gurvich krnc vyalyi conjecture 1 total amoeba confinement

Vladimir Gurvich, Matjaž Krnc and Mikhail Vyalyi, Growing Trees and Amoebas’ Replications, arXiv:2401.07484v1/v2 and DOI 10.1007/s00025-025-02421-6, state Conjecture 1 in Section 2.1 of arXiv v1 and on page 11 of the published PDF: if an amoeba A=(H,m) has maximum degree greater than its number of roots k by more than two, then A is totally confined.

The formal target keeps the complete ordinary finite-host reading: every finite simple tree H, every multiplicity function m, every extension length l, every finite initial tree T containing H, every multiplicity-preserving copy and transported multiplicity placement, every simultaneous family of distinct arms, and every minimum growth ordered by total host edges. Length zero, zero multiplicities and k=0 remain included. Total confinement requires a finite host containing T in which every such copy is dead.

The result addresses the complete named conjecture, including its universal quantifiers over copies, initial hosts, arms and total-edge minimizers. The source's infinite union-limit questions are separate and are not part of this finite assertion.

The source leaves this finite total-confinement assertion open. The proof must construct a finite host for an arbitrary initial T, realize all transported multiplicity arms simultaneously for every copy, and show that every minimum-total-edge growth coincides with the containing host. A bounded example, a selected growth, an infinite-host cardinal argument or a narrower arm model would leave the target unresolved.

The formalization uses a finite layered tree. A safe branching degree avoids all intervals [deg_H(v), deg_H(v)+m(v)) at once. The core receives a finite branching budget and a finite depth buffer. Unique paths in a tree place every copy within the core radius, supply the required fresh first edges jointly, and continue every arm without return or collision. The total-edge comparison then forces surjectivity and adjacency reflection for every minimizing containing map. The arbitrary initial copy is preserved by the core embedding.

A finite H, multiplicity function m and length l satisfying the strict degree condition, together with a finite initial T containing H, for which no finite confining supertree exists would refute the result. A counterexample must use all copies, transported multiplicities, simultaneous arms and total host-edge minimization. Failure of a different construction or an infinite-host variant would not refute this statement.

The settling declaration is D5/S3/Combinatorics/Graph/Walks/AmoebaHost.total_confinement. Its definitions and supporting theorems are in AmoebaConfinement and AmoebaHost; the emitted Blueprint pages mirror the closed Lean statements. The canonical scoped report covers both modules and records the standard axiom closure Classical.choice, Quot.sound and propext. The Library note D5/L/Combinatorics/gurvichkrncvyalyi2025amoebas records the verified DOI, URLs and source locators.

The source mapping is the published Section 2 definitions and Conjecture 1, with the arXiv v1/v2 versions as corresponding source records. The construction retains every original finite quantifier and makes no claim about infinite cardinal minimization.

What the settlement shows

• Settlement — proved, kernel-checked: D5/S3/Combinatorics/Graph/Walks/AmoebaHost.total_confinement proves the complete finite-host Conjecture 1 under the stated degree condition.
• Mechanism — proved, kernel-checked: exists_safe_degree supplies a branching degree outside every multiplicity demand interval; exists_buffered_host supplies a finite layered host with the required core budgets and depth; sprout_copy, hasExtension_one and hasExtension_of_buffer give simultaneous fresh arms and their finite continuations; dead_of_hasExtension turns total-edge minimality into coincidence with the same containing host.
• Boundary — retained: the theorem includes l=0, m=0 and k=0, all transported multiplicity placements and all minimum total-edge growths. It concerns finite simple tree hosts. Infinite union limits and cardinal minimization remain separate questions.
• Escape audit — unfinished: the eight public theorem targets named by the preregistration have no declared_validated Reg mirrors. The concrete audit boundary and missing dependent-family evidence are disclosed under issue #23364. No registration is represented as validated.

The literature screen is bounded to the cited arXiv versions, published paper, direct predecessor and the recorded repository search. Worldwide absence, priority, novelty and completeness of all indexed or posted work are ASSUMED-UNVERIFIED. No such claim is needed for the formal theorem or its canonical resolution metadata.