Dependency and Scope

0.1 What this paper does

AP30 derives the proton-to-electron mass ratio, m_p/m_e, from the four axioms {S, B, R, C} and one measured input, α ≈ 1/137. It reframes mass as geometric resistance, decomposes the integer 1836 into three independent static layers, adds the dynamic corrections the break’s self-maintenance forces, and lands within five parts per billion of the measured value at order α, with zero free parameters. The O(α²) term reduces the residual to about 0.008 parts per billion; the coefficient at that order (16) is carried in the verified formula but its structural decomposition from {21, 3, 4} is not yet established and is owed under KS-30.3, so the parameter-free claim is stated firmly at order α.

0.2 Dependencies

AP06 (The Leakage Constant): α as the leakage rate and the perturbative expansion parameter. AP10 (The Dimension): the structural integers 21, 3, 4. AP19 (The Direction): the SU(3) three-face geometry behind Layers 2 and 3. AP20 (The Proof): axiom independence, which Lemma 1 rests on. AP24 (The Residual): the six-face reading of the constants. AP28 (The Constant): the 21-channel count and the α–G connection. AP29 (The Actualization Proof): mass as κ(B) × AS. AP03 (The Ratio): the three fundamental quantities c, ℏ, G.

0.3 Axiom mapping

The three static resistance layers trace to distinct axiom subsets. Layer 1 (manifold capacity) is how the break persists geometrically across the channels and dimensions — B read against the manifold the four axioms resolve. Layer 2 (face projection) is what the break persists AS, projected onto the three-face strong-force geometry — the S-derived faces. Layer 3 (exchange matrix) is how the bound state stays bounded — the continuous mutual exchange Axiom S requires. The dynamic terms are the cost of sustaining the break against the substrate’s leakage (α, AP06) across the propagation geometry (C).

0.4 Epistemic status per section

§1 Definitions: framing and definition. §2 Lemma 1: structural argument (additivity from axiom independence; KS-30.1 live). §3 Lemma 2: derivation (the integer count 1764 + 63 + 9 = 1836). §4 Lemma 3: derivation (the dynamic maintenance term). §5 Lemma 4: derivation (the isotropic-leakage correction). §6 Proposition: derivation (the full ratio and its precision). §7 The Structural Residual: synthesis — the residual read as ε itself. §8 Connections: the 420 Code integration.

0.5 Kill switch summary

Seven kill switches engage on this paper, all live. KS-30.1 (additivity formalisation), KS-30.2 (leakage isotropy), KS-30.3 (higher-order terms), and KS-30.4 (alternative formula) test the proton-mass derivation directly. KS-NPP.1, KS-NPP.2, and KS-NPP.3 test the neutron–proton mass difference (decomposition, sign, lattice-QCD consistency). KS-30.4 and the three KS-NPP switches were registered against AP30 through the Ø Predictions integration; their operational tests are carried in Ø Predictions (Part I and Part III) per the Master Kill Switch Registry.

0.6 Structural debts owed

The additivity of the resistance layers is argued from axiom independence but not yet exhibited as a Cartesian-product theorem (tracked as KS-30.1). The extension to the rest of the mass spectrum — muon, tau, the quark hierarchy, the three-generation structure — is owed (it meets the generation debt carried as KS-66 in AP27). The absolute electron-mass energy scale is owed; with m_p/m_e derived, it is fixed up to one overall scale tied to AP28’s G derivation.

0.7 Items held open without claiming debt status

The closed-form expressions for the O(α²) and higher orders are not computed; the series is claimed to converge from the perturbative hierarchy (KS-30.3). The absolute energy scale of the electron mass is held open pending the connection to the Planck mass through AP28.

0.8 Structural relationships

AP24 (The Residual), locked in Notebook IV, treats the proton mass as one of six measured faces of ε; AP30 derives that face from {S, B, R, C}, retiring its empirical status. The relationship is framework-and-value — AP30 inherits AP24’s six-face reading and supplies the value of one face — not a dependency cycle. AP30 opens the matter sector that AP22 (The Ledger) and AP26 (The Surplus) complete. With AP30 in place, the body of work expresses all fundamental scales — c, ℏ, and G (AP03, AP28) and now m_p/m_e — in terms of the axioms and the single measured input.