The base-read offsets are invisible to the return criterion too

Category: 7-SigmaLayer. BACKLOG #182 (out of #19, #175, #177).

#175 showed that two transverse coordinate-reading shears with base-read offsets g₁, g₂ compose to the affine cat map affineCat (g₁ + g₂, g₂), conjugate to the cat stroke by a translation of the fibre, and that the offsets are invisible to the Lyapunov reading of #19's criterion. #177 replaced that reading by a derivative-free one: which fibre configurations come back, and how many. This file carries #175 across: return growth is conjugation-invariant (HasReturnGrowth.conj), so for every pair of offsets the offset shears have the same return count as the cat stroke at every stroke and return growth 2 log φ, and at no other rate.

What the base says at the base point changes which configurations return, never how many.

• hasReturnGrowth_shearXOff_comp_shearYOff, returnCount_shearXOff_comp_shearYOff;
• ★★★ fibre_affine_returns — the package.

Honest scope

• ⚠️ Narrows, does not decide: the whole affine class of candidate fibre maps passes #19's return criterion at one rate; which class nature uses is #19.
• The offsets are real numbers, pointwise in the base, as in #175; no base dependence is modelled.

References: specs/BACKLOG.md #182, #19, #175, #177; SigmaLayer/FibreAffineShears.lean (shearXOff, shearYOff, shearXOff_comp_shearYOff, fibre_affine_shears), SigmaLayer/FibreReturnCriterion.lean (catStroke_hasReturnGrowth), Mathlib/Dynamics/ReturnCriterionConj.lean (HasReturnGrowth.conj, affineCat_hasReturnGrowth).

theorem CSD.SigmaLayer.hasReturnGrowth_shearXOff_comp_shearYOff (g₁ g₂ : ℝ) : HasReturnGrowth (shearXOff g₁ ∘ shearYOff g₂) (2 * Real.log Real.goldenRatio)

Two offset shears have return growth 2 log φ, whatever the offsets.

theorem CSD.SigmaLayer.returnCount_shearXOff_comp_shearYOff (g₁ g₂ : ℝ) (t : ℕ) : returnCount (shearXOff g₁ ∘ shearYOff g₂) t = returnCount catStroke t

Two offset shears have the cat stroke's return count at every stroke.

theorem CSD.SigmaLayer.fibre_affine_returns (g₁ g₂ : ℝ) : HasReturnGrowth (shearXOff g₁ ∘ shearYOff g₂) (2 * Real.log Real.goldenRatio) ∧ (∀ (t : ℕ), returnCount (shearXOff g₁ ∘ shearYOff g₂) t = returnCount catStroke t) ∧ ∀ (h : ℝ), HasReturnGrowth (shearXOff g₁ ∘ shearYOff g₂) h → h = 2 * Real.log Real.goldenRatio

★★★ The base-read offsets are invisible to the return criterion: for every pair of offsets the two transverse shears have return growth 2 log φ, the cat stroke's return count at every stroke, and return growth at no other rate. Which conjugacy class nature uses is BACKLOG #19.