SigmaLayer/MeasurementConstraints: what a dynamical measurement witness must satisfy

Category: 7-SigmaLayer (the record layer — constraints on the unbuilt dynamical layer).

Glossary: https://glossary.constraintsurfacedynamics.com/collapse/ Plain-language, CSD-role and formal statements of collapse, with this module as its Lean anchor. Kept symmetric by scripts/check-glossary.sh.

Glossary: https://glossary.constraintsurfacedynamics.com/measurement-problem/ Plain-language, CSD-role and formal statements of the measurement problem, with this module as its Lean anchor. Kept symmetric by scripts/check-glossary.sh.

Why this exists, and what it is not

GlobalRecordClosure supplies a context-fixed partition, but globalBasin_ae_total shows the basins cover Σ up to a null set — so a.e. point already carries a record, and there is no apparatus-ready state of positive measure. A flow cannot create a record in such a space. The proposed repair (external review, 2026-08-01) separates the hidden selector from a pointer register:

Σ_meas = ℂℙⁿ⁻¹ × T²_λ × T²_R (real dimension 2n+2 — even, compact, a Kähler product)

with a ready region R₀ ⊆ T²_R disjoint from the pointer regions Bᵢ(M) ⊆ T²_R, and a two-time propagator Φ carrying Sᵢ(M) × R₀ into Bᵢ(M).

This file builds none of that. It derives necessary conditions on any such witness, from measure preservation and continuity alone — before a Hamiltonian exists. The point is to fail cheaply if the architecture is inconsistent, the way the ℂℙⁿ⁻¹ × S¹ parity argument would have failed cheaply had anyone run it (specs/reconstruction-status.md §2a).

⚠️ Nothing here asserts that a witness exists. These are constraints a witness must meet.

What is proved

• pointer_region_measure_ge — measure preservation forces μ_sel(Sᵢ) · μ_R(R₀) ≤ μ_R(Bᵢ). The check comes back NEGATIVE: this does not obstruct. ⚠️ Not formalised here: the selector sectors should carry equal Liouville weight 1/n — the moment coordinates sum to 1 (momentMap_sum_eq_one) and μ_FS is U(N)-invariant, so they are exchangeable and each integrates to 1/n. Granting that, the bound reads μ_R(Bᵢ) ≥ μ_R(R₀)/n, satisfiable with room to spare. The theorem below is proved in full generality and does not depend on that evaluation; only this numerical gloss does. Reported as a green light, not as a strong constraint — see the honest reading below.
• ready_region_measure_le — summing gives μ_R(R₀) ≤ ∑ᵢ μ_R(Bᵢ) ≤ 1. Weak, and stated so that the weakness is on the record rather than dressed up.
• no_everywhere_correlation — the constraint with teeth. If the correlation held everywhere rather than a.e., a connected S ×ˢ R₀ would have connected image, which cannot meet two disjoint open pointer regions. So an everywhere-version is impossible for any continuous Φ whenever n ≥ 2.

★ The finding that matters for the implementation

The "almost everywhere" in the correlation theorem is load-bearing, not a technical convenience. no_everywhere_correlation shows the exceptional set is necessarily non-empty: it must contain the seams between the selector sectors, and its image is what threads through the gaps between the pointer regions. Two consequences for anyone building the witness:

1. Any candidate H_int advertised as producing an exact correlation on all of Σ_sel × R₀ is wrong, and can be rejected without examining it.
2. The construction must say what happens on the seam. A witness that leaves the boundary unspecified has not addressed the hardest part of its own statement.

Honest reading of the negative result

The measure constraints being satisfiable is worth something — it means the architecture is not dead on arrival, so the concrete H_int is worth attempting. It is not evidence that a witness exists. Measure preservation is a weak invariant; the real obstructions, if any, will be symplectic and topological, and only no_everywhere_correlation here is of that kind.

References

SigmaLayer/GlobalBasin.lean (globalBasin, globalBasin_ae_total — the a.e.-totality that forces the selector/register split); SigmaLayer/DeIsolationFlow.lean (the open H_int(M) obligation these constrain); specs/BACKLOG.md.

The measure constraint

theorem CSD.RecordLayer.pointer_region_measure_ge {Xsel : Type u_1} {Xreg : Type u_2} [MeasurableSpace Xsel] [MeasurableSpace Xreg] {μs : MeasureTheory.Measure Xsel} {μr : MeasureTheory.Measure Xreg} [MeasureTheory.SFinite μs] [MeasureTheory.SFinite μr] [MeasureTheory.IsProbabilityMeasure μs] {Φ : Xsel × Xreg → Xsel × Xreg} (hΦ : MeasureTheory.MeasurePreserving Φ (μs.prod μr) (μs.prod μr)) {S : Set Xsel} {R₀ B : Set Xreg} (hB : MeasurableSet B) (hcorr : S ×ˢ R₀ ⊆ Φ ⁻¹' Set.univ ×ˢ B) : μs S * μr R₀ ≤ μr B

A pointer region must be at least as large as the selector weight it receives.

If the propagator Φ preserves the Liouville measure and carries S ×ˢ R₀ into the states whose pointer coordinate lies in B, then μ_sel(S) · μ_R(R₀) ≤ μ_R(B).

The proof is the one-line measure-theoretic core of the whole consistency question: the correlation requirement is exactly S ×ˢ R₀ ⊆ Φ ⁻¹' (univ ×ˢ B), and measure preservation turns the measure of that preimage into μ_R(B). No injectivity of Φ is needed — only that it preserves measure.

theorem CSD.RecordLayer.ready_region_measure_le {Xsel : Type u_1} {Xreg : Type u_2} [MeasurableSpace Xsel] [MeasurableSpace Xreg] {n : ℕ} {μs : MeasureTheory.Measure Xsel} {μr : MeasureTheory.Measure Xreg} [MeasureTheory.SFinite μs] [MeasureTheory.SFinite μr] [MeasureTheory.IsProbabilityMeasure μs] {Φ : Xsel × Xreg → Xsel × Xreg} (hΦ : MeasureTheory.MeasurePreserving Φ (μs.prod μr) (μs.prod μr)) {S : Fin n → Set Xsel} {R₀ : Set Xreg} {B : Fin n → Set Xreg} (hB : ∀ (i : Fin n), MeasurableSet (B i)) (hcorr : ∀ (i : Fin n), S i ×ˢ R₀ ⊆ Φ ⁻¹' Set.univ ×ˢ B i) (hweights : ∑ i : Fin n, μs (S i) = 1) : μr R₀ ≤ ∑ i : Fin n, μr (B i)

The ready region is bounded by the total pointer budget. Summing pointer_region_measure_ge over a family of selector sectors whose weights sum to 1.

⚠️ This constraint is weak and is recorded as such. It says only μ_R(R₀) ≤ ∑ᵢ μ_R(Bᵢ), and since the Bᵢ are disjoint subsets of a probability space the right-hand side is at most 1. It does not meaningfully restrict the construction; it is stated so that the weakness is visible rather than inferred.

The topological constraint — the one with teeth

theorem CSD.RecordLayer.no_everywhere_correlation {Xsel : Type u_1} {Xreg : Type u_2} [TopologicalSpace Xsel] [TopologicalSpace Xreg] {Φ : Xsel × Xreg → Xsel × Xreg} (hcont : Continuous Φ) {S : Set Xsel} {R₀ : Set Xreg} (hconn : IsPreconnected (S ×ˢ R₀)) {U V : Set (Xsel × Xreg)} (hU : IsOpen U) (hV : IsOpen V) (hUV : Disjoint U V) (hsub : Φ '' S ×ˢ R₀ ⊆ U ∪ V) (hmeetU : ∃ x ∈ S ×ˢ R₀, Φ x ∈ U) (hmeetV : ∃ x ∈ S ×ˢ R₀, Φ x ∈ V) : False

★ An everywhere correlation is impossible.

If Φ is continuous and the source S ×ˢ R₀ is preconnected, its image is preconnected, so it cannot meet two disjoint open pointer regions. Since a measurement with n ≥ 2 outcomes requires the image to reach at least two of them, no continuous propagator can satisfy the correlation requirement everywhere on a connected ready set.

★ Therefore the a.e. qualifier in the correlation theorem is mathematically necessary, not a technical convenience. The exceptional set must be non-empty — it contains the seams between selector sectors, and its image is what threads through the gaps between pointer regions. Any proposed H_int advertised as giving an exact, everywhere correlation is wrong on these grounds alone, without inspecting it.

Stated for two regions, which is all an impossibility needs.

The collapse no-gos (2026-08-01)

Constraints on the Lüders half of the dynamical layer, derived before any collapse witness is built — the same cheap-failure discipline as above. The upshot of the pair: exact pointwise collapse is impossible for a measure-preserving dynamics, and approximate collapse is paid for in ready-state improbability. So any exact-Lüders witness must implement collapse as a measure-zero relocation (the epistemic Dirac slice moves) rather than a contraction of positive-measure sets.

The continuous-collapse twin of no_everywhere_correlation — a continuous map cannot carry a connected ready set into disjoint neighbourhoods of the N basis vertices — is an instantiation of that theorem (take U, V to be the neighbourhoods), not a new statement; no separate declaration is added for it.

theorem CSD.RecordLayer.no_exact_collapse {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {Φ : X → X} (hΦ : MeasureTheory.MeasurePreserving Φ μ μ) {C T : Set X} (hT : MeasureTheory.NullMeasurableSet T μ) (hTnull : μ T = 0) (hsub : C ⊆ Φ ⁻¹' T) (hCpos : μ C ≠ 0) : False

★ No-go A: exact pointwise collapse is impossible.

If a measure-preserving Φ sends a positive-measure set C (the collapsing states) into a null target T (e.g. {[e₁],…,[e_N]} × anything, null because μ_FS is nonatomic), that contradicts measure preservation outright: 0 < μ C ≤ μ (Φ⁻¹ T) = μ T = 0.

Consequence: "conditioned on the outcome, the base moves to [eᵢ]" can hold only on a Liouville-null set of states — which the epistemic δ_[ψ] ⊗ Haar slice is. Collapse must be relocation of a null slice, not contraction of a positive-measure one.

theorem CSD.RecordLayer.collapse_accuracy_bound {Xsel : Type u_1} {Xreg : Type u_2} [MeasurableSpace Xsel] [MeasurableSpace Xreg] {μs : MeasureTheory.Measure Xsel} {μr : MeasureTheory.Measure Xreg} [MeasureTheory.SFinite μs] [MeasureTheory.SFinite μr] [MeasureTheory.IsProbabilityMeasure μr] {Φ : Xsel × Xreg → Xsel × Xreg} (hΦ : MeasureTheory.MeasurePreserving Φ (μs.prod μr) (μs.prod μr)) {S B : Set Xsel} {R₀ : Set Xreg} (hB : MeasurableSet B) (hcorr : S ×ˢ R₀ ⊆ Φ ⁻¹' B ×ˢ Set.univ) : μs S * μr R₀ ≤ μs B

★ No-go B: collapse accuracy is bought with ready-state improbability.

The mirror of pointer_region_measure_ge with the roles of base and register exchanged: if the correlation drives the base into an ε-target B (a small neighbourhood of a vertex), then μ_sel(S) · μ_R(R₀) ≤ μ_sel(B). Summed over outcomes (with the selector weights summing to 1) this reads μ_R(R₀) ≤ ∑ᵢ μ_sel(Bᵢ) — the apparatus-ready region must shrink as the collapse targets do, and exact collapse (ε → 0) forces a null ready set.

This is Landauer's cost appearing as a measure inequality, and it retroactively justifies the corpus's Dirac-calibration convention: an exactly calibrated apparatus is a null set because it has to be.