1 Introduction The aim of this paper is to rewrite a previously comparative and partially conjectural bridge betweentwo frameworks as a mathematically controlled reduced-sector result. The rst framework is thegenerating-law program, whose scalar branch K(λ) is known to satisfy an exact cubic algebraicrelation together with a Möbius trans-series completion law. The second framework is RecognitiveConsciousness (RC), formulated operator-algebraically through conditional expectations, modularow, and KMS equilibrium.Remark 1.1 (On the cubic generating law). The cubic stationarity equation (2) was rst identi edin Paper 4 1 as the equation governing the spectral peak κmax(λ) of the C(λ) family. The presentpaper derives it from the quartic reduced potential (1), providing a new intrinsic derivation from theRCaxioms that does not require the generating-law spectral program as input. The two derivationsproduce the same cubic, con rming their consistency.The central claim defended here is not that the two full theories are already proved identical asunreduced operator-algebraic objects. That stronger statement remains open. The claim provedhere is narrower and rigorous:Within the intrinsic one-scalar reduced recognition sector of RC, the unique minimal2reduced spectral law is the quartic potential whose stationarity equation is exactly thecubic generating law.This is the mathematically stable point at which the bridge becomes genuinely useful. It yieldsan explicit scalarization of recognition depth, an exact modular derivation of the Möbius completionfactor, a proof of the minimality principle in the reduced sector, and an intrinsic explanation of thecoe cient shift 2λ + 3 = (2λ+1)+2.Organisation. Section 2 isolates the hypotheses imported from the companion RC manuscripts.Section 3 proves the canonical scalar lift. Section 4 derives the one-mode modular kernel and Möbiuslaw. Section 5 proves the minimality theorem for the intrinsic one-scalar reduced sector. Section 6constructs the quartic potential and recovers the cubic generating law. Section 7 proves the reducedsector bilateral-operator theorem. Section 8 presents the unreduced candidate spectral action andone-mode compression test. Section 9 states the precise scope, signi cance, and remaining openproblems.
Hillard et al. (Thu,) studied this question.