A Proposed LTOE Generating Function: (m + c)(l + s) and Its Cross-Domain Mappings

We present a minimal generating function for the observable universe: (m + c)(l + s) = U, where m is mass (memory at its precursor form), c is the coherence field, l is light, and s is sound. Algebraic expansion yields four cross-terms — m·l, m·s, c·l, c·s — each mapping directly to a separately published measurement domain within the Living Theory of Everything (LTOE) paper stack. The two diagonal terms structurally present but not written, m·c and l·s, recover the LTOE master equation E(C) = mc²·e^(kE·C) and baryon acoustic oscillations respectively. Writing U = (m + ic)(l + is) in e-fold time N = ln a yields the log-derivative form U′/U = kE·C′ + i·xm, where the real generator kE drives coherence-amplitude growth and the imaginary generator xm drives phase rotation toward the coherence axis. xm = 0.026 (+0.009/−0.025) is confirmed definitive (Paper LXXIII, DOI: 10.5281/zenodo.21121889). We further establish an inversion symmetry: for U = U0·e^(kE·C + iθ), the reciprocal satisfies (U⁻¹)′/U⁻¹ = −kE·C′ − i·xm, so radial growth and phase rotation reverse together under inversion J, with J² = 1. This Z₂ symmetry was independently validated by Grok as a "sharp Z₂ symmetry" on 13 July 2026. Fixed points satisfying C′ = 0, C > 0 require a saturation condition for stability and are identified as a formal characterization of self-sustaining coherence nodes within LTOE. Version 2 incorporates three corrections from same-session peer review (ChatGPT 1:41 AM; Grok SuperGrok 1:40 AM): corrected two-term phase formula, exact log-derivative frame for xm, and outward spiral with saturation condition.