The Atlas: Phase-Locked Geometric Neural Manifolds on Mixed-Curvature Hyperbolic–Toroidal Product Spaces

Abstract

We give a complete formal specification of Atlas, a phase-locked geometric representation substrate whose state space is the mixed-curvature Riemannian product M= Hn ×Tn. The hyperbolic factor carries semantic lineage (exponential volume growth, low-distortion hierarchy); the flat toroidal factor carries cyclic rhythm, its fundamental group π1(Tn)∼= Zn supplying integer winding charges provably immune to continuous metric perturbation. We (i) construct the composite geodesic metric dM = whd2H + wt d2T with golden-ratio φ−k angular decay and prove its total weight mass is exactly φ; (ii) prove homotopy-invariance of the discrete winding vector under bounded perturbation and formalize dual-chart disagreement as the separator of lineage drift from harmonic rhythm over the lemniscate lobes; (iii) derive the asymmetric log-odds regulator κ = κ(1−κ)z, whose exact solution is the logistic sigmoid — an intrinsic (0, 1) interlock — and prove the φ2 ≈2.618×penalty asymmetry under the conservation identity SC·SR = s2;(iv) derive the superposition holding loss Lk = exp(ρDk)−1, prove (Lemma 3) the strict bound supk Lk < e−1 ≈1.71828 for all |vk|≤1, ρ ∈(0, 1), and prove Theorem 9 showing > 38.2% reduction in drift settling time (T Atlasdrift ≤1/ϕ T Euclideandrift); and (v) specify the post-quantum Rosen Bridge: ML-KEM-768 + X25519 hybrid KEM with HKDF-SHA-256 transcript binding, the COROS envelope (AES-256-GCM + AES-CTR whitening + constant-time 256-byte padding), and a counterfree forward window Ω= 32 on the hyperbolic geodesic ratchet with a qΩ2−128 forgery bound. Every gate is a checkable inequality.