Documentation

CsdLean4.RecordLayer.PointerFrequency

SigmaLayer/PointerFrequency: the ε-Born frequency layer #

Category: 7-SigmaLayer (dynamical measurement) — specs/BACKLOG.md B3a (the small half of the recorded "smooth-witness Lüders composition + ε-Born LLN" row).

The smooth horn proves a single-shot sandwich: the outcome sector of the ready-conditioned preparation carries mass between rⱼ − 2ε and rⱼ + 2(N−1)ε (pointer_born_lower/pointer_born_upper). What it did not say is what an experimenter sees — that repeated independent trials produce these numbers as frequencies. The exact horns have had that since LF1 (freq_tendsto_of_iid); the smooth horn did not.

★ pointer_born_frequency closes it: for i.i.d. trials of the smooth witness's preparation, the relative frequency of outcome j converges almost surely, and its limit lies in the ε-window

rⱼ − 2ε ≤ lim ≤ rⱼ + 2(N−1)ε.

The proof is short because nothing new is needed: freq_tendsto_of_iid is already generic over (measurable space, probability measure, measurable event), and the smooth witness supplies all three — pointerPrep is a probability measure (isProbabilityMeasure_pointerPrep), the outcome sector is measurable (outcomeSector_measurable). The only real work is carrying the ℝ≥0∞ sandwich across ENNReal.toReal, which is safe because a probability measure's sector mass is finite.

⚠️ Honest scope. (i) The limit is bracketed, not pinned: that is the ε-horn's price, and the bracket is exactly as wide as the single-shot sandwich. Sending ε → 0 sharpens the window but not within a fixed witness — each ε is a different propagator. (ii) The independence hypothesis is the LLN's, taken in the same indicator form the corpus uses everywhere (freq_tendsto_of_iid); no new probabilistic content is claimed. (iii) This is about the statistics of repeated preparations, not about repeated measurement of one system — sequential measurement on a single system is the swap/join witnesses' territory (csd_sequential_born).

References #

specs/BACKLOG.md B3a; LF1/GeneralFrequency.lean (freq_tendsto_of_iid, the generic strong law used unchanged); RecordLayer/PointerBorn.lean (pointerPrep, the sandwich); RecordLayer/JointFlowTransfer.lean (A1, which transports the sandwich to joint flows — and hence transports this too).

source
noncomputable def CSD.RecordLayer.pointerSectorProb {N : ℕ} (c : ContextField N) (hc : ∀ (j : Fin N), Continuous fun (p : LF4.CPN N) => c.rate p j) (ε : ℝ) {δ : ℝ} (hδ : δ ≤ 1 / 2) (p : LF4.CPN N) (q₀ : Pointer N) (j : Fin N) :
ℝ

The sector mass of the smooth witness's preparation, as a real number.

Equations
Instances For
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    theorem CSD.RecordLayer.pointerSectorProb_mem_window {N : ℕ} [NeZero N] (c : ContextField N) (hc : ∀ (j : Fin N), Continuous fun (p : LF4.CPN N) => c.rate p j) {ε δ : ℝ} (hε : 0 < ε) (hδpos : 0 < δ) (hδ : δ ≤ 1 / 2) (p : LF4.CPN N) (q₀ : Pointer N) (j : Fin N) :
    c.rate p j - 2 * ε ≤ pointerSectorProb c hc ε hδ p q₀ j ∧ pointerSectorProb c hc ε hδ p q₀ j ≤ c.rate p j + 2 * (↑N - 1) * ε

    The ε-sandwich, transported to real numbers. Safe because pointerPrep is a probability measure, so the sector mass is finite.

    source
    theorem CSD.RecordLayer.pointer_born_frequency {N : ℕ} (c : ContextField N) (hc : ∀ (j : Fin N), Continuous fun (p : LF4.CPN N) => c.rate p j) {ε δ : ℝ} (_hε : 0 < ε) (_hδpos : 0 < δ) (hδ : δ ≤ 1 / 2) (p : LF4.CPN N) (q₀ : Pointer N) (j : Fin N) {Ω : Type u_1} [MeasurableSpace Ω] {Pr : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure Pr] (X : ℕ → Ω → PointerArena N N) (hX : ∀ (n : ℕ), Measurable (X n)) (hlaw : ∀ (n : ℕ), MeasureTheory.Measure.map (X n) Pr = pointerPrep p q₀ δ) (hindep : Pairwise (Function.onFun (fun (f g : Ω → ℝ) => ProbabilityTheory.IndepFun f g Pr) fun (n : ℕ) => (X n ⁻¹' (pointerProtocol c hc ε hδ).outcomeSector j).indicator fun (x : Ω) => 1)) :
    ∀ᵐ (ω : Ω) ∂Pr, Filter.Tendsto (fun (m : ℕ) => (∑ k ∈ Finset.range m, (X k ⁻¹' (pointerProtocol c hc ε hδ).outcomeSector j).indicator (fun (x : Ω) => 1) ω) / ↑m) Filter.atTop (nhds (pointerSectorProb c hc ε hδ p q₀ j))

    ★ The ε-Born frequency layer. For i.i.d. trials of the smooth witness's preparation, the relative frequency of outcome j converges almost surely, and the limit sits inside the single-shot ε-window. This is what an experimenter running the smooth horn would actually measure.