The Number of Waves Is the Resolution of the System: Externality of Counting in Closed Systems and Steady-State Underfilling

This paper presents two claims. Claim 1 (externality of counting): the number of waves N-cal is nothing but the number of terms of the closure sum_{k=1}^{N-cal} x_k^2 = 0 - the maximum harmonic index with respect to the base frequency when one writes the sum as R^2, that is, the resolution of the system itself - and it is not a quantity the system determines internally. This is not an assumption but a consequence of zero closure (the zero right-hand side designates no privileged basis, fundamental oscillation, or decomposition depth), the anonymity of components, the unreadability of phase, and the flattening of the closure tower (no internal readout can distinguish a reorganization of harmonics): no internal readout exists whose result is N-cal, so N-cal belongs to the specification of the representation, not to the state. The externality of N-cal is equivalent to the preservation of anonymity (representation independence). Claim 2 (conditional ceiling and underfilling): once a counting convention C = (omega_0, epsilon) - a frequency resolution and a relative amplitude floor - is given, the number of readable waves acquires a ceiling n_max(C) = min(floor(Omega/omega_0), 1/epsilon^2). The ceiling is transiently attainable under the iterated dynamics, but the steady state does not reach it and stabilizes below it. Both pillars come from closure: the amplitude-floor ceiling is a pigeonhole consequence of the fixed sum of allocation ratios and disappears in open systems; the frequency-slot ceiling comes from the bounded band of the discrete update (Nyquist type). That a ceiling exists is a consequence of closure (convention invariant); what the ceiling is equals a convention (relative). Numerically, in the unfrozen dynamics of the N-body relational-wave model (N = 8, 12, 16), no convention among time series and 352-cell convention grids ever exceeded the ceiling; transient contact with the ceiling was observed (N = 8); and steady-state filling ratios remained at 0.1-0.4. The mechanism is the coexistence of amplitude thermalization (participation ratio rising from 0.52 to 1.0) and band-edge frequency condensation (occupied frequency classes dropping from 8 to 3). Reanalysis of one and the same run under a grid of conventions changes the readout count from 1 to 7 - the count is an attribute of the readout - and lowering the floor without limit raises the ceiling without limit, which is why the number of waves appears to grow indefinitely. Structural counterparts in standard theory (resolution dependence of parton number, observer dependence of particle number) are discussed as an alternative mapping, not a replacement.