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| Main Authors: | , |
|---|---|
| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
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| Subjects: | |
| Online Access: | https://doi.org/10.5281/zenodo.20086492 |
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Table of Contents:
- <p>1 Introduction<br> The aim of this paper is to rewrite a previously comparative and partially conjectural bridge between<br>two frameworks as a mathematically controlled reduced-sector result. The rst framework is the<br>generating-law program, whose scalar branch K(λ) is known to satisfy an exact cubic algebraic<br>relation together with a Möbius trans-series completion law. The second framework is Recognitive<br>Consciousness (RC), formulated operator-algebraically through conditional expectations, modular<br>ow, and KMS equilibrium.<br>Remark 1.1 (On the cubic generating law). The cubic stationarity equation (2) was rst identi ed<br>in Paper 4 [1] as the equation governing the spectral peak κmax(λ) of the C(λ) family. The present<br>paper derives it from the quartic reduced potential (1), providing a new intrinsic derivation from the<br>RCaxioms that does not require the generating-law spectral program as input. The two derivations<br>produce the same cubic, con rming their consistency.<br>The central claim defended here is not that the two full theories are already proved identical as<br>unreduced operator-algebraic objects. That stronger statement remains open. The claim proved<br>here is narrower and rigorous:<br>Within the intrinsic one-scalar reduced recognition sector of RC, the unique minimal<br>2<br>reduced spectral law is the quartic potential whose stationarity equation is exactly the<br>cubic generating law.<br>This is the mathematically stable point at which the bridge becomes genuinely useful. It yields<br>an explicit scalarization of recognition depth, an exact modular derivation of the Möbius completion<br>factor, a proof of the minimality principle in the reduced sector, and an intrinsic explanation of the<br>coe cient shift 2λ + 3 = (2λ+1)+2.<br>Organisation. Section 2 isolates the hypotheses imported from the companion RC manuscripts.<br>Section 3 proves the canonical scalar lift. Section 4 derives the one-mode modular kernel and Möbius<br>law. Section 5 proves the minimality theorem for the intrinsic one-scalar reduced sector. Section 6<br>constructs the quartic potential and recovers the cubic generating law. Section 7 proves the reduced<br>sector bilateral-operator theorem. Section 8 presents the unreduced candidate spectral action and<br>one-mode compression test. Section 9 states the precise scope, signi cance, and remaining open<br>problems.</p>