Origin Note Coefficients 11D of Quantum G Theory

Abstract

This note documents, in detail and with explicit attribution, the provenance ofthree results used in Quantum G Theory: the weight \(g_s=4/3\) with which theFreund--Rubin energy enters the depth of the four-dimensional minimum, theweight \(w_{\rm int}=2/3\) of the internal flux energy, and the critical-pointtheorem \(V_{\min}=-\sum_i B_i\,(n_i-n_R)/n_R\,\rho^{-n_i}\) from which bothfollow ---an exact algebraic identity, not an approximation nor a perturbativeexpansion---. Those ingredients that are standard literature ---theKaluza--Klein reduction with breathing mode, the power-law potential for fluxcompactifications, the Freund--Rubin sign reversal at fixed charge, theMaldacena--Nu\~nez-type no-go and the Casimir channel--- are expressly separatedfrom what is own elaboration: the closed forms \(g_s=2(d-1)/(d+2)\) and\(w_{\rm int}=2(p-1)/(d+2)\), and the statement of the theorem in the form givenhere. Also documented are the normalization chain that fixes \(m=\tau_0/3\), thedistinction between \(w_{\rm int}\) and the flux-amplitude ratio \(\tau_0/g_1\)---two distinct objects with the same numerical value--- and the obstruction ofthe singlet flux ansatz at the level of the field equations. Every step isverified symbolically in the reproducibility package (groups \texttt{[K]} and\texttt{[L]}).