Truncation Levels and Spacetime E: Satisfiability, Unsatisfiability, and an Explicit Truncation-Level Chain for the Bridge Hypothesi

This paper audits the premise structure of the conditional truncation theorem,shifting the burden of proof from the conclusion to the satisfiability decision of thepremises. Within the constraint class jointly determined by information conservationand program-level computability, four levels of results are given. First, a semantictrichotomy at the definitional level: truncation events are distinguished into saturation points, legal truncations, and structural truncations, three logically separatedconcepts; a saturation point is a purely intrinsic counting event at which the carrierinjection condition fails, and the chain can still be legally extended after the saturationpoint. Second, a no-go theorem: within the decidable chain class, the clause in thestrong-generation definition stating that “a new state cannot be recovered by finiterecoding” is a trivially empty predicate under the point reading, and is incompatible with program-level computability under the uniform reading, so this clause isunsatisfiable; the tightness of the two ends of the scope is given respectively by aconstant-encoding counterexample and a random real-prefix chain. Third, disjunctive compression: the global single finite-description reading satisfies a boundednessdichotomy, the unbounded growth of the growth bridge directly conflicts with thisdichotomy, the first joint condition is unsatisfiable, and the three-branch disjunctionis compressed into two branches. Fourth, satisfiability classification: at the two levelsof single premises and joint conditions, explicit witnesses, explicit counterexamples,or unsatisfiability proofs are given cell by cell; the second joint condition is witnessedby a fixed-carrier prefix chain, the truncation point is precisely characterized by theintrinsic invariant dk as k∗ = max{k : dk ≤ N}, and the counting-type saturationupper bound t ≤ N −n0 is attained on unit-increment split-free chains, hence is tight;the third joint condition is given a degenerate witness by the constant chain and anondegenerate associativity witness by the uniform coupling tower, while the nonassociative alternative-algebra witness is registered as open. Eight groups of machine-checkprotocols are executed under exact integer arithmetic, and all pass; among them, theoracle-chain side is registered at the proof level and does not enter machine-check decision. The deterministic conclusions of the entire paper are limited to discrete integerindices, information conservation under the injective reading, and program-level computability under the Turing-machine reading; continuous indices, random indices, andthe intrinsic computation of dk on general chains are registered as open problems.