Observational Indistinguishability: An Axiomatic Limit on Verification from Within a Bounded Perceptual Domain

  We give an axiomatic account of one narrow claim: an observer with finite-resolution sensors cannot, from within its
  accessible domain, verify that its perceptions track an external reality rather than a perceptually complete rendering
  of one. We formalize an observer as sensor channels, resolution thresholds, a temporal rate, and an accessible state
  domain, and define a rendering to be perceptually complete when its error lies everywhere below the observer's
  detection thresholds. Under a deterministic-readout idealization, three near-immediate results follow: no admissible
  procedure distinguishes a perceptually complete rendering from reality; no finite aggregation of such observations
  does either; and, for a single observer under irreversible action, any distinguishing procedure requires leaving the
  accessible domain, which by definition dissolves the observer. We then give the stochastic generalization that real,
  noisy sensors demand: distinguishability is governed by the statistical distance between the induced observation
  distributions and the sample budget (Neyman–Pearson, Chernoff–Stein), so the limit relocates from the single-shot
  threshold to the best attainable resolution, and no further.

  We separate verification (a sound internal certification) from knowledge (which, on reliabilist semantics, a veridical
  observer may possess without being able to verify it); the thesis is therefore not skeptical. We situate the
  framework against Cartesian skepticism, Putnam's semantic externalism, Quine–Duhem underdetermination, Moorean and
  contextualist anti-skepticism, Bostrom's simulation argument, Chalmers' virtual realism, the information-theoretic
  data-processing inequality and Blackwell sufficiency, and — on the physical side — the detectability results of
  Beane–Davoudi–Savage and the quantum floor on resolution (Heisenberg), together with the Bell constraint on what a
  complete rendering of a quantum observer must reproduce, answering the strongest objection from each. The defensible
  residue is modest: verification-from-within is bounded by the observer's best attainable resolution. Whether
  perceptually complete renderings must always hide a discrepancy — the Hidden-Discrepancy Schema — is presented
  honestly as proto-formal and left open, with three documented obstructions.