13 / C3-K: The Absolute Occupancy Worldvolume, Subthreshold Flux Variation, and the One-Moment Critical Gap

C3-J proved that counting re-entries is gauge-dependent. This round switches entirely to the absolute shell identity $q$ and a fixed threshold $\beta$, independent of any moving frontier, defining $A_{q,\sigma}(\beta)=\{t:a_q^\sigma(t)\ge\beta\}$ — a definition that is completely gauge-invariant. The Absolute Active-Worldvolume Budget (Theorem 4.1): using an annular Bernstein lower bound together with the global energy inequality, it proves $\sum_{q,\sigma}\lambda_q|A_{q,\sigma}(\beta)|\le CE_0/(\nu^3\beta^2)$ — a genuinely gauge-invariant, finite weighted occupancy measure. But hypothetical blow-up still requires the support to escape to infinite frequency, so the C3-K congestion signature is "finite weighted occupancy plus unbounded shell support," not "total blow-up." The same method yields the Weighted Hysteretic Activation Count (§12), but this is immediately shown to be entirely compatible with an infinite genealogy where $q_n\to\infty$, even if each individual shell is used only once — a gauge-invariant version of the Zeno no-go. It further proves finiteness of local subthreshold energy turnover (Theorem 17.1): as long as neighboring shells stay subthreshold, the absolute variation has a finite global budget, so infinite turnover, if it exists, must concentrate in the critically active neighborhood. The paper's most important explanation is in §22–24: ordinary energy transfer uses $|\dot e|$, but critical pair production uses $\lambda|\dot e|$ — carrying one extra power of frequency weight (the One-Frequency-Moment Gap). An abstract counterexample (§23) shows this directly: $X_n=\lambda_n^{-1}$ is summable, but $Y_n=\lambda_nX_n=1$ is not — this precisely explains why every previous attempt to close the problem with an energy budget has ultimately failed: not because the bookkeeping wasn't fine enough, but because it was missing exactly one whole frequency moment. The document also gives the One-Moment Occupancy Barrier (§28): global energy controls the $\lambda^1$ occupancy moment, but the natural time rate of critical renewal lives at $\lambda^2$ — with precisely one whole frequency moment missing in between.