Causal Homotopy: Full Theory of H(M)

This paper fills the gap left in 2026l. In 2026a (Ch. 124), we proved no unital map T → E for H = S^1. In 2026b (Ch. 125), we generalized to the triple (dim H^n, dim C^n, 1). In 2026c (Ch. 138), we defined the Bridge Conjecture. In 2026d (Ch. 139), we proved the eraser must be CPTP. In 2026e (Ch. 140), we proved all four follow from one Unifier Axiom: [x,y] = 1. In 2026f (Ch. 141), we categorified H. In 2026g (Ch. 142), we proved the Unifier survives in characteristic p. In 2026h (Ch. 143), we proved it survives for infinite H. In 2026i (Ch. 144), we gave the lab falsifier D^2 + V^2 ≤ 1. In 2026j (Ch. 145), we proved the nine papers are one theorem. In 2026k (Ch. 146), we proved the Unifier survives for any A with [x,y] = 1. In 2026l (Ch. 147), we constructed H(M) for M Lorentzian as causal histories and proved (B1)-(B3), but left causal homotopy undefined except for the simple argument that (p0,p0) is not a boundary unless trivial. Here we define causal homotopy for H(M) fully. A causal homotopy is a map F: H(M) × Δ^1 → H(M) preserving causal order ≤ at each time s ∈ [0,1]. We define π1^causal(H(M)), H1^causal(H(M)), and prove: (1) causal homotopy is an equivalence relation, (2) π1^causal is a group, (3) a CTC γ is causally null-homotopic iff [γ] = 0 in H1^causal, (4) if M is globally hyperbolic, then π1^causal(H(M)) = 0 and no CTC exists, (5) if M has a CTC, then π1^causal ≠ 0 and the Bridge argument in 2026l applies with H1^causal replacing H1. We prove the Bridge does not need ordinary homotopy; it needs causal homotopy. This does not prove chronology protection in full GR. It proves the homotopy theory needed for the Bridge. Tseglin 2026m = Ch. 148. Cites Tseglin 2026a DOI 10.5281/zenodo.23977165, 2026b DOI 10.5281/zenodo.23980498, 2026c DOI 10.5281/zenodo.23980738, 2026d DOI 10.5281/zenodo.23981860, 2026e DOI 10.5281/zenodo.23982063, 2026f DOI 10.5281/zenodo.23982359, 2026g DOI 10.5281/zenodo.23985216, 2026h DOI 10.5281/zenodo.23991367, 2026i DOI 10.5281/zenodo.23991442, 2026j DOI 10.5281/zenodo.23991535, 2026k DOI 10.5281/zenodo.23991603, 2026l DOI 10.5281/zenodo.23991665.