The record layer transports to a de-isolation arena

Category: 7-SigmaLayer (the record layer — Paper C A7). BACKLOG #107, out of #106's surviving route.

Rows #102–#106 are statements about globalBasins of a ContextField under epistemicMeasure. The pointer the corpus actually constructs lives elsewhere: ShearDeIsolation.lean's outcome sectors are preimages of those basins under a de-isolation propagator, on a fibred arena with its own preparation measure (readyPrep [ψ] = epistemicMeasure [ψ] ⊗ readyMeasure). #107 asks for the bridge, so that the stability and robustness results apply to the flow-carved sectors rather than to posited cells.

The bridge is one condition, and this file isolates it.

The condition, and why it is the right one

• ReadsEpistemic p μ Φ — the arena's preparation measure pushes forward along Φ to the epistemic measure. Every statement below is a transport along that single identity, so nothing is assumed about the arena's type, its fibre, or its dynamics;
• ★★★ readsEpistemic_of_measurePreserving — and the condition reduces to a preparation-preserving flow. On a product arena Σ × (ready register) with a probability measure on the register, if the de-isolation flow F preserves the preparation measure then reading Σ after the flow reads the epistemic measure. That is exactly the shape readyPrep has and exactly the property a propagator is built to have, so the bridge is not an extra posit about the arena — it is measure preservation;
• ★★★ readsEpistemic_readyPrep_of_measurePreserving — and the corpus's own preparation measure is that product. readyPrep p = epistemicMeasure p ⊗ readyMeasure N literally, so for the shear arena the bridge is a single property of the propagator and nothing else: it preserves readyPrep.

What transports

• ★ measure_preimage_eq — any measurable record event has the same probability in the arena as in Σ, and ★ isProbabilityMeasure_of_readsEpistemic: the arena's preparation is a probability measure for free;
• ★★ measure_preimage_globalBasin — the flow-carved sector carries the Born weight, which is the shape of shear_sector_born obtained here from the pushforward alone;
• ★★ measure_preimage_robustBasin_ge — the robust part of a sector is at least rate − δ, and ★★★ one_sub_le_robust_fraction_preimage — #106's robust fraction holds for the flow-carved sector: within the sector the realised record survives a write of size δ on all but δ / rate of it;
• ★★★ measure_preimage_recordString_ne_le — #103's relabelling bound holds for the arena: the arena's microstates whose record string a write of size δ changes have preparation measure at most k · N · δ.

So once the flow preserves the preparation, the record layer's stability results are statements about the pointer the de-isolation flow produces.

Honest scope

⚠️ The shear propagator is not shown to satisfy this. The corpus has the per-sector Born identity (shear_sector_born) but not the pushforward identity, and readsEpistemic_readyPrep_of_measurePreserving reduces the gap to exactly one hypothesis — MeasurePreserving F (readyPrep p) (readyPrep p) for the shear propagator — which is not proved here, and sits upstream of ShearWitness item 1 (the propagator's Hamiltonian generation is stated, not formalised). That is BACKLOG #108, and nothing below claims the connection is made for the shear construction specifically.

⚠️ A pushforward is not a dynamics. ReadsEpistemic says where the preparation goes, not how. No generator, no interaction Hamiltonian, no time parameter appears, and the de-isolation obligation of DeIsolationFlow.lean is untouched by anything here.

⚠️ Transport preserves scope, including the limits. Everything carried over keeps its own caveats: #103's bound is robustness and not invariance, #106's fraction is conditional on the realised outcome and claims no cell is large, and the write is a translation of the record coordinate. Pulling them back along Φ adds nothing and removes nothing.

⚠️ One preparation. The transport is stated at a fixed base point p; nothing here varies the preparation or integrates over it.

References: records-to-spacetime-scoping.md ST-3; future-work.md; PointerDynamics.lean (#106), PointerConcentration.lean (#105(a)), MacrostateStability.lean (#103), ShearDeIsolation.lean, DeIsolationFlow.lean; specs/BACKLOG.md #107, #108, #106, #103, #102.

The bridge condition

structure CSD.RecordLayer.ReadsEpistemic {N : ℕ} {α : Type u_1} [MeasurableSpace α] (p : LF4.CPN N) (μ : MeasureTheory.Measure α) (Φ : α → LF4.KSigma N) : Prop

The arena reads the epistemic measure: its preparation measure pushes forward along Φ to the epistemic measure at p. One identity, and every transport below is an instance of it.

• measurable_map : Measurable Φ

  The reading map is measurable.

• pushforward : MeasureTheory.Measure.map Φ μ = epistemicMeasure p

  The preparation pushes forward to the epistemic measure.

theorem CSD.RecordLayer.isProbabilityMeasure_of_readsEpistemic {N : ℕ} {α : Type u_1} [MeasurableSpace α] {p : LF4.CPN N} {μ : MeasureTheory.Measure α} {Φ : α → LF4.KSigma N} (h : ReadsEpistemic p μ Φ) : MeasureTheory.IsProbabilityMeasure μ

★ The arena's preparation is a probability measure for free.

theorem CSD.RecordLayer.measure_preimage_eq {N : ℕ} {α : Type u_1} [MeasurableSpace α] {p : LF4.CPN N} {μ : MeasureTheory.Measure α} {Φ : α → LF4.KSigma N} (h : ReadsEpistemic p μ Φ) {S : Set (LF4.KSigma N)} (hS : MeasurableSet S) : μ (Φ ⁻¹' S) = (epistemicMeasure p) S

★ Every measurable record event has the same probability in the arena as in Σ.

theorem CSD.RecordLayer.readsEpistemic_of_measurePreserving {N : ℕ} {p : LF4.CPN N} {β : Type u_2} [MeasurableSpace β] (ν : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure ν] {F : LF4.KSigma N × β → LF4.KSigma N × β} (hF : MeasureTheory.MeasurePreserving F ((epistemicMeasure p).prod ν) ((epistemicMeasure p).prod ν)) : ReadsEpistemic p ((epistemicMeasure p).prod ν) (Prod.fst ∘ F)

★★★ The condition reduces to a preparation-preserving flow. On a product arena Σ × (ready register) whose register carries a probability measure, a de-isolation flow preserving the preparation measure reads the epistemic measure when Σ is read after the flow.

This is the shape readyPrep = epistemicMeasure ⊗ readyMeasure has, so the bridge is not a new posit about the arena: it is measure preservation of the propagator.

theorem CSD.RecordLayer.readsEpistemic_readyPrep_of_measurePreserving {N : ℕ} {p : LF4.CPN N} {F : LF4.KSigma N × LF4.KTorus → LF4.KSigma N × LF4.KTorus} (hF : MeasureTheory.MeasurePreserving F (readyPrep p) (readyPrep p)) : ReadsEpistemic p (readyPrep p) (Prod.fst ∘ F)

★★★ And the corpus's own preparation measure is that product. readyPrep — the ontic preparation of the shear arena — is epistemicMeasure p ⊗ readyMeasure N by definition, so a de-isolation flow on that arena which preserves it reads the epistemic measure.

This is the bridge for the construction rather than for an abstract product arena: what #108 owes is one hypothesis about the shear propagator, MeasurePreserving F (readyPrep p) (readyPrep p), and nothing further about records, cells or basins.

What transports

theorem CSD.RecordLayer.measure_preimage_globalBasin {N : ℕ} {α : Type u_1} [MeasurableSpace α] {p : LF4.CPN N} {μ : MeasureTheory.Measure α} {Φ : α → LF4.KSigma N} (h : ReadsEpistemic p μ Φ) (c : ContextField N) (i : Fin N) : μ (Φ ⁻¹' globalBasin c i) = ENNReal.ofReal (c.rate p i)

★★ The flow-carved sector carries the Born weight. The shape of shear_sector_born, obtained from the pushforward alone.

theorem CSD.RecordLayer.measure_preimage_robustBasin_ge {N : ℕ} {α : Type u_1} [MeasurableSpace α] {p : LF4.CPN N} {μ : MeasureTheory.Measure α} {Φ : α → LF4.KSigma N} (h : ReadsEpistemic p μ Φ) (c : ContextField N) {δ : ℝ} (hδ : 0 ≤ δ) (i : Fin N) : ENNReal.ofReal (c.rate p i - δ) ≤ μ (Φ ⁻¹' robustBasin c δ i)

★★ The robust part of a sector is at least rate − δ.

theorem CSD.RecordLayer.one_sub_le_robust_fraction_preimage {N : ℕ} {α : Type u_1} [MeasurableSpace α] {p : LF4.CPN N} {μ : MeasureTheory.Measure α} {Φ : α → LF4.KSigma N} (h : ReadsEpistemic p μ Φ) (c : ContextField N) {δ : ℝ} (hδ : 0 ≤ δ) (i : Fin N) (hrate : 0 < c.rate p i) : 1 - δ / c.rate p i ≤ (μ (Φ ⁻¹' robustBasin c δ i)).toReal / c.rate p i

★★★ #106's robust fraction holds for the flow-carved sector. Within the sector the realised record survives a write of size δ on all but δ / rate of it — with no concentration hypothesis, as in #106.

theorem CSD.RecordLayer.measure_preimage_recordString_ne_le {N : ℕ} {α : Type u_1} [MeasurableSpace α] {p : LF4.CPN N} {μ : MeasureTheory.Measure α} {Φ : α → LF4.KSigma N} (h : ReadsEpistemic p μ Φ) {k : ℕ} (c : Fin k → ContextField N) {δ : ℝ} (hδ : 0 ≤ δ) : μ (Φ ⁻¹' {x : LF4.KSigma N | recordString c (sigmaShift δ x) ≠ recordString c x}) ≤ ↑k * (↑N * ENNReal.ofReal δ)

★★★ #103's relabelling bound holds for the arena. The arena's microstates whose record string a write of size δ changes have preparation measure at most k · N · δ.