Gaussian charge fluctuations of the eight-state native clock

We derive the Gaussian charge-fluctuation limit of independent copies of the complete eight-state native clock in its product ground state, with the native occupation operator and additive clock dynamics. The limiting charge field has variance 1/4, four positive-frequency modes and a canonical conjugate constructed from the charge velocity. Its time covariance recurs exactly after eight native ticks. This recurrence provides an exact obstruction to identifying the native product-clock covariance, by any nonzero scalar rescaling, with the covariance of a nonzero square-integrable profile of the full transverse cubic photon field: the latter has an atomless frequency measure. Matching equal-time Gaussian widths therefore cannot determine the physical photon source coefficient. Independent spatial smearing retains the four flat frequency bands; interacting emergence is outside the result’s scope. The general product-state quantum central-limit method is established. The contribution is the explicit native-clock calculation and its dynamical consequence for source calibration. Independent preparation and additive dynamics are premises, and the physical source magnitude and event energy–duration identification remain open. The accompanying source archive contains the derivation, source snapshots and provenance, and a reproducible exact checker with 34 passing checks. The limit and spectral comparison are analytical results, not newly Lean-formalized theorems. AI-use disclosure: OpenAI Codex assisted with mathematical development, exact checks and drafting. Jonathan Washburn is the author; no external peer review or independent human verification is claimed.