SigmaLayer/ShearDeIsolation: the de-isolation interaction of the constructed flow

Category: 7-SigmaLayer (the record layer — the Q12 successor question, step 2b′ assembly).

What this closes

specs/q12-fibre-mechanism-scoping.md (successor question, corrected 2026-08-26) states the standing obligation as DeIsolationFlow.lean's: exhibit a pointer p = readout ∘ flow(H_int(M)) whose basins carry the Born measures — with the scoping that the cell shapes are bookkeeping while the fibre, the rates and the selection are not. The corpus had two DeIsolationInteraction witnesses (cdfDeIsolationInteraction, raceDeIsolationInteraction), both with defined cells: the pointer is the ψ-indexed cell family, and no dynamics carves it.

This module supplies the third witness, and it is the one the obligation asks for in shape:

• ★★ shearDeIsolationInteraction — a DeIsolationInteraction (readyPrep [ψ]) ψ whose pointer is the total readout of the constructed de-isolation propagator: cellPointer over the flow-carved outcome sectors Ωᵢ = Φ_{0→1}⁻¹(Bᵢ) of the shear protocol, driven by the context-fixed momentContext basins.
• ★ cellPointer_outcomeSector_eq_readout — that pointer is readout ∘ flow, literally: pointer x = (readout (Φ_{startTime→readoutTime} x)).getD i₀ for every point. The p = readout ∘ flow(H_int(M)) shape is a theorem of the construction, not a gloss.
• ★★ shear_sector_born — the dynamical Born on the shear arena: the flow-carved outcome sector's readyPrep measure is the moment-map weight, via measure_outcomeSector_eq_of_correlates — so basin_rate is discharged from the constructed propagator, not assumed as a hypothesis field and not read off defined cells. (The swap-arena analogue is swap_sector_born; this is the bankless mirror on Σ_sel × T²_R.)
• readyPrep_selReady, readyPrep_selReady_cover — the selector-and-ready sectors carry the moment-map weights and exhaust the canonical ready preparation.

Where the ψ-dependence lives — the reason this is not bookkeeping

The pointer is one context-fixed map: the readout arcs (pointerArc), the basins (globalBasin (momentContext N)), and the propagator are all preparation-independent. The state enters only through the ontic preparation measure readyPrep [ψ] = epistemicMeasure [ψ] ⊗ readyMeasure — ignorance of the microstate in the prepared region, the Papers A/D typicality story. The Born identity readyPrep [ψ] (Ωᵢ) = ‖ψ i‖² is a theorem of that preparation plus the constructed dynamics. Contrast the CDF/race witnesses, where the ψ-dependence sits inside the pointer itself.

The basins are not literally cdfCell on the abstract [0,1) fibre: they are the outcome sectors on the fibred arena Σ_sel × T²_R, whose fibre coordinate is the uniformly-distributed register the scoping note's volume_circleCell reading refers to. Per the 2026-08-26 scoping, that is the bookkeeping difference; the fibre, the rates (moment map) and the ontic selection (which Ωᵢ the microstate occupies) are exactly what is carried.

⚠️ Honest scope — what this does NOT close

1. The Hamiltonian generation is stated, not formalised — unchanged from ShearWitness item 1. The propagator is explicit, measure-preserving, and discharges CorrelatesOn / PointerInvariantOn; that it is the time-T_M flow of H_int(t) = g(t)·(ι+1)·δ·p_R is a symplectic-geometry calculation Mathlib cannot state (no manifold Hamiltonian-flow API; the permanently scoped row of reconstruction-status.md §2a). basin_rate is discharged from the constructed propagator; the "flow of H_int(M)" reading carries that standing caveat. Do not cite this as a formalised H_int. What remains of D1 is exactly that formalisation gap, plus the witness-not-derivation caveat below.
2. The coupling is engineered (the ontic von Neumann shape, coupled to the outcome index) — a witness that a de-isolation interaction with the required readout exists on the arena, not a derivation that a physically natural interaction must take this form (ShearWitness items 2–3).
3. The seam. The everywhere-form of the correlation is impossible (no_everywhere_correlation); the witness's correlation holds on the selector-and-ready sectors, whose union exhausts the ready preparation (readyPrep_selReady_cover) — the exceptional set is the null seam, exactly where the constraint said it must live.

References

specs/q12-fibre-mechanism-scoping.md (the successor question this answers in its honest form); specs/record-layer-plan.md §3c (step 2b′); specs/future-work.md; RecordLayer/MomentMapRace.lean (DeIsolationInteraction, bornRate_eq_momentMap, the two prior witnesses); RecordLayer/DeIsolationFlow.lean (the obligation, map_pointer_apply); RecordLayer/ShearWitness.lean (shearProtocol, selReady, shear_correlates); RecordLayer/DynamicBorn.lean (basinIndex, measure_basinIndex_fibre); RecordLayer/SwapClosure.lean (readyPrep, swap_sector_born — the assembly pattern); Mathlib/MeasureTheory/CellPointer.lean (cellPointer, measure_cellPointer_preimage).

The selector-and-ready sectors under the canonical ready preparation

theorem CSD.RecordLayer.readyPrep_selReady {N : ℕ} [NeZero N] (p : LF4.CPN N) (i : Fin N) : (readyPrep p) (selReady (basinIndex (momentContext N)) i) = ENNReal.ofReal (LF4.momentMap p i)

The canonical ready preparation weights the selector-and-ready sector by the moment map. The sector factors as (basin fibre) × (ready arc); the fibre carries the moment-map weight (measure_basinIndex_fibre + globalBasin_prob) and the ready arc has full conditional measure.

theorem CSD.RecordLayer.readyPrep_selReady_cover {N : ℕ} [NeZero N] (p : LF4.CPN N) : ∑ i : Fin N, (readyPrep p) (selReady (basinIndex (momentContext N)) i) = 1

The selector-and-ready sectors exhaust the canonical ready preparation.

The dynamical Born on the shear arena

theorem CSD.RecordLayer.shear_sector_born {N : ℕ} [NeZero N] (p : LF4.CPN N) (i : Fin N) : (readyPrep p) ((shearProtocol (basinIndex (momentContext N)) ⋯).outcomeSector i) = ENNReal.ofReal (LF4.momentMap p i)

★★ The flow-carved basin carries the moment-map weight. The outcome sector Ωᵢ = Φ_{0→1}⁻¹(Bᵢ) — the initial states the constructed propagator carries into the pointer arc for i — has readyPrep measure exactly the moment-map weight. Via measure_outcomeSector_eq_of_correlates, so the discharged correlation theorem (shear_correlates) is genuinely consumed: the Born weight of the basin is transported by the interaction, not posited for it. The bankless mirror of swap_sector_born.

The pointer IS readout ∘ flow

theorem CSD.RecordLayer.cellPointer_outcomeSector_eq_readout {Sigma : Type u_1} [MeasurableSpace Sigma] {K : ℕ} (P : MeasurementProtocol Sigma K) (i₀ : Fin K) (x : Sigma) : MeasureTheory.cellPointer P.outcomeSector i₀ x = (P.readout (P.evolve P.startTime P.readoutTime x)).getD i₀

★ The total pointer of the outcome sectors is the readout composed with the flow. For any measurement protocol, cellPointer over the flow-carved outcome sectors computes (readout (Φ_{startTime→readoutTime} x)).getD i₀ at every point: the p = readout ∘ flow shape of the step-2b′ obligation, as a pointwise theorem rather than a reading.

The de-isolation interaction of the constructed flow

noncomputable def CSD.RecordLayer.shearDeIsolationInteraction {N : ℕ} [NeZero N] (ψ : EuclideanSpace ℂ (Fin N)) (hψ0 : ψ ≠ 0) (hψ : ‖ψ‖ = 1) (i₀ : Fin N) : DeIsolationInteraction (readyPrep (Projectivization.mk ℂ ψ hψ0)) ψ

★★ The de-isolation interaction of the constructed flow. The third DeIsolationInteraction witness — and the first whose pointer is the readout of the constructed de-isolation propagator rather than a defined cell family: the pointer is cellPointer over the flow-carved outcome sectors Ωᵢ = Φ_{0→1}⁻¹(Bᵢ) (pointwise = (readout ∘ Φ_{0→1}).getD i₀, cellPointer_outcomeSector_eq_readout), and basin_rate is discharged from the dynamics (shear_sector_born), not assumed.

The pointer, the readout arcs, the basins and the propagator are all context-fixed; ψ enters only through the ontic preparation measure readyPrep [ψ]. ⚠️ The standing caveat is ShearWitness item 1, carried not laundered: the propagator's Hamiltonian generation is stated, not formalised (no manifold symplectic API in Mathlib) — so read this as basin_rate discharged from the constructed de-isolation propagator, with the H_int(M) origin of that propagator the corpus's permanently-scoped symplectic residue.

Equations

• One or more equations did not get rendered due to their size.

theorem CSD.RecordLayer.shearDeIsolationInteraction_pointer {N : ℕ} [NeZero N] (ψ : EuclideanSpace ℂ (Fin N)) (hψ0 : ψ ≠ 0) (hψ : ‖ψ‖ = 1) (i₀ : Fin N) (x : LF4.KSigma N × LF4.KTorus) : (shearDeIsolationInteraction ψ hψ0 hψ i₀).pointer x = ((shearProtocol (basinIndex (momentContext N)) ⋯).readout ((shearProtocol (basinIndex (momentContext N)) ⋯).evolve (shearProtocol (basinIndex (momentContext N)) ⋯).startTime (shearProtocol (basinIndex (momentContext N)) ⋯).readoutTime x)).getD i₀

★ The instance's pointer is literally readout ∘ flow. Certifies the step-2b′ shape for the witness above: at every arena point, the pointer reads the record the propagator has created (startTime = 0, readoutTime = 1 for the shear protocol).

theorem CSD.RecordLayer.shearDeIsolation_born {N : ℕ} [NeZero N] (ψ : EuclideanSpace ℂ (Fin N)) (hψ0 : ψ ≠ 0) (hψ : ‖ψ‖ = 1) (i₀ i : Fin N) : (MeasureTheory.Measure.map (shearDeIsolationInteraction ψ hψ0 hψ i₀).pointer (readyPrep (Projectivization.mk ℂ ψ hψ0))) {i} = ENNReal.ofReal (‖ψ.ofLp i‖ ^ 2)

The Born conclusion, from the flow. The constructed de-isolation propagator's pointer pushes the canonical ready preparation forward to the Born distribution — outcome i with probability ‖ψ i‖². DeIsolationInteraction.born applied to the flow-carved witness: the conditional's antecedent is now populated by dynamics.