The Margin Carry Law

Abstract

The Margin Carry Law A Falsifiable Cross-Octave Law for Structural Margin PreservationOctober 2026By Kevin L. Brown This paper addresses a restricted question. Given a system organized into adjacent scale levels — termed octaves, where each level is constructed from, and constrained by, the level below it — what lower bound can be stated for the structural margin that remains after transport from one level to the next? The paper proposes such a bound, states the conditions under which it is asserted, and specifies the criterion under which it is to be rejected. The structural margin δ_o at octave o is defined as the fraction of that level's capacity that remains uncommitted after the level's internal obligations are met. The Margin Carry Law consists of two clauses. Clause (i) bounds the margin at the receiving octave from below: δ_{o+1} ≥ δ_o · ρ_o, with ρ_o = m_o · A_{o+1} / (A_{o+1} + D_map,o), where m_o is the fraction of margin retained through the transport step, A_{o+1} is the intake total at the receiving octave, and D_map,o is the mapping-distortion term for the transition from octave o to octave o+1. Clause (ii) bounds the receiving octave's cumulative intake over its lifetime: Σ_n B_{o+1}(n) ≤ A_{o+1}, where B_{o+1}(n) is the intake drawn at turn n, summed over completed turns with the terminal partial turn included as specified in the paper. Under the stated definitions the carry factor satisfies ρ_o < 1, so the law does not permit transport to generate margin. Iterated over a chain of octaves, clause (i) yields a product bound: the margin at the terminal octave is bounded below by the initial margin multiplied by the carry factors of the individual transitions. The law is proposed, not proved. The paper distinguishes its contents by status throughout — derived, proposed, computed, and open — and asserts no universal transport theorem. A countermodel presented in the paper shows that the formation of a two-octave structure does not by itself imply clause (i), so the law is not a consequence of the definitions alone. Its falsification criterion (K-SCALE) is stated in advance: a built instance that satisfies the paper's stated conditions and whose receiving-octave margin falls below the carry floor falsifies the law within its stated domain. The supporting computation is a single built two-octave instance evaluated in exact rational arithmetic. In that instance the retention factor is m ≈ 0.959814, the required floor is δ_o · ρ_o ≈ 0.463014, and the receiving octave's margin is δ = 55311/114658 ≈ 0.482400, so clause (i) holds with a positive surplus of approximately 0.0194 over the floor. The instance value is a designated record value (record y′_S[MI-G-001], card MC-G-001, observation horizon N_obs = 3) and is reported together with the defining data on which it depends, including the formation threshold convention adopted in the paper. A verification script accompanies the record and recomputes the rational quantities reported for the instance, including the formation thresholds, the corrected intake shares, and the countermodel values. The principal limitation is stated explicitly. Flat channels — zero modes, directions along which a level carries no structure that the transport step can act on — are not resolved by the present construction. A computation reported in the paper gives near-misses of approximately 0.0063 against the floor in the zero-mode case; this computation is not a built counterexample and does not satisfy K-SCALE. The zero-mode case, multi-transition (multi-hop) tests, and positive-cost model instances are listed in the paper's open register. Status: a self-contained, falsifiable proposal with one built instance and a named open edge; independent checking and attempted falsification are the intended next steps. Keywords: scale transport; structural margin; cross-octave bounds; falsification criteria; exact rational arithmetic; iterated bounds; zero modes