feat: the bipartite regression for the successor contract (#196)

Graphon.lean

@@ -49,6 +49,7 @@ import Graphon.RelPooledLatents
 import Graphon.RelPooledExtension
 import Graphon.RelPooledAcceptance
 import Graphon.RelRankSuccessorContract
+import Graphon.RelBipartiteRegression
 import Graphon.RelSingletonPeel
 import Graphon.RelFixingAlgebra
 import Graphon.RelRankAlgebra
@@ -268,7 +269,8 @@ in Lean 4 using Mathlib.
 * `Graphon.RelPooledLatents` — R4 converse (#107), stage 1 of the pooled-latent extension gate: the core instantiated at `PoolVertex S`. `PooledRankLatentIndex`/`PooledRankLatentSpace`/`pooledRankLatentSource`; `pooledRankLatentRelabel`, the action of the **full mixed** pooled family `∀ s, Equiv.Perm (PoolVertex S s)` — permutations moving vertices between halves included — with its identity and composition laws and exact source invariance; and `restrictOriginalLatents`, the measurable restriction to latents indexed by original supports, whose codomain is the rank-`n` cube itself because the index type is the carrier-parametric one. Naturality is stated in the **honest moved-window form** valid for every pooled permutation, with the commuting square available only as the split corollary. Law-free: no `RankRepresentation`, recovery, screening, or coupling appears, and those enter at later stages of the gate
 * `Graphon.RelPooledExtension` — R4 converse (#107), stage 2 of the pooled-latent extension gate: **the pooled rank extension**. `PooledRankExtension C` carries exactly three fields — the joint law on the pooled structure space times the pooled latent cube, its exact restriction to `C.P` along the two original restrictions, and invariance under the **full** pooled permutation family (mixed permutations included, the load-bearing quantifier). **No independence field**: an independent pool would recreate the defect of the rejected factor coupling. `RankRepresentation.pooledExtension` is the cheap constructor, and both of its laws are `map_prodMap_restrict_self` in disguise — writing `pv` for `poolVertexEquiv` and `ov` for `originalVertex`, the transport is `comap pv` on structures and restriction along `pv` on latents; `restrictOriginal ∘ transport` is `comap (pv ∘ ov)` with `pv ∘ ov` a **self-injection** of the original carrier, and `relabel ρ ∘ transport = transport ∘ relabel κ` for the conjugate `κ = pv ∘ ρ ∘ pv⁻¹`, a **permutation** of it. The structure deliberately carries **no mixed-window field**: the joint mixed-window marginal, and current-rank recovery and screening on mixed pooled supports, are separate derived consequences of the three fields rather than part of the primitive, and nothing route-specific belongs here
 * `Graphon.RelPooledAcceptance` — R4 converse (#107), stage 3 of the pooled-latent extension gate, organizing result: **the joint restriction theorem**. For *every* sortwise embedding `e : ∀ s, Vinfinite S s ↪ PoolVertex S s`, restricting a pooled rank extension jointly — structure and latents along the same embedding — returns the representation exactly: `Q.law.map (Prod.map (RelStructure.restrict e) (latentRestrictOver e n)) = C.P`. Being **joint** is the point: it recovers the `(X, U_{<n})` law and says strictly more than a structure-only window theorem. Proved by joint-cylinder extensionality plus finite agreement with a mixed pooled permutation — on a cylinder the combined vertex support is finite, the assignment `originalVertex v ↦ e v` there is a finite partial injection of the pooled carrier, and `exists_perm_extend_of_injOn` extends it; the full mixed invariance then absorbs the permutation, which is also what lets it be chosen independently on each sort with no finite-support or uniform-bound issue. The four route-neutral consequences follow from it: `map_snd` (the pooled latent marginal is the pooled i.i.d. source), `toStationaryExtension` (the structure marginal is a `StationaryExtension M`), `lower_recovers` (local recovery on every pooled support below rank `n`, the decoder conjugated through the local and block measurable equivalences), and `screening` (screening on every pooled support of rank `n`, a pullback through `measurePreserving_pooledJointEquiv` followed by `CondIndepFun.comp` on the codomains and `CondIndepFun.congr_cond` with `comap_measurableEquiv_comp` on the conditioning algebra — no conditional-expectation reasoning). `PooledRankExtension.map_poolVertexEquiv` is the canonical specialization; bundling that restriction as the measurable equivalence `pooledJointEquiv` and cancelling it yields **`PooledRankExtension.law_eq`, a uniqueness theorem** — every pooled rank extension *is* the cheap one. Marginals, the `StationaryExtension` structure, recovery and screening therefore transport from `C.P` through one canonical law identity rather than requiring separate measure arguments
-* `Graphon.RelRankSuccessorContract` — R4 converse (#107), **interface only**: the shared witness both successor constructions must produce, and the two identically typed statements they target. `RankSuccessor C` carries the next representation together with **exact truncation compatibility**, `next.P.map (Prod.map id (rankLatentProjection (Nat.le_succ n))) = C.P` — the observable that makes the statement mention `C` at all. A theorem returning merely `Nonempty (RankRepresentation (n + 1))` could ignore `C` and produce an unrelated representation, which is the same underdetermination that sank the earlier shell attempt. The witness carries **no independence field** beyond `RankRepresentation`'s own, and **no equality between the two routes' outputs** is asserted — they prove one statement by different means and will not produce canonically equal representations. `AustinSuccessor` and `KallenbergSuccessor` are the same proposition by construction, so whichever lands first discharges the induction while the other remains an independent proof. Regression: `truncation_zero` shows the truncation equation is **automatic at rank zero** — the rank-zero latent cube is a single point, so both sides are determined by their structure marginals, which the representation axioms already pin, and the base case imposes nothing extra. The witness exposes no pooled carrier, so whether a construction genuinely used boundary-crossing permutations is observed by the pooled gate's `map_restrict_embedding` rather than by any final-output check here; each route's intermediate construction consumes that theorem. The adversarial bipartite and i.i.d. regressions live in a separate shared module
+* `Graphon.RelRankSuccessorContract` — R4 converse (#107), **interface only**: the shared witness both successor constructions must produce, and the two identically typed statements they target. `RankSuccessor C` carries the next representation together with **exact truncation compatibility**, `next.P.map (Prod.map id (rankLatentProjection (Nat.le_succ n))) = C.P` — the observable that makes the statement mention `C` at all. A theorem returning merely `Nonempty (RankRepresentation (n + 1))` could ignore `C` and produce an unrelated representation, which is the same underdetermination that sank the earlier shell attempt. The witness carries **no independence field** beyond `RankRepresentation`'s own, and **no equality between the two routes' outputs** is asserted — they prove one statement by different means and will not produce canonically equal representations. `AustinSuccessor` and `KallenbergSuccessor` are the same proposition by construction, so whichever lands first discharges the induction while the other remains an independent proof. Regression: `truncation_zero` shows the truncation equation is **automatic at rank zero** — the rank-zero latent cube is a single point, so both sides are determined by their structure marginals, which the representation axioms already pin, and the base case imposes nothing extra. The witness exposes no pooled carrier, so whether a construction genuinely used boundary-crossing permutations is observed by the pooled gate's `map_restrict_embedding` rather than by any final-output check here; each route's intermediate construction consumes that theorem. Adversarial examples are kept in route-independent regression modules, outside this interface
+* `Graphon.RelBipartiteRegression` — R4 converse (#107/#196): the **bipartite regression** for the successor contract, a hand-built rank `1 → 2` witness over `digraphSig` whose purpose is to test that `RankSuccessor` is an expressive specification *before* either route is attempted. Each vertex carries an i.i.d. colour on its own fresh singleton coordinate and the edge is the parity, so the diagonal is constantly `false` for free. Three independent things make it a regression rather than a construction: the array law is defined **from the fresh singleton layer alone** and the rank-one coupling as the independent product `bipartiteLaw × rankLatentSource 1`, so `rankTwoCoupling_truncation` compares two separately described couplings and genuinely consumes the source factorization; the rank-two block decoder recovers **both** directed coordinates `X_uv` and `X_vu` as the same parity, exhibiting symmetry as a property of this law rather than of the signature; and `not_indepFun_rankTwoCoupling` is **numerical** — the edge and colour-parity events coincide with probability `1/2` against a product of marginals `1/4`, which a constant edge could not achieve even though it too would be "a function of the colours". `bipartiteSuccessor : RankSuccessor rankOneRep` is the witness itself
 * `Graphon.RelRankInjectionInvariance` — R4 converse (#107), the isolated proof risk of the pooled-latent extension gate: **joint invariance under arbitrary sortwise self-injections**. `RankRepresentation.invariant` is stated for *finitely supported permutations*, but a pooled object built cheaply through `poolVertexEquiv` needs the joint law invariant under every self-injection; `RankRepresentation.map_prodMap_restrict_self` supplies exactly that, so full mixed pooled invariance later costs no additional mathematics. The route is finite-cylinder extensionality on the **joint** space, which had no machinery before — rank one's joint invariance goes through only because the rank-one latent action is trivial. `rankLatentIndexInj`/`rankLatentReindex` extend the latent action from permutations to injections (only injectivity is used — a support keeps its cardinality); `exists_finSuppPerm_agree_on_finset` matches an injection to a finitely supported permutation on any finite tagged-vertex support; `ext_of_prod_cylinders` is the joint extensionality, rectangles of coordinate cylinders on both factors. **No finiteness/`Fintype` hypothesis beyond `RankRepresentation`'s ambient countability.** Equality is tested on *coordinate* cylinders — finitely many `RelCoord`s and finitely many latent indices — whose combined vertex support is one finite `Finset (Σ s, Vinfinite S s)` and so touches only finitely many sorts; the injection is matched there by extending it on each active sort, taking the identity elsewhere, and maximizing finitely many support bounds. The coarser `restrictFin` cylinder family would instead force a uniform all-sort bound that no self-injection need admit
 * `Graphon.RelRankCoding` — R4 converse piece 3 (#107): factor-law coding of the lower-rank factor. **Not** the inductive hypothesis of a working recursion: `RankCoding n → ShellProperty n` is **false**, refuted by Austin's random complete bipartite graph `X_uv = z_u ⊕ z_v` (arXiv:0801.1698 §3.6), where `lowerRankAlgebra 2` is trivial modulo the law but every triangle satisfies `X₁₂ ⊕ X₁₃ ⊕ X₂₃ = 0`, so the exact layers are pairwise but not mutually independent — while a `RankCoding 2` exists because the rank-2 factor law is a point mass. The example separates the true two-set theorem from mutuality, and shows the gap is not closable by coupling: a relatively independent joining over `lowerFactorMap` attaches latents that are conditionally independent of the structure given a trivial factor, whereas a representation needs the hidden colours, correlated with the array yet not recoverable from it. `RankCoding n` represents the rank-`n` factor by latents — a measurable coding map carrying the latent source to the factor law and intertwining the two relabeling actions **almost everywhere**. The a.e. form is forced: `lowerFactorSpaceEquiv σ n` fixes the *image* of `lowerFactorMap n`, not the whole Bool-cube, since it permutes distinct basis indices that name the same event, so the strict version is false already at rank one. `ShellProperty n` is the conclusion the recursion consumes, bundling mutual conditional independence of the exact layers over the rank-`n` supports **with** per-support locality; neither half implies the other, since mutual independence given the whole lower-rank factor permits dependence on all of it rather than only through `boundaryMap A`. `RankCoding.rankOne` is the base case, built from the #140 randomization adapter transported along `rankLatentOneEquiv`; its equivariance clause is genuinely exercised rather than vacuous, because at rank one the latent relabeling is the identity while the factor equivalence is not
 * `Graphon.RelFixingAlgebra` — R4 converse piece 2a (#107): the **law-independent** factor-algebra layer — `SortwiseFixing` (the `A`-fixing stabilizer of finitely supported sortwise permutations, closed under `1`/`*`/`⁻¹`/conjugation), the raw `RelStructure.fixingAlgebra` (events invariant under the `A`-fixing group; deliberately *not* "generated by relations inside `A`", which loses hidden vertex information), monotonicity, `fixingAlgebra_empty = invariantAlgebra` (near-definitional), and the **transport equality** `fixingAlgebra_comap_relabel : comap (relabel σ) (fixingAlgebra A) = fixingAlgebra (image σ A)` via stabilizer conjugation — no completions, no law; the conditional-independence theorem is its own later PR

Graphon/InfiniteSampler.lean

@@ -142,16 +142,6 @@ probability is the clamped edge product; integrating out the positions gives
 
 section Marginal
 
-/-- The lower-interval mass of the uniform distribution on `[0,1]`. -/
-theorem uniform01_Iic {c : ℝ} (hc : c ∈ Set.Icc (0 : ℝ) 1) :
-    uniform01 (Set.Iic c) = ENNReal.ofReal c := by
-  rw [uniform01, Measure.restrict_apply measurableSet_Iic]
-  have h : Set.Iic c ∩ Set.Icc 0 1 = Set.Icc 0 c := by
-    ext x
-    simp only [Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc]
-    exact ⟨fun h => ⟨h.2.1, h.1⟩, fun h => ⟨h.2, h.1, h.2.trans hc.2⟩⟩
-  rw [h, Real.volume_Icc, sub_zero]
-
 /-- The joint law of the first `k` vertex positions is the finite i.i.d. product. -/
 theorem vertexSource_map_fin [IsProbabilityMeasure μ] (k : ℕ) :
     (vertexSource μ).map (fun (x : ℕ → α) (a : Fin k) => x ↑a) =