The Structure of Coherent Construction: A Bivalent Trajectory Theory

Fixed-Scope Coherence, Tensor Determination, and Irrecoverable Failure This paper develops a bivalent trajectory theory of coherent construction over a finite fixed-scope substrate. It is the trajectory-theoretic member of a layered AASC foundational programme. The paper records the standing and admissibility structure required by the trajectory layer, then derives the exact path geometry forced once that substrate is fixed. The paper does not re-derive the upstream kernel, admissibility boundary, or retained-dependency substrate. Its contribution is narrower and more precise: it establishes how legal construction histories behave when standing must be preserved throughout their development. Within the declared regime, a trajectory is coherent exactly when every state along it retains standing: Coherent(tau) iff for every i, S(tau_i) The resulting theory is rigid. Coherence is binary, tensor-determined, invariant under identity-preserving transformation, closed under prefix extraction and legal composition, and irrecoverable once a standing-failing state appears. Fixed-scope substrate and local closure The theory is developed over a primitive substrate consisting of: a fixed state space; an admissibility predicate; a standing predicate; a licensed same-scope transition relation; a standing-equivalence relation; and a tensor-content assignment. At fixed scope, standing and admissibility are extensionally equivalent: A(x) iff S(x) States are standing-equivalent when they carry the same tensor content: x ~ y iff Ten(x) = Ten(y) The paper distinguishes anchors, tensors, and skin. Anchors are non-negotiable standing preconditions. Tensors carry standing-relevant compatibility content. Skin consists of post-admission descriptive surplus, including presentation, bookkeeping, and narrative. The local structural closure theorem proves that no admissibility-preserving auxiliary representation can provide a stricter standing-bearing refinement beneath tensor content while the fixed substrate remains unchanged. If two states have the same tensor content, every admissibility-preserving standing representation must assign them the same standing result. In plain terms, within the fixed substrate: no deeper standing generator is available; any auxiliary standing classifier collapses to tensor-determined bookkeeping; additional metadata may remain, but it cannot create a new standing-bearing layer; the theorem is local to the fixed substrate and does not settle cross-scope or substrate-changing extensions. Coherent trajectories A trajectory is a finite, nonempty sequence of states connected by licensed same-scope transitions. Legality concerns whether the transitions are licensed. Coherence adds the stronger requirement that standing is preserved at every state. The paper proves the following positive structural results: coherent trajectories remain coherent under prefix extraction; coherent trajectories are closed under legal composition; identity-preserving, tensor-preserving transformations preserve coherence; coherence is constant across trajectories with the same tensor content at corresponding stages; skin-only changes cannot alter the coherence status of a legal trajectory. Thus the coherent trajectory class is closed under the principal structural operations available inside the fixed substrate: coherent trajectory -> coherent prefix and coherent trajectory + coherent composable trajectory -> coherent composite These are internal fixed-scope results. They do not authorize gluing across distinct substrates, scopes, or incompatible construction envelopes. Bivalence and the exclusion of partial coherence Every legal trajectory is either coherent or incoherent. There is no primitive intermediate coherence status: Coherent(tau) or not Coherent(tau) The result does not prohibit descriptive summaries such as defect counts, collapse depth, or the position of the earliest failure. It does establish that such quantities are secondary descriptions of a prior binary structural fact. They are not primitive degrees of coherence. This distinction prevents coherence from being treated as a scalar score, a graded quality measure, or a retrospective evaluation attached to an otherwise completed construction. Collapse and irrecoverability The negative side of the theory is governed by failure witnesses. A legal trajectory is incoherent precisely when at least one state fails to retain standing: not Coherent(tau) iff there exists i such that not S(tau_i) The central no-deferred-coherence theorem states that if an incoherent trajectory occurs as a prefix of a legal trajectory, the full trajectory remains incoherent. Later continuation cannot erase the earlier failure: incoherent prefix + legal extension -> incoherent extension The stronger irrecoverability theorem proves that any legal extension of an incoherent trajectory remains incoherent, provided the original history is retained as a prefix of the same fixed-scope lineage. The paper also proves downstream propagation of standing failure. Once a state fails standing, every later state reachable through the same legal trajectory also fails standing. The earliest failure therefore controls the entire downstream tail. This establishes a fail-closed path structure: failure is witnessed at a definite state; the witness persists under extension; the failure propagates downstream; no later continuation restores coherence; favorable later descriptions cannot repair the inherited trajectory. Terminal certification and path memory Although coherence is defined over complete trajectories, the fixed fail-closed substrate yields a terminal-state criterion: Coherent(tau) iff S(last state of tau) This is not a reduction of trajectory coherence to ordinary static endpoint appraisal. Terminal standing certifies the preceding trajectory only because the standing predicate is history-sensitive and earlier failure cannot remain hidden under legal continuation. The endpoint criterion therefore compresses retained history rather than discarding it. Two legal trajectories in the same fixed substrate with the same literal terminal state have the same coherence status. The result does not extend automatically to merely similar endpoints drawn from different scopes or state spaces. The paper also proves exact factorization under composition: Coherent(tau1 composed with tau2) iff Coherent(tau1) and Coherent(tau2) Consequently, coherence cannot be produced by aggregation. An incoherent prefix cannot be repaired by appending a coherent suffix, and an incoherent suffix cannot be domesticated by placing it after a coherent segment. Principal structural consequences Within the declared finite fixed-scope regime, coherent construction is: bivalent rather than graded; tensor-determined rather than skin-dependent; stable under prefix extraction; closed under legal composition; invariant under identity-preserving transformation; endpoint-certifiable under the fail-closed standing predicate; non-aggregative under composition; irrecoverable after standing failure; internal to one fixed substrate and scope. The theory therefore differs from static consequence systems, probabilistic coherence programmes, revision-based repair models, and dynamic-information approaches in one specific respect: path status is not an external appraisal imposed after construction. It is determined by standing preservation along the legal construction history. Scope and nonclaims The results are exact relative to the declared fixed substrate and finite-trajectory regime. The paper does not claim to establish: cross-scope transport or comparison; branching developments; infinite trajectories; interacting-agent dynamics; probabilistic enrichment; alternative substrates with different scope data; a global theory of every possible construction process; a new primitive metric or scalar measure of coherence; restoration of coherence through later reinterpretation or repair. The local no-lower-generator theorem concerns admissibility-preserving auxiliary representations on the same fixed substrate. It does not by itself establish that the substrate is derivable from a weaker base, nor does it settle broader governance-level closure questions. Place in the AASC arc This paper supplies the internal trajectory layer following the upstream kernel, admissibility, and anchor-tensor-skin results: Non-Degenerate Construction and the Kernel of Admissibility The Structure of Admissibility The AMetric Boundary as a Unique Regime-Invariant Constraint Anchor, Tensor, and Skin The present paper: The Structure of Coherent Construction The paper’s role is to derive the finite fixed-scope path structure once that substrate has been established. It provides the bivalent, fail-closed trajectory interface required for later work on retained history, endpoint certification, and compositional construction. Keywords coherent construction; bivalent trajectory theory; fixed-scope substrate; standing and admissibility; tensor determination; anchor tensor skin; identity-preserving transformation; legal construction paths; prefix closure; composition closure; no deferred coherence; irrecoverable failure; terminal-state certification; path memory; AASC