail. The 420 Code is published copyleft. Free forever. No paywall. No gatekeepers.

Where this paper fits in The 420 Code

The 420 Code is organised across eight Notebooks. This paper opens Notebook II — Spacetime. The chapter sequence runs AP03 (this paper) → AP05 (The Break) → AP10 (The Dimension) → AP08 (The Identity).

Notebook II derives the structure of space, time, and gravity from the axiom. Notebook I (The Premise) established the axiom, named the four conditions {S, B, R, C}, and proved the Embedding Hypothesis. With EH proven, Notebook II runs unconditionally. The fundamental constants c and G, the metric signature, the dimensional count N = 3, and Einstein’s field equations all follow from {S, B, R, C} once the conditional has been removed.

AP03 sits first in Notebook II because it does the constants. Without c and G, the arena does not exist. The chapter shows that c and G are not independent inputs but conjugate outputs of the same break — both determined by the single breaking amplitude ε. This sets up the rest of Notebook II: AP05 walks the seven-step chain from the axiom to the obligation and reformulates the cosmological identity as junction conditions, AP10 derives N = 3, AP08 derives Einstein’s field equations. Each chapter inherits AP03’s structural setting that both speed limits come from the same source.

AP03 is logically independent of AP01 and AP40 — its two postulates (E0, E1) are self-contained and its derivations require only standard representation theory and scalar field theory. The paper can be read on its own. Within The 420 Code, AP20 adds the structural upgrade: AP20’s forcing proof on Axiom B turns the conjugacy of c and G from conjecture to theorem.

How to read this paper

The reading order is front to back. Section 0 names the dependencies and the scope. Sections 1 through 8 carry the structural argument and the formal mathematics. Section 9 is the falsification register. Section 10 is the synthesis — what this paper closes, what it leaves open, where the philosophical-register companions live, and how The 420 Code reads with AP03 in place.

Two voices alternate as the material requires. Narrative passages — openings, transitions, body anchors, the bootstrap problem, the Planck-units whisper, the Big Bang as door — speak directly. Formal sections — postulates, theorems, the action, kill switches — speak in the precision the structure requires. The gear-changes are intentional.

A note on what this paper does and does not do. AP03 derives c from a pre-geometric action, proves the binary decomposition of the broken phase from the ℤ₂ representation theorem, identifies the structural identity of the cosmological origin and the black hole event horizon as the same marginal surface ε = 0, and lands a qualitative connection between small ε and the observed dark-to-visible energy ratio. It does not derive G from first principles within its own action — the derivation runs from the structural condensate density of section 7 plus AP20’s forcing proof. It does not predict the quantitative dark-sector partition; only the qualitative direction (small ε → most content in Mode 0). The substrate parameters α and β in the kinetic term are introduced but not derived from a more fundamental theory. What the paper does is install the conjugacy of c and G as a structural fact and locate the cosmological and black-hole boundaries on the same marginal surface.