feat: rank-one mutual conditional independence (#107) - #158
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Statement A of the rank-one checkpoint: the singleton exact-anchor layers are
MUTUALLY conditionally independent given the invariant sigma-algebra,
iCondIndepFun (fun v => exactMap {v}) invariantAlgebra M.law
for an arbitrary exchangeable law — no dissociation, no NoNullary.
Proved entirely from the merged two-set condIndep_fixingAlgebra, with no de
Finetti machinery. Peel one vertex v off the finite family; the accumulated
events bundle into one fixingAlgebra s event by monotonicity; {v} ∩ s = ∅, so
the two-set theorem conditions on fixingAlgebra ∅ = invariantAlgebra — the same
algebra at every stage — and the product composes against the induction
hypothesis.
The peel is available only at rank one. For supports of rank n > 1 the peeled
support meets the accumulated union in POSITIVE rank, so each stage would
condition on a larger algebra, and conditional independence is not preserved
under enlarging the conditioning. That is the collider/intersection obstruction,
and it is avoided here rather than solved.
Also adds measurable_exactMap_fixingAlgebra, the sharpening of
measurable_exactMap that the peel needs.
Audit 227 -> 229.
* Move `measurable_exactMap_fixingAlgebra` into `Graphon/RelFactorLaws.lean` beside `exactMap` and `measurable_exactMap`. It is general factor API — any argument bundling exact-layer events over a support needs it — and future rankwise work should not import the rank-one theorem to obtain it. * Say 'the conditional-independence half of the multi-sorted de Finetti statement'. Sort-orbit equality of the conditional kernels is the other half and is deliberately absent here. * Weaken the higher-rank claim from 'meets the accumulated union in positive rank' to 'need not be disjoint'. A positive-rank intersection is possible and breaks the induction, but is not inevitable for every family.
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Statement A of the rank-one checkpoint, proved entirely from the merged two-set theorem. No de Finetti representation, no mixing measure, no identification of a mixing measure with an invariant conditional distribution, and nothing ported from the exchangeability package.
This is the conditional-independence half of the multi-sorted de Finetti statement; sort-orbit equality of the conditional kernels is the other half and is deliberately absent here. Mutual, not pairwise, and stated over the whole vertex index —
iCondIndepFunalready quantifies over finite subfamilies, so finite families are a corollary. Arbitrary exchangeable law: noIsDissociated, noNoNullary. Mathlib'siCondIndepFuntakes dependent codomains, sov ↦ exactMap {v}with its varyingExactSpace {v}is a valid target as it stands.The peel
Via
iCondIndepFun_iff_condExp_inter_preimage_eq_mul, thenFinset.induction_on:v;fixingAlgebra s-event, since each isfixingAlgebra {i}-measurable andfixingAlgebrais monotone;{v} ∩ s = ∅becausev ∉ s, socondIndep_fixingAlgebra {v} sconditions onfixingAlgebra ∅ = invariantAlgebra— the same algebra at every stage;Why this does not generalize
The third step is what fails above rank one: for supports of rank
n > 1the peeled support need not be disjoint from the accumulated union, and where it is not, that stage conditions on a larger algebra — and conditional independence is not preserved under enlarging the conditioning. A positive-rank intersection is not inevitable for every family, but it is possible, which is enough to break the induction. That is the collider/intersection obstruction recorded on #107 — avoided here, not solved. The module docstring says so explicitly, so the argument is not mistaken for a route to the general case.Also adds
measurable_exactMap_fixingAlgebra—exactMap Ais measurable forfixingAlgebra A, not merely for the ambient algebra — which is what lets the accumulated events be bundled. It lives inGraphon/RelFactorLaws.leanbesideexactMap, as general factor API: rankwise work should not import the rank-one theorem to obtain it.Deliberately not in this PR: kernel equality within each sort orbit (provable separately from exchangeability and conditional-distribution uniqueness), and everything about I(1), the uniform-factor coupling, and transfer.
Gates:
lake buildclean (3363 jobs), zero sorries, axiom audit 227 → 229 (two commits on the branch), documentation drift checks pass.#107 remains open.