Coherence Selects No Transverse First-Order Dynamics: Presymplectic Identification of the Space of Cuts, Canonical 3-Geometries (Flat Lagrangian Torus, Geodesic Berger Spheres, Never Einstein), Flat Non-trivial Holonomy, and a Geodesic Inertial Law without Arrow — RCD Program, Act I, Technical Note I-F0 (v0.1)
Year: 2026
Type: article
Abstract: Technical Note I-F0 of the RCD program, Act I (continues I-D/I-E0, DOI 10.5281/zenodo.24024984; I-C, 10.5281/zenodo.24023903; I-B1, 10.5281/zenodo.23990720; I-B0, 10.5281/zenodo.23965760; Act I, 10.5281/zenodo.23961107). Question: can the geometry of the space of cuts derived from coherence alone, (M_d, ρ, G₀), in I-E0 select by itself a non-trivial dynamics and an internal notion of duration — and if not, which exact ingredient is missing? Finite bank, 17 control states (d = 2, 3, 4); brief by ChatGPT, addendum with predictions frozen and hashed before running; run 69/69 (a first run with three failures is archived with its diagnosis); an independent run by ChatGPT with its own code and method (45/45) matches every common figure. Results. (I-F0A) Presymplectic identification: the orbit of cuts is exactly the moment-map fibre 𝒪_ρ = μ⁻¹(p) of the central torus, p_a = n_aλ_a; the restricted Fubini–Study form has rank 2Σ(n_a − 1) and kernel of dimension |J| − 1 (relative phases); quotient ∏(ℂP^{n_a−1}, p_a ω_FS) (Marsden–Weinstein, inherited); Kähler only with one active block, a Lagrangian torus for non-degenerate states. (I-F0B) No-Transverse Equivariant First-Order Dynamics: every G₀-equivariant vector field on 𝒪_ρ, state-dependent or not, is tangent to the kernel (proof by isotropy); corollary: every H = f(ρ), including the modular flow K = −log ρ, generates a Killing flow inside the kernel; the same fixed algebra M_d^{G₀} = W*(ρ) that forbids a new cut forbids a transverse first-order dynamics. (I-F0C) The three 3-geometries of the bank are a flat torus diag(p) − ppᵀ and geodesic spheres of ℂP² of radius arccos√λ₂ (Berger): Ric = (2/p + 2/3)I + diag(4/3, 4/3, −8/3), R = 2 + 6/p, universal shape–phase gap 4 (Gauss equation) and Einstein defect 32/3 — never Einstein; derived symbolically from the immersion and checked against the bank by finite differences. (I-F0D) The torus has zero symplectic form but a flat, non-trivial phase holonomy 2πΣm_a p_a and a non-zero extrinsic Bargmann phase; ambient geodesics stay inside the torus iff the spectrum admits a balanced partition Σ s_a p_a = 0. (I-F0E) The canonical metric does select a second-order geodesic inertial law, transverse but reversible: coherence selects neither its initial momentum nor its orientation. The no-go is therefore against autonomous first-order dynamics, not against all dynamics; what is missing for time is the initial momentum, its orientation and a causal signature. Inherited mathematics (Marsden–Weinstein, Takagi, Cecil–Ryan, Bohr–Sommerfeld, action–angle) is declared as such; what is RCD's own is the chain coherence → canonical space of cuts → concrete fibre → common obstruction. No physical space, time or gravity is claimed. Scripts with SHA-256 manifest; all run incidents and all charges of the adversarial rounds documented. English and Spanish versions; the Spanish text prevails.
Author: Cerezo García, Arturo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225646617
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