RecordLayer/BackReactionNecessity: where the preparation can enter a Σ-local measurement

Category: 7-SigmaLayer (the record layer — the shape of Posit 1's discharge, BACKLOG #19-A).

The record layer's preparation is epistemicMeasure p = δ_p ⊗ vol_{T²}: the epistemic state is a point of the calculation space ℂℙ^{N−1}, and all ontic ignorance lives in the fibre T², uniformly. A measurement is a flow on Σ read against a readout. This file asks the one question that shape forces: through which door can the preparation reach the outcome statistics?

The theorem

Call a map Φ : Σ → Σ fibre-preserving if it fixes the base point and acts on the fibre over each base point by a volume-preserving map (FibrePreserving, the Category-1 notion), and call a readout fibre-only if it is univ ×ˢ R for a set R ⊆ T² of fibre points.

• ★ epistemicMeasure_preimage_univ_prod: under δ_p ⊗ vol, a fibre-preserving map sends mass vol R into a fibre-only readout — for every p. The statistics cannot see the preparation.
• ★★ no_fibrePreserving_born: hence no fibre-preserving map with fibre-only readouts has Born statistics vol(Φ⁻¹ R_i) = momentMap p i (two basis rays already disagree).
• ★★★ exists_fst_ne_of_born: contrapositively, a Σ-local measurement with Born statistics and a fibre-only readout must move the base point — back-reaction on the calculation space is necessary, not optional.
• ★ globalBasin_not_fibreOnly: the corpus's own basins are not fibre-only: globalBasin reads the rate at the ontic base point, which is exactly the other door.
• shearEvolve_fibrePreserving: the constructed de-isolation witness is fibre-preserving, so it takes that other door — its readout reads the selector, not the fibre alone.

What this says about Posit 1 (BACKLOG #19)

The preparation reaches the outcome through one of two doors and no third: the readout reads the base (the cell law as the corpus has it: a context-fixed partition whose arcs are sized by the rate at the ontic point), or the flow moves the base (back-reaction, the register acting on the calculation space). Every landed witness takes the first door with a fibre-preserving flow; the back-reacting jointLift moves the base only along the torus directions, with the rates conserved, so it takes the first door too. A derivation of the rates from the dynamics, which is what #19 asks for, is therefore a theorem about the second door: a flow whose base motion is what sizes the sectors. That is a Σ-local statement — manifold-locality of a flow on Σ — and it is where the epistemic point, the state QM calculates with, is acted on by the ontic register.

Honest scope

• Nothing here decides #19, names an interaction (R-015) or formalises generation (R-016).
• "Fibre-preserving" is stated pointwise over every base point, which is what the witnesses satisfy (measurePreserving_pshift). A flow preserving μ_FS ⊗ vol need only preserve the fibre measure over almost every base point; the almost-everywhere version is not stated.
• The readout is on the fibre T² of Σ itself. The pointer-arena witnesses carry a second torus T²_R for the register and read that; the same argument applies verbatim with readyPrep p = epistemicMeasure p ⊗ readyMeasure, and is not repeated.

References: specs/BACKLOG.md #19, #137; specs/POSITS.md Posit 1; RecordLayer/GlobalBasin.lean (epistemicMeasure, globalBasin, globalBasin_prob); RecordLayer/ShearWitness.lean (shearEvolve, measurePreserving_pshift); RecordLayer/JointLift.lean (jointLift); Mathlib/MeasureTheory/FibrePreserving.lean.

The two basis rays that disagree

noncomputable def CSD.RecordLayer.basisRay {N : ℕ} (i : Fin N) : LF4.CPN N

The ray of the i-th standard basis vector.

Equations

• CSD.RecordLayer.basisRay i = Projectivization.mk ℂ (EuclideanSpace.single i 1) ⋯

theorem CSD.RecordLayer.momentMap_basisRay_self {N : ℕ} (i : Fin N) : Projectivization.momentMap (basisRay i) i = 1

theorem CSD.RecordLayer.momentMap_basisRay_of_ne {N : ℕ} {i j : Fin N} (h : j ≠ i) : Projectivization.momentMap (basisRay i) j = 0

Fibre-preserving maps against fibre-only readouts

theorem CSD.RecordLayer.epistemicMeasure_preimage_univ_prod {N : ℕ} {Φ : LF4.KSigma N → LF4.KSigma N} (hΦ : MeasureTheory.FibrePreserving Φ MeasureTheory.volume) (hm : Measurable Φ) (p : LF4.CPN N) {R : Set LF4.KTorus} (hR : MeasurableSet R) : (epistemicMeasure p) (Φ ⁻¹' Set.univ ×ˢ R) = MeasureTheory.volume R

★ Under δ_p ⊗ vol, a fibre-preserving map sends mass vol R into a fibre-only readout, for every preparation p.

theorem CSD.RecordLayer.no_fibrePreserving_born {N : ℕ} (hN : 2 ≤ N) {Φ : LF4.KSigma N → LF4.KSigma N} (hΦ : MeasureTheory.FibrePreserving Φ MeasureTheory.volume) (hm : Measurable Φ) {R : Fin N → Set LF4.KTorus} (hR : ∀ (i : Fin N), MeasurableSet (R i)) : ¬∀ (p : LF4.CPN N) (i : Fin N), (epistemicMeasure p) (Φ ⁻¹' Set.univ ×ˢ R i) = ENNReal.ofReal (Projectivization.momentMap p i)

★★ No fibre-preserving measurement with fibre-only readouts has Born statistics.

theorem CSD.RecordLayer.exists_fst_ne_of_born {N : ℕ} (hN : 2 ≤ N) {Φ : LF4.KSigma N → LF4.KSigma N} (hm : Measurable Φ) (hfib : ∀ (b : LF4.CPN N), MeasureTheory.MeasurePreserving (fun (y : LF4.KTorus) => (Φ (b, y)).2) MeasureTheory.volume MeasureTheory.volume) {R : Fin N → Set LF4.KTorus} (hR : ∀ (i : Fin N), MeasurableSet (R i)) (hborn : ∀ (p : LF4.CPN N) (i : Fin N), (epistemicMeasure p) (Φ ⁻¹' Set.univ ×ˢ R i) = ENNReal.ofReal (Projectivization.momentMap p i)) : ∃ (x : LF4.KSigma N), (Φ x).1 ≠ x.1

★★★ Back-reaction is necessary. A measurement on Σ whose fibre maps preserve volume over every base point, read against fibre-only readouts with Born statistics, moves some base point: the register acts on the calculation space.

The corpus's witnesses take the other door

theorem CSD.RecordLayer.globalBasin_not_fibreOnly {N : ℕ} (hN : 2 ≤ N) : ¬∃ (R : Set LF4.KTorus), globalBasin (momentContext N) ⟨0, ⋯⟩ = Set.univ ×ˢ R

★ The basins read the base. globalBasin (momentContext N) i is not a fibre-only set: its epistemic measure is the rate at the preparation, which two basis rays already distinguish.

theorem CSD.RecordLayer.shearEvolve_fibrePreserving {Xsel : Type u_1} {K : ℕ} (idx : Xsel → Fin K) (s t : SigmaLayer.OnticTime) : MeasureTheory.FibrePreserving (shearEvolve idx s t) MeasureTheory.volume

The constructed de-isolation witness is fibre-preserving: it translates the fibre over each selector point by a volume-preserving shift. Its readout reads the selector.