Global Executable Uniqueness Theorem: Why Only One Reality Compiles

Abstract Version 3.0 of this paper claimed a triple filter---source, executability, ergodicity---converging on the unique generator D = 5. The ergodic layer is asserted as a theorem retracted: its compatibility condition restated executability verbatim (the “third filter” was the second); the Ergodic Compilation Theorem contained a non sequitur (joint non-ergodicity does not imply node confinement); and pure Branch 1 is provably never transitive. An erratum fixes the ambiguous typesetting of the carrier differential. The replacement is an execution layer of four exact theorems---the complete temporal theorem of the carrier: uniform, not mixing; never frozen; never returning. Theorem A (Bernoulli kernel). The two-layer Branch-1 map is a register bijection preserving the uniform product measure, successive layers independent and uniform---per bit-plane. The ensemble half of the v3.0 programme, now exact. Theorem B (no-go). On any finite loopless graph the map is never a single cycle on non-zero states; on even-coordination carriers (FCC, k = 12) it carries an explicit fixed-point and 2-cycle family---the vacuum family. Strong ergodicity is unattainable, not merely unproved. Theorem C (liveness). The phase-accumulator arithmetic guarantees Θi ≥ 1/13 on every trajectory, permanently. Freeze and Θ → 0 are arithmetically impossible under the current microcode. Theorem D (non-closure). Under the Existence Tax, the global state of the world never repeats. The registers are a permutation (all trajectories are cycles), but the budget ledger Σ lies outside the permutation, grows unconditionally, and births new nodes (Genesis); the arrow of time is the tax ledger. A world that closed would fail at the tick of closure in both directions of the architecture: as unpaid existence (the ledger would have to not grow, contradicting A7) and as an already fully read stream (a periodic stream compresses to its period; zero residual description gain). The classical doctrine of eternal return is thereby not refuted but shown non-executable; its explanation is exactly one axiom deep. Only one loop exists in the architecture---the static, definitional loop of the source (closure 1 − x2 = x, whose unique solution grades the scale system {φ−1, φ−2, φ−3} = closure point, voxel, Existence Tax); the world is its non-terminating expansion. To exist is not to have closed. Theorem C initially stood in conflict with two corpus claims (the v3.0 singularity corollary; PHOTON-2’s horizon closure). The conflict is resolved by identifying the horizon as a break at the first seam of the reading pipeline: no commit ⇒ no photon ⇒ no qualia “light.” A commit is not a firing of the accumulator; it is a firing translated. At the horizon the carrier still executes (Θ = 1/13: an unswitchable processor without a monitor), but nothing converts. The singularity is the limit of the model’s action, not a region of the carrier; Genesis is the same boundary seen in the other temporal direction. The main result survives as a double filter: Ωreal = Ωsource ∩ Ωexec = {5}, with a complete elementary proof of the generator gap (ηD ≥ 1+√2 for D ≠ 5; margin ×28), and two robustness results: Threshold Robustness (the selection is invariant under every threshold in (δ5, δ2)) and the reading of the window’s φ-calibration as a budget identity in tariff units---the tariff φ−3 being now derived by Paper Zero v1.3 as the unique self-consistent price of distinction. Finally, the paper assembles the corpus into its target: the Φ-Uniqueness Theorem. Lock 1 (Source; Zero v1.3): the canonical formulation of reality forces closure, yielding the unique generator φ, the quadratic class, and the graded scale system. Lock 2 (Carrier): among quadratic generators exactly one is executable, robustly. Lock 3 (Execution): the translation is lossless, unstoppable, and non-closing. One named hinge: one tick of Φ costs the self-consistent price (ΔΣ = φ−3, ARCH-1 A7). The translation admits exactly one executable instantiation: generator φ, price φ−3 per tick, tempo one hop per tick, bijective, live, open.Φ is φ-valued. Keywords: executable uniqueness; ergodicity retraction; Bernoulli kernel; non-transitivity; liveness; non-closure; existence tax; golden ratio; threshold robustness; horizon; commit; translation; Ontological Resolution Theory.