The κ0 RC = κ0 LQC Hypothesis: An Operational Falsification Protocol (Paper 7 of the LQG–LQC Intertwiner Series)

1 Introduction 1.1 TheroleofHypothesis(H)intheseriesTheLQG–LQCIntertwinerSerieshas, throughPapers I–V, establishedaparameter-freespectral theoryof theLoopQuantumCosmologybouncefromSU(2)geometryalone. ThecentraloutputisthebouncewindowfunctionW(κ)= κ3(κ2−2)Csinh(πκ/2) , κ≥κ0=√2,whereC=3.6032935039138... istheHillardconstant(Paper8)andκ0=√2istheexactspectralthresholdderivedfromtheSU(2)Casimirk(k+1)=2atspink=1.Paper 6provedthat thevonNeumannalgebraMLQCgeneratedbythe cosmologicalperturbationmodesaboveκ0 is theuniquehyperfiniteType III1 factor,withKMSstateat inverse temperatureβB=π√2andmodularflowσt(a(κ))=e−iκta(κ). Paper6thenconstructedaparallelalgebraMRCfortheRecognitiveConsciousness(RC)Framework—anindependentlymotivatedalgebraicmodelofconsciousrecognition(see[16] fortheformalmathematical foundationsand[15] fortheempiricalevidence)inwhichencountersbetweenquantumsubsystemsAandBgeneratemodesabovearecognitionthresholdκ0RC>0.2The central claim of Paper 6 — that MLQC = MRC as concrete von Neumann algebras(Corollary 4.3) — is conditional on Hypothesis (H):Hypothesis (H). The recognition threshold of the RC Framework and the spectral thresholdof the LQC bounce coincide:κ0RC = κ0LQC = √2.This identification has the status of a physical proposal, not a derived result. Paper 6 isexplicit: “Hypothesis (H) is a falsifiable physical proposal, explicitly not a derived result.The falsification protocol is the subject of Paper 7 [in preparation].”The present paper is that protocol.1.2 Why a dedicated falsification paper is neededHypothesis (H) is not falsifiable as a bare mathematical statement: both κ0RC and κ0LQC aredefined within their respective frameworks, and until κ0RC is given an operational measurement procedure, the claim κ0RC = √2 has no empirical content.The challenge is that κ0RC is defined within the RC Framework, which describes recognition encounters between quantum subsystems as a priori relational — not accessible todirect spectroscopic measurement in the way that, say, a CMB multipole is. An operationalfalsification protocol must therefore:1. Identify empirical proxies for κ0RC within the RC Framework’s own observational program;2. Map those proxies to a numerical estimate of κ0RC;3. State predictions that follow specifically from κ0RC = √2 (as opposed to any otherthreshold value); and4. Specify decision criteria that distinguish confirmation from falsification from no-signal.This paper executes each of these steps in turn.1.3 Structure of the paperSection 2 restates Hypothesis (H) in its full algebraic context and identifies the preciseconsequences of each outcome for the broader program. Section 3 operationalizes κ0RC fromthe RC Framework’s empirical signature system. Section 4 states four falsifiable predictionsP1–P4 that are specific to the value √2. Section 5 presents the full falsification protocol.Section 6 presents Table 3 — the three-outcome decision structure cited in Paper 6. Section 7discusses implications and open questions.