Bridge Necessity, Nonterminality, Selection Jump, and Bridge Trichotomy

We develop an abstract metatheory for mathematical statements equipped with an \emph{exact selected closure presentation} over a background theory. A selected presentation fixes distinguished closure channels---for example bad-witness channels, stagewise certificate channels, or arithmetized channels---and thereby makes terminality relative to those channels mathematically precise. At the single-instance level we prove a channel-exhaustion principle: once a base theory proves that a sentence is equivalent to each of its selected channels, every extension of that base theory proves the sentence if and only if it proves any selected channel. The main results are family-level barrier theorems. First, finite prefixes of stagewise channels are uniformly insufficient: no fixed finite initial segment is a general finitary bridge. At the same time, for each represented decidable stagewise predicate \(R\), Robinson arithmetic already proves the single-sentence equivalence \[ \forall M\,\forall n<M\,\exists c\,R(n,c) \;\leftrightarrow\; \forall n\,\exists c\,R(n,c), \] so the genuinely infinitary step is the passage from the numeral family \(\{\Pref_R(\overline M):M\in\N\}\) to that universal closure via the \(\omega\)-rule. Second, for undecidable co-c.e.\ selected families \[ A=\{x:\forall w\,\neg B(x,w)\} \] with decidable bad-witness predicate \(B\), there is no decidable one-shot positive certifier and no partial computable exact-domain compiler into finite positive certificates. Third, using a primitive recursive non-halting predicate \[ U(e,t)\equiv \neg \KleeneT(\pi_0(e),\pi_1(e),t), \] we construct a universal \(\Pi^0_1\)-selected class \[ \PiTrue=\{e:\forall t\,U(e,t)\}, \] prove it \(\Pi^0_1\)-complete under primitive recursive injective many-one reducibility, and derive both external classifier barriers and an internal certification-collapse theorem: in any consistent recursively axiomatizable theory \(T\supseteq \ISone\), a same-theory terminality predicate adequate along a \(\Pi_1\)-universal embedding yields the full \(\Pi_1\)-reflection scheme \(\RFN_{\Pi_1}(T)\), and is therefore impossible. Fourth, we prove an exact threshold theorem for Tarski barriers: an arithmetic exact terminality predicate exists on a truth-faithfully embedded sentence fragment if and only if the truth set of that fragment is arithmetical. This yields a selected Tarski barrier for truth-universal classes and a selected diagonal barrier for diagonally universal classes; full internal biconditional schemes along truth-universal images collapse directly to inconsistency. Fifth, we isolate a distinct structural mechanism, the \emph{selection jump}. For every stagewise-local selected class of the form \[ \Suc_{\Cert}(e)\equiv \forall n\,\exists t\bigl(\KleeneT(e,n,t)\wedge \Cert(n,\KleeneU(t))\bigr) \] with decidable local verifier \(\Cert\), the existence of one successful seed forces \(\Pi^0_2\)-universality: every nonempty such class is automatically \(\Pi^0_2\)-complete. Sixth, we replace the weak language of ``singular bridges'' by a precise fixed-proposition \emph{bridge trichotomy}. For a proposition \(P\) with an exact two-sided decidable selected package, the bare semantic notion of an isolated bridge sentence is vacuous; the meaningful fixed-proposition bridge object is the effective bridge class \[ \{e:\forall n\,\exists t\,(\KleeneT(e,n,t)\wedge R_P(n,\KleeneU(t)))\}, \] and that class is either empty or \(\Pi^0_2\)-complete. Any same-theory adequate terminality principle on a direct \(\Pi_1\)-universal image of that class yields a Gödel--Turing ladder via reflection collapse. Seventh, passin