Generative Structural Theory v1.57 — Unit-Budget Reachability of Bidirected Trees with a Tripod Core on the Original Z3 Growth Sequence

A growth sequence is a sequence of finite relations R_n on node sets V_n with V_n ⊆ Vₙ₊₁, R_n ⊆ Rₙ₊₁ and every stage strongly connected; a selection sequence chooses at every stage an inclusion-minimal strongly connected spanning sub-relation x_n of R_n, and its deletion budget bounds the number of pairs of x_n that xₙ₊₁ drops; the number of pairs added is not bounded. On the growth sequence of v1.36 Example Z3, whose stage n has the nodes 0, ..., 2n and 2n + 2 and all pairs between them, v1.40-v1.56 showed that deletion budget 1 reaches, from every starting stage and every inclusion-minimal strongly connected initial structure, the bidirected tree P_T of a sequence of classes of trees, ending with every locally finite tree (v1.48), every tree with finitely many nodes of infinite degree (v1.49-v1.54) and every tree whose infinitely many nodes of infinite degree all lie on one ray (v1.55) or on one double ray (v1.56). Here the least subtree H of T containing the set B of the nodes of infinite degree, the core, is a tripod: the union of three rays with a common first node c, any two of which meet in c only, each holding infinitely many nodes of B. The branch point c may be of infinite degree or not. Every component of T without the nodes of H is a locally finite tree, finite or infinite, joined to H by one edge, with no condition on the sizes, the depths or the labels of the components, on the gaps between the nodes of B along the rays or on the order in which the nodes enter. The infinite core now branches, with three directions to be extended. With the root c, the settled part of the core is kept as the union of three finite intervals of the rays starting at c, and the three untreated infinite parts beyond its three ends are held by three relations, the core sides, none of which is ever part of the locally finite side. A first turn at c makes the three core sides (Lemma D_Ψ): if c is in B, one separation at c with three moved parts, which also makes the root side of c; if c is not in B, the root turn of the procedure of v1.48 at c, which settles the finite branches at c and separates its infinite children, those outside the core starting the locally finite side. A core turn in any direction extends the settled part by one node in that direction, on a core tree that holds only the untreated part in that direction, while the core sides of the other two directions, each joined at its own end, the root sides and the locally finite side are protected and only receive new nodes (Lemma K_Ψ): at a node not in B by a turn of the procedure of v1.48 (Lemma C_Ψ), and at a node b of B by the advance part of such a turn followed by one separation, whose kept part becomes the new root side of b (Lemma J_Ψ). Round ρ = 1, 2, ... has the core turns at the ρ-th nodes of the three rays, in a fixed order, and then one root turn at each node of B in the settled part, the root sides being counted after the three core turns; each root turn settles a finite branch or introduces an infinite one by the separation lemma of v1.48 and is followed, whenever the locally finite side has a relation, by one turn of v1.48 on a locally finite tree that grows at the core turns and at the introductions (Lemma E_Ψ); a round with no root turn, possible only in the first rounds when c is not in B, has one such turn if the locally finite side has a relation. The root turn at b with index k falls in round max{1, a(b)} + k - 1, a(b) being the round in which b is settled (a(c) = 0 when c is in B). One turn runs at a time and at most one elementary step at a transition (Lemma U_Ψ). A global invariant gives the covering, the joints and the minimality at the turn boundaries (Lemma V_Ψ). Every wait is finite and every round ends after finitely many stages, with at most 3ρ + 1 root turns; once the locally finite side has a relation, every later round has at least one turn of it under the least-label rule, and every node is settled at a finite stage (Lemma F_Ψ); no settled pair is ever deleted (Claim Ψ3). For every such tree, budget 1 reaches P_T from every starting stage and every initial structure, with lower and upper limits both P_T (Theorem Ψ): each pair of P_T is present, and each other pair absent, from some stage on that depends on the pair, and no stage common to all pairs is claimed. With v1.48-v1.56 this holds for every tree with node set N_0 whose nodes of infinite degree are finitely many, or infinitely many and all on one ray or all on one double ray, or whose least connecting subtree is such a tripod (Corollary Ψ6), and the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Corollaries Ψ5 and Ψ6). Unit-budget reachability is not established here for trees with infinitely many nodes of infinite degree whose least connecting subtree has a branch point and is not such a tripod. Online constructions and other growth sequences are not treated. The proofs of Theorem Ψ and Corollaries Ψ5 and Ψ6, with Procedure Ψ, were adopted by the review on condition of four corrections, made in this text, in an AI-assisted reading that takes the cited basic lemmas of earlier versions as known. No novelty is claimed. The proofs are presented as AI-assisted symbolic arguments, without machine enumeration or numerical computation on the relations studied.