4D from the Sharp Terminal Retained Image: Terminal Self-Rereading, Exact Retention, and Viewpoint Reconciliation

We prove a four-dimensionality theorem for a universe specified only as a terminal self-rereading object.  The root axiom is fixed in plain form: the universe can read itself, retain exactly what is publicly rereadable, and reconstruct itself again without external support, indefinitely.  Its formal unpacking gives finite public rereading, exact retention, indefinite non-static self-reconstruction, public viewpoint reconciliation, and terminal closure.  These clauses are not auxiliary physical assumptions; they are the typed content required for the primitive verbs ``read,'' ``retain,'' ``reconstruct,'' and ``close without external support'' to have mathematical force.  No background spacetime, manifold, metric, dimension, recovery map, rank-two carrier, Fisher geometry, or four-dimensional public carrier is assumed.  The proof uses standard theorem families: behavioural quotient theory constructs the retained object; coalgebraic final semantics supplies terminal rereading; image-kernel factorisation constructs the sharp image; finite generated statistical response geometry supplies finite public response-germ quotients; the minimal branch-retaining quotient theorem forces the renewal window to have exactly one present quotient line and one successor-innovation quotient line; the terminal public viewpoint image is defined as the full finite-public automorphism image inside the finite Fisher--Hellinger orthogonal group, so compactness follows by the closed subgroup theorem rather than from an arbitrary subgroup assumption; and the sharp reconciliation carrier is the minimal nontrivial public rotation quotient of the terminal viewpoint circle, hence has real rank two after finite ineffective-kernel removal.  The behavioural part of the theorem constructs the sharp $(2,2)$ public comparison image.  Fisher--Hellinger information geometry then equips the internally generated public response-cotangent quotient with a nondegenerate metric.  The resulting public response carrier decomposes canonically as two renewal-window directions plus two sharp viewpoint-reconciliation directions, hence has real dimension four.  Once the exact successor orientation is read as a unit renewal direction in this positive four-dimensional carrier, the standard reflection construction \(g^{\mathrm L}=g_{\FH}-2U^\flat\otimes U^\flat\) supplies a canonical Lorentzian comparator of signature \((-+++)\).  Because the primitive public data are finite statistical response laws rather than deterministic hidden labels, the retained object is naturally closer to an operational state space than to a classical phase space.  The present theorem derives the four-dimensional public response carrier and its local Lorentzian comparison interface, not causal dynamics, field equations, or the full quantum formalism.