The "Exact Copy" Problem in Infinite Space: Continuity, Verifiability, Dynamical Amplification, and Criteria of Individuation

The Level I multiverse argument in cosmology holds that if space is infinite and matter is distributed ergodically, then somewhere far away there must be infinitely many observers exactly identical to us. This paper splits “exactly identical” into four independent questions: the ontic layer (are two objects exactly the same?), the epistemic layer (can measurement establish that?), the dynamical layer (does the same initial state lead to the same evolution?), and the identity layer (is the same state the same individual?). The main results are as follows. (1) Ontic layer: in a model with a continuous state space and an atomless conditional law, any countable family of regions in infinite space is almost surely pairwise distinct (Theorem 2); at finite resolution, spatial ergodicity implies that every class of positive probability recurs infinitely often, and this can be stated directly as a theorem on a stationary random field over the continuum, with no partition of the universe required (Theorems 1″ , 3). The two conclusions point in opposite directions but do not contradict each other. (2) Whole histories: a history within a finite spacetime window is still a finite-state object at finite resolution, so a copy that is “the same from birth to death at every moment” has positive probability at finite resolution and probability zero in the exact sense; its difference from a “same at a single moment” copy is one of degree (the recurrence interval grows exponentially with the length of the history), not of kind (Proposition 4). (3) Epistemic layer: a measurement of finite resolution can only falsify “sameness”, never confirm it (Proposition 5); in the quantum case the mechanism is the distinguishability of non-orthogonal states and sample complexity, not “the measuring tool cannot be made infinitely small”: the uncertainty relation does not forbid arbitrarily precise measurement of a single observable, and elementary particles of the same species are strictly identical. (4) Dynamical layer: for threshold spiking networks we prove a “margin theorem” (Proposition 6): a perturbation smaller than the margin changes no spike, and the margin is tight. In a whole-brain model based on the fruit-fly connectome (127,400 neurons, 14.69 million synapses) the margin is about 10−4 mV (median 2.1 × 10−4 mV over 12 conditions, range 4 × 10−6 to 3.3 × 10−4 mV); perturbations below the margin, including float64 rounding, are absorbed completely; once the margin is crossed, in 75% of cases the microscopic spike trains amplify, within a median of 60 ms, to the same difference rate as when the stimulus realization is replaced, while the output distribution of the downstream motor neuron MN9 is not significantly different from that obtained by changing the random seed of the stimulus. Thus “the initial state is the same at finite resolution” does not imply “the history is the same at the microscopic level”. (5) Identity layer: under the criteria of exact physical identity and of numerical (historical) identity, the claim that “there is only one me” holds; under a functional criterion the existence of copies is a conditional conclusion; counting the field environment as part of the “structure” does not change this unless consciousness depends sensitively and without bound on the environment.