SigmaLayer/ShearDiscontinuity: the measurement propagator is provably not continuous

Category: dynamical measurement / classification (machine-checking an external review's claim, 2026-08-02).

The claim, and why it matters

The external review of the dynamical arc observed: the shear witness's measurement-interval propagator displaces the register by shearAmt (basinIndex x) — an amount depending on a discrete index that jumps across basin seams — so the propagator cannot be continuous, and therefore cannot be the time slice of any ordinary (continuous, let alone smooth Hamiltonian) flow. The earlier classification of "Hamiltonian generation" as scoped on Mathlib's missing manifold API was therefore misjustified: writing down H_int = g(t)·Â_sel ⊗ P̂_R does not make the witness its flow — the flow of that (bounded, self-adjoint) generator is continuous, and the witness is not.

This module machine-checks the review's claim: ★ shearEvolve_not_continuous.

What this does and does not say

• It confirms the witness is a measurable, measure-preserving, piecewise map — not a continuous flow. The scope claims of ShearWitness.lean (which never asserted continuity) stand; the classification of the Hamiltonian-origin row is what changes.
• ⚠️ Corrected 2026-08-02 (second external review) — the original text here overread the no-go. no_everywhere_correlation forces an exceptional (non-correlating) set: a continuous propagator cannot have its whole connected ready image inside ⋃ᵢ Bᵢ while meeting two of them. It does not force the propagator to be discontinuous — a continuous (even smooth, globally Hamiltonian) propagator that sends the seams to transition states outside ⋃ᵢ Bᵢ remains mathematically open. What this module proves is only that this witness is discontinuous. Corrected 2026-08-04 (codebase audit). — the continuous route was delivered twice on 2026-08-03: the ε-corridor pointer witness (continuous_pointerEvolve, SigmaLayer/PointerWeights.lean) and the exact-Born third horn (nullSeamClosure, SigmaLayer/NullSeamWitness.lean, whose seams map to the kissing state outside every record region). This module's negative result stands, and is what makes its witness one horn of a trilemma rather than a defect.
• The two recorded repair routes (specs/BACKLOG.md): (1) smooth corridor regularisation — records and Born correct up to ε, priced by collapse_accuracy_bound; (2) classify the measurement dynamics as piecewise Hamiltonian with null seam set (standard in dynamics: billiards, impact systems), justified physically by the forced-seam no-go. Choosing between them is an authorial decision, reopened 2026-08-02.

Supporting results

• Projectivization.connectedSpace_of_isConnected_nonzero + the instance here: ℂℙ^{N-1} is connected (nonzero vectors of a rank-2N > 1 real space are connected), hence so is KSigma N — the topological input that makes a locally constant nontrivial index impossible.
• vertex_mem_globalBasin — explicit basin inhabitants at the vertices (every fibre point of a vertex lies in its full-width cell), giving basinIndex at least two values when N ≥ 2.
• pshift_shearAmt_inj — the N shear displacements are pairwise distinct points of the register torus, so the propagator's register marginal takes ≥ 2 values.

References

SigmaLayer/ShearWitness.lean (shearEvolve, pshift, shearAmt); SigmaLayer/MeasurementConstraints.lean (no_everywhere_correlation — the forced-seam no-go); SigmaLayer/DynamicBorn.lean (basinIndex, basinIndex_eq_of_mem); SigmaLayer/DegenerateLuders.lean (vertexPoint, momentMap_vertex); Mathlib/LinearAlgebra/Projectivization/Topology.lean (connectedness, staged); specs/reconstruction-status.md §2a (A2 row); specs/BACKLOG.md (the reopened row).

ℂℙ^{N-1} and KSigma N are connected

instance CSD.RecordLayer.instConnectedSpaceCPNOfNeZeroNat {N : ℕ} [NeZero N] : ConnectedSpace (LF4.CPN N)

Basin inhabitants at the vertices

theorem CSD.RecordLayer.vertex_mem_globalBasin {N : ℕ} [NeZero N] (j : Fin N) (θ : LF4.KTorus) : (vertexPoint j, θ) ∈ globalBasin (momentContext N) j

Every fibre point over a vertex lies in that vertex's basin: the vertex's cell has full width, and rep lands in Ioc 0 1 by construction.

The shear displacements are pairwise distinct

theorem CSD.RecordLayer.shearAmt_mem_Ico (K : ℕ) (i : Fin K) : shearAmt K i ∈ Set.Ico 0 1

theorem CSD.RecordLayer.pshift_shearAmt_inj (K : ℕ) (r : LF4.KTorus) {i j : Fin K} (h : pshift (shearAmt K i) r = pshift (shearAmt K j) r) : i = j

Distinct outcomes displace the register to distinct points.

The headline

theorem CSD.RecordLayer.shearEvolve_not_continuous {N : ℕ} [NeZero N] (hN : 2 ≤ N) : ¬Continuous (shearEvolve (basinIndex (momentContext N)) 0 1)

★ The shear witness's measurement propagator is not continuous. Over the full measurement interval, the propagator displaces the register by shearAmt (basinIndex x). If it were continuous, the register marginal at a fixed ready register would be a continuous map from the connected space KSigma N onto ≥ 2 distinct points, whose fibres would give a clopen partition — impossible. So the witness is a measurable, measure-preserving, piecewise map, not a time slice of any continuous flow. (Correction 2026-08-02: the no-go forces an exceptional non-correlating set, not discontinuity — see the module header; a continuous propagator with seams mapped outside the pointer regions was open when that was written and is now delivered: continuous_pointerEvolve and nullSeamClosure, both 2026-08-03. Corrected 2026-08-04 (codebase audit).)