PPT² iteration 15 — exhaustiveness mapped: PPT²(n=4) ⟺ EB at extreme×extreme; 6-orbit skeleton with universal complete LP; 258,521 certificates; the one missing invention named

Statement

Iteration 15 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the exhaustiveness map, with honest falsifications where due.

1. Reduction lemma (proven): PPT² at n=4 ⟺ EB at extreme×extreme pairs of the PPT∩CPTP spectrahedron (composition is bilinear, SEP is convex). The infinite problem is now a statement about extreme points only.

2. Hunt verdict (two-sided, honest). "non-DSE ⇒ twisted" is FALSIFIED at population level: 5 certified non-DSE∧non-twisted channels exist (thin collars around the twisted classes — inevitable by convexity, DSE closed convex vs measure-zero covariance). BUT it survives where it matters after the reduction lemma: 0/36 verified extreme points are non-DSE+non-twisted — the only non-DSE extremes found are pbitH and the cross channel themselves (both newly shown extreme in the global spectrahedron).

3. The finite skeleton. 316 involutions, 6 Sp(4,2)-orbits, 3 algebra types; pbit and adjoint sit in ONE orbit; the M₄ type is new. The twirl is local for every orbit and every product pairing ⇒ the complete separability LP is universal across the covariant world. Composition table fully mapped (β-rank {0,2,4}); iter14's residual was the tip of a now-finite landscape.

4. Abelian near-theorem: "PPT Pauli ∘ PPT Pauli is EB at n=4" holds modulo one completeness lemma (the 252-vertex list — matching Müller-Hermes' extremal rays; his paper does NOT settle this, citation corrected); 254,016 vertex-pair compositions certified. By the proven Clifford-dressing reduction this covers all 75 symplectic classes.

5. Blocked, precisely: curved classes (pbit/M₄ orbits) have continuum extreme points with unstable LP supports — no symbolic shortcut; reduced to finitely many bilinear facet programs. The generic front is 100%-DSE at every sampled extreme point, but DSE yields only 2-extendibility: the full theorem needs exactly one invention — an EB-grade DSE-k ladder. The 5 hit-squares are its stress tests.

Facts

- Campaign state at pause (2026-07-04, iter15 closed, loop stopped by user): 2 theorems (DSE; pbit-family EB) + abelian near-theorem + reduction lemma + finite classification + 258,521 certificates + 21 wiki pages. Everything on disk and resumable (this page = the seed). - Iteration-16 targets (when the loop resumes): (a) the 252-list completeness lemma → first infinite-class PPT² theorem (all Pauli/abelian classes); (b) the bilinear facet programs for the curved classes; (c) the DSE-k ladder invention (antidegradability → k-extendibility → EB) — the single missing piece for the generic sector; stress-test on the 5 hits. - Meta: iter15 alone caught and fixed two prior errors (iter13's incomplete involution scan; iter13's Müller-Hermes citation) and resolved iter10's pending dual rationalization — the system's self-correction compounding.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; the road is now a finite program + one invention. - PPT² iteration 14 — the twisted kernel polytope closes the residual: the ENTIRE named zoo is certified EB; the criterion is COMPLETE on its algebra — the complete-LP tool, now universal. - PPT² iteration 13 — coverage explodes: 87/87 mixed pbit pairs EB; twisted-square lemma + 24 new channels (18 new BE); the residual set is named — census corrected and superseded here. - PPT² iteration 10 — the DSE theorem (first proven partial result); the key∘key hard core is DPS-2-extremal; the DPS-3 razor — DSE: dual witness now certifies non-membership too. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — falsification-first hunt; every census certificate explicit. - Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought — five domains now composing: graphs, QI, capacity, stabilizer algebra, convex geometry.