feat(Cosmology): comoving distance, redshift, Hubble-Lemaitre law, transverse distance and horizons

[KevorkianPhilippe]

AI disclosure

AI disclosure. This PR was generated with Claude Fable 5.1 (Claude Code) under my
supervision, following AI-POLICY.md and AGENTS.md. I have read every definition and lemma
statement and vouch that they state what the docstrings say. The statements were fixed before
the proofs were written and checked symbolically (sympy) against the Friedmann equations of
Physlib.Cosmology.FLRW.Basic; no textbook is cited, the TODO items of the file are the
specification.

Stacked on #1638 (flrw-dynamics-7): the diff includes the earlier commits of this series; only the last commit is new here.

Resolves six of the ten TODOs of Physlib/Cosmology/FLRW/Distances.lean and the
change-of-variable TODO of Basic.lean (the only change to Basic.lean: that TODO is removed).

Declarations added (namespace Cosmology.FLRW.FriedmannEquation):

• comovingDistance: χ = c ∫_t^{t₀} dτ / a; deriv_comovingDistance: ∂ₜ χ = - c / a
  (fundamental theorem of calculus, a continuous and positive).
• redshift: 1 + z = a(t₀) / a(t); one_add_redshift; deriv_redshift: ∂ₜ z = -(1 + z) H;
  deriv_eq_mul_hubbleConstant: ∂ₜ a = a H; deriv_comovingDistance_eq_mul_deriv_redshift:
  ∂ₜ χ = (c / (a(t₀) H)) ∂ₜ z, the relation dχ = c dz / H behind the distance integrals in z.
• properDistance, deriv_properDistance (Hubble-Lemaître law ∂ₜ D = H D); hubbleRadius,
  deriv_properDistance_eq_iff (recession velocity c exactly at R_H).
• spatialGeometryOfCurvature, transverseComovingDistance, and the closed forms
  transverseComovingDistance_of_pos, _zero, _of_neg (curvature radius 1 / √|K|).
• particleHorizon, eventHorizon, particleHorizon_einsteinDeSitter (3 c t₀^(2/3) t^(1/3)).

Not included, said in the module docstring: the redshift law E ∝ 1 / a of a photon (needs the
metric and null geodesics) and χ(z) as an integral in z (needs H as a function of z).

Reviewer map: A (FTC), B (redshift and change of variables), C, D (conventions of
SpatialGeometry), E.

Update (Sept 11). Applied the review remarks of #1642 across the series: no isolated general-purpose lemma (the one-off HasDerivAt and rpow helpers are inlined in the proofs that used them), and the three small bridges between Time.deriv and Mathlib's HasDerivAt now live next to the definition of the time derivative, in Physlib/SpaceAndTime/Time/Derivatives.lean (deriv_comp_val, deriv_eq_of_hasDerivAt, hasDerivAt_mk_of_differentiableAt, each introduced by the first PR of the series that needs it). Section letters shifted accordingly.

Update (Sept 17). Merged the review of #1635 (statements at t : Time, Time.deriv_comp_val dropped in favour of Time.deriv_eq_of_hasDerivAt); API-map.yaml now records what this PR completes.

🤖 Generated with Claude Code

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