Dependency and Scope

0.1 — What this paper does

AP03 derives the conjugacy of the two fundamental speed limits of physics — c and G — from a single symmetry-breaking event with amplitude ε. Both constants emerge from the same break. Neither is independent. Neither is fundamental on its own.

Three claims, each derived or structurally forced.

First. At the moment of symmetry breaking, only the breaking amplitude ε is available as a scale. No spacetime, no mass-energy, no pre-existing coupling — nothing has emerged yet. Both c and G must therefore inherit from ε. This is a structural argument from the bootstrap problem: you cannot use the rules of the house to explain how the house was built.

Second. The minimal break produces exactly two sectors. The pre-state has ℤ₂ symmetry — the simplest non-trivial symmetry group. By the representation theory of ℤ₂ (section 8), the broken phase decomposes into exactly two irreducible sectors: a propagation sector (Mode 1) carrying c, and a resistance sector (Mode 0) carrying G. Not three. Not one. Two.

Third. The cosmological origin and the black hole event horizon are the same marginal surface — the surface where ε locally returns to zero — traversed in opposite temporal directions. The Big Bang is the break opening; the event horizon is the break closing. Hawking radiation is the universe refusing to permit a permanent pocket of restored symmetry within a globally broken phase.

The paper makes one further structural claim, qualitatively. The dark-to-visible energy ratio (~95:5) is a consequence of small ε: the smaller the break, the more content remains in Mode 0 (the resistance sector), the less spills into Mode 1 (the propagation sector that carries electromagnetic interaction). The qualitative direction is robust; the quantitative mapping requires a model of the initial fluctuation spectrum that is not derived here.

0.2 — Dependencies

The axioms {S, B, R, C} (formal statement carried at AP20 section 3.3; completeness and minimality obligations fenced at KS-P.1 / KS-P.2). AP03 was originally written before the canonical formalisation; the mapping from AP03’s postulates onto the canonical axioms is made explicit in section 0.3 below. Axiom S is Symmetry; AP03’s E0 specifies the minimal symmetry. Axiom B is the unique break; AP03’s E1 instantiates it as the spontaneous breaking ℤ₂ → trivial with amplitude ε. Axiom R is the persistence of the record; the broken vacuum is what persists. Axiom C is Constraint — locality; the bounded propagation it produces is why c is finite, G is positive, and ε is small and bounded.

AP20 (The Proof). Load-bearing for the conjugacy upgrade. AP20’s Paper D proves Axiom B forces exactly one free parameter. With this proof in place, the AP03 conjecture E-G (G depends on ε) becomes a theorem consequence: G cannot be independent of ε, because if it were, the axioms would admit at least two free parameters. AP20 takes AP03’s central conjecture and makes it structurally necessary.

AP06 (The Leakage Constant). Load-bearing for the numerical identification of ε. AP06 derives the cycle form 1:1 + 1×ε @ AS [+1/137 / −1/137] and identifies ε numerically with the fine-structure constant α ≈ 1/137. The gravitational hierarchy ke²/(G·m_e²) ≈ 10⁴²–10⁴³ verification in section 7.5 uses ε ≈ 1/137 from AP06.

AP24 (The Residual). Provides the constraint Λ that pins the dependence of mₚ on ε through the relation mₚ ∝ √ε · √Λ. AP03 derives the proportionality mₚ ∝ √ε; AP24 fixes Λ so the relationship is dimensionally complete.

AP08 (The Identity). Dependent on AP03, not the reverse. AP08 derives Einstein’s field equations from the record algebra; the gravitational coupling G that appears in AP08’s field equations is the same G that AP03 conjectures (and AP20 forces) to be determined by ε. The disclosed circularity in AP08’s G = 2κ/m_e² is structural debt D1 of The 420 Code, substantially reduced by the AP20 forcing proof + the section 7 derivation but not fully closed.

0.3 — Axiom mapping

AP03 was written before the canonical axiom set {S, B, R, C} was formalised. The mapping below shows how AP03’s postulates instantiate the four axioms.

Axiom S (Symmetry) ↔ Postulate E0. S is the two-sector base condition — identity under exchange, the precondition for distinction to be meaningful. E0 specifies the minimal symmetry of the pre-state: ℤ₂, the simplest non-trivial group. The pre-state whose existence Axiom S asserts is what E0 endows with the minimal symmetry capable of nontrivial breaking. AP03 chooses ℤ₂ by parsimony; the choice is tested via KS-R.2 (binary decomposition) — a richer pre-state symmetry would produce more than two irreducible representations, so any falsification of the binary decomposition is also a falsification of ℤ₂ minimality.

Axiom B (Exactly one break) ↔ Postulate E1. B requires exactly one irreducible symmetry-breaking event. E1 instantiates this as the spontaneous breaking ℤ₂ → trivial with amplitude ε. The minimality of ℤ₂ ensures the break is irreducible: there is no intermediate subgroup between ℤ₂ and the trivial group. AP20’s forcing proof on Axiom B is what upgrades AP03’s E-G conjecture (G depends on ε) to a theorem.

Axiom R (The record persists) ↔ The broken vacuum. R requires that the break leave a persistent record. In AP03, the broken vacuum with propagation speed c = √(β/α) and condensate energy density ρ₀ = ε²Λ⁴/(4λ) is the persistent record. The two sectors (Mode 0, Mode 1) persist because the broken vacuum is stable; stability is what makes the record a record.

Axiom C (Constraint) ↔ Finiteness of c and positivity of G. C requires that all quantities remain bounded. In AP03: c is finite (set by the substrate ratio β/α); G > 0 (theorem from E-G: G = κ/(εΛ²)); ε itself is small and bounded.

Note: AP03’s postulates are more specific than {S, B, R, C}. They choose a particular symmetry group (ℤ₂) and a particular mechanism (SSB of a scalar field). The axioms {S, B, R, C} are the abstract structure; AP03 is one specific instantiation.

0.4 — Epistemic status per section

section 1 (The Two Speed Limits). FRAMING. Names the two structural speed limits of physics and proposes they are siblings.

section 2 (Before the Arena). STRUCTURAL ARGUMENT. The bootstrap problem and the forcing of ε as the only available scale at the moment of symmetry breaking. Claim E-Bootstrap.

section 3 (The Minimal Break). DERIVATION. The ℤ₂ representation theorem applied to the broken phase. Postulate E0; Theorem 3.1 (proof in section 8); Claim E2 with KS-R.2.

section 4 (The Conjugacy of c and G). DERIVATION + STRUCTURAL CRITIQUE. Planck units as evidence of conjugacy; the derivation of c from the pre-geometric action (Result A.1); Theorem E-G (G = κ/(εΛ²)) via AP20 forcing proof + the section 7 derivation; the dimensionless formulation as the correct quantitative target.

section 5 (The Origin and the Horizon). DERIVATION. The structural identity Claim E-Identity: cosmological origin and event horizon as the same marginal surface, traversed in opposite directions.

section 6 (The Dark Sector Connection). QUALITATIVE FRAMING. Conjecture E5 that the dark-to-visible energy ratio is determined by ε, qualitatively consistent but not quantitatively derived.

section 7 (The Action Principle). FORMAL DERIVATION. The minimal ℤ₂-symmetric pre-geometric action; Result A.1 (Emergence of c); the form of G’s dependence on ε.

section 8 (The ℤ₂ Representation Theorem). FORMAL PROOF. Theorem B.1 (the only two irreducible representations of ℤ₂ are the trivial and the sign representation).

section 9 (Kill Switches). FORMAL FALSIFICATION REGISTER. Six kill switches engaged: KS-R.1 through KS-R.6.

section 10 (What This Establishes and What Remains Open). SYNTHESIS.

0.5 — Kill switch summary

Six kill switches are engaged in this paper. The full register is in section 9.

KS-R.1: Independent variability of c and G (gravitational conjugacy). LIVE — EMPIRICAL.

KS-R.2: Non-binary origin. Universe requires more than two fundamental sectors. LIVE — EMPIRICAL.

KS-R.3: Dark ratio independence (mechanism unrelated to c/G). LIVE — EMPIRICAL.

KS-R.4: Horizon inequivalence (cosmological origin and black hole horizon structurally distinct). LIVE — HARD.

KS-R.5: Planck scale mismatch (Planck onset incompatible with mₚ ∝ √ε). LIVE — EMPIRICAL.

KS-R.6: Dimensionless scaling violation (α · α_G at fixed mass varies with ε, or α/α_G hierarchy is not consistent with α²). LIVE — HARD.

0.6 — Structural debts owed

Two structural debts are named explicitly. These are unresolved derivations tied to specific structural claims, tracked at the registry level.

D1 (registry-level, partially closed by AP08 + this paper). The first-principles derivation of G from the axioms without external scaffolding. AP08 derives Einstein’s field equations from the record algebra and identifies G through The Lock (G = 2κ/m_e²), with the disclosed structural circularity that κ is itself defined as Gm_e²/2. AP03 contributes by establishing that G cannot be independent of ε (via AP20’s forcing proof) and by deriving the explicit form G = κ/(εΛ²) from the structural condensate density of section 7. The induced-gravity calculation used in section 7.5 is named in the literature as Sakharov-Visser; it leans on the standard QFT cutoff structure. The 420 Code does not import that calculation as its source — the load-bearing inputs are the AP20 forcing proof and the structural condensate density, both internal to The 420 Code; the literature calculation is correspondence, confirming the same form from a different route. The closure is partial. The full first-principles derivation — G computed from {S, B, R, C} alone, with the structural coefficient κ derived from the broken vacuum response without any reliance on the standard cutoff machinery — remains open. This debt is registry-level, not AP03-specific; AP03’s contribution to the partial closure is recorded here.

D-AP03.1. The substrate parameters α and β in the kinetic term ℒ = ½α(∂_τφ)² − ½β(∇φ)² − V(φ) are introduced but not derived from a more fundamental theory. Their ratio gives c² = β/α, but neither α nor β individually is computed from the axioms. This is the deepest open problem in AP03 — distinct from and prior to the G derivation. A truly pre-geometric formulation would derive the distinction between temporal and spatial coordinates from the axioms rather than presupposing it in the kinetic term.

0.7 — Items held open without claiming debt status

Two further items are bracketed inside this paper. They are real and live, but they are not structural debts in the same sense as D1 and D-AP03.1.

E5 quantitative. The dark-to-visible energy ratio as a function of ε. The paper establishes the qualitative direction (small ε → most content in Mode 0) but does not derive the quantitative mapping. Three issues need resolution before the connection becomes quantitative: the phenomenological distinction between dark matter (clusters gravitationally) and dark energy (does not) within Mode 0; the epoch-dependence of the 95:5 ratio over cosmological history; and the specific functional form linking ε to the partition. This is open derivation work; the qualitative claim stands.

E-Hawking sufficiency. Hawking radiation is structurally motivated in section 4 as the necessary consequence of imposing a global ε > 0 condition on a locally ε → 0 region. The paper does not derive the Hawking spectrum from the AP03 action; it argues that the global condition forces leakage and identifies the spectrum derived elsewhere with that leakage. The empirical test of the connection rides on KS-R.4 (horizon inequivalence): if the cosmological origin and the black-hole horizon are shown to be structurally distinct, the structural account of Hawking radiation falls with it.

0.8 — Structural relationships

AP03 opens Notebook II of The 420 Code. AP05, AP10, and AP08 follow as the remaining chapters of Notebook II — the structure of space and time, the dimensional count, and Einstein’s field equations — all inheriting the structural setting AP03 establishes: c and G as conjugate outputs of a single break.

AP20 (Notebook I) upgrades AP03’s conjecture E-G to a theorem via the forcing proof on Axiom B. With AP20 in place, the conditional claims (E4, E-Identity, E-Hawking) inherit the upgrade. AP03 returns the favour by giving AP20’s forcing proof its first quantitative payoff outside the foundational proof itself.

AP06 (Notebook IV) identifies ε numerically with α ≈ 1/137. AP24 (Notebook IV) constrains Λ. Together they pin AP03’s scaling relations to specific numbers: the gravitational hierarchy ke²/(G·m_e²) ≈ 10⁴²–10⁴³, mₚ in terms of ε and Λ, the Planck scale as the scale of the break itself.

AP17 (Notebook VI) relates the 0-pole / 1-pole structure of cosmological geometry to AP03’s Mode 0 / Mode 1 sectors. AP18 (Notebook VI) derives the MOND floor a₀ ≈ cH₀ from the axioms, downstream of the small ε.