58 / C6-I: Scale-Normalized Critical Debt, Capacity-at-Infinity Compactification, and Barrier-Accumulation Cycles

C6-H left behind two negative results — not every reserve going to zero qualifies as a physical node, and a finite kinetic-energy budget is structurally unable to supply a consistent cost for a scale-invariant boundary event — and proposed switching to a "critical-barrier debt" instead. C6-I formally does three things: systematically computes the Navier–Stokes scaling degree of every major quantity built up across the C6 series, establishes a general-purpose criticalization operator, and determines which quantities are genuine critical currency, which "raw divergences" are merely artifacts of scale rescaling, and whether critical barriers can really accumulate into a contradiction. C6-I.1 Criticalization Operator defines: for any event quantity with scaling degree $d_Q$ ($Q_\lambda=\lambda^{d_Q}Q$), paired with an event scale $r\mapsto r/\lambda$, define $Q^{crit}=r^{d_Q}Q$ — proved exactly invariant under synchronized rescaling. Using this tool it recomputes the main quantities the series has built up so far: C6-F's shared-source-bridging intermediate load ($M_J$, degree $1\to\mathfrak M_J^{crit}=rM_J$), the operator's positive mass ($P_J$, degree $3\to\mathfrak P_J^{crit}=r^3P_J$), the same-instant core load, and the operator-times-derivative product cost (both factors degree $5/2$; C6-I.3 gives a fully scale-invariant critical junction inequality — "C6-F's TS-to-forcing bridge loses no scale strength once exactly criticalized"). More importantly, it checks one by one that the already-published external theorems themselves land exactly at degree $0$: Miller's intermediate criterion (verified exactly via its own exponent relation $2/p+3/q=2$), Miller's operator criterion (exponent $p=2/(1+\alpha)$, C6-I.4), the Cheskidov–Dai high-frequency shell tolerance (up to a dyadic shell-index shift), the pressure $L^{3/2}$ quantity, and the CKN local quantity — confirming that the entire C6 mechanism is fully compatible with the literature's own scaling structure, not talking only to itself. It also honestly draws a critical boundary line: criticalization restores scale consistency, not global additivity — a local critical quantity of degree zero still gives no guarantee of summing to something finite across arbitrarily many nested scales. This formally leads into the round's second, and more important, result. C6-I.6 Raw-Infinity No-Go corrects an imprecision C6-H itself left behind: if a raw capacity $C$ has positive scaling degree $d_C>0$, then "raw $C\to\infty$" along a sequence of events shrinking in scale can be purely a UV-rescaling artifact — even for the very same normalized event, it will appear to diverge simply because $C_n=\lambda_n^{d_C}C_0\to\infty$ — so C6-H's $B_{CAP^\infty}$ must be replaced with a genuinely scale-invariant $B_{CAP^{crit,\infty}}$ (using the criticalized capacity or a dimensionless relative capacity); it verifies that the capacity blow-up of the Duhamel quantity ($\Gamma^{-1}$) and of operator efficiency is already dimensionless and genuinely legitimate, and gives the corrected, fully scale-invariant middle-gap critical dichotomy (C6-I.7). The updated physical boundary alphabet still has six names, but $LOAD$ and $CAP^\infty$ now both use criticalized/relative definitions — "the boundary ontology is finally scale-consistent." The round's real central impact is the second no-go: even with correct criticalization, genuinely scale-invariant, with a fixed positive critical cost at every generation, a finite-time contradiction still cannot be forced — a geometric scale ladder $r_n=r_0a^{-n}$ paired with a parabolic window $|J_n|=\theta_nr_n^2$ automatically gives $\sum|J_n|<\infty$ (finite total physical time), while even if the critical dissipation at each generation satisfies $r_n^{-1}D_n\ge d_0>0$, the raw energy cost is only $D_n\ge d_0r_n$, which likewise gives $\sum r_n<\infty$ — C6-I.8 Critical Zeno Compatibility Lemma and C6-I.10 Barrier-Zeno No-Go therefore prove: finite global kinetic energy, finite remaining physical time, and infinitely many fixed positive critical-cost events can all be simultaneously compatible — explicitly flagged as not a construction of a genuine N–S singular solution, but a scaling-type no-go against an entire class of cycle-elimination arguments. The same logic extends to "critical-barrier accumulation" itself: even if a killer-barrier coordinate $b_n$ must satisfy $\ge b_{crit}>0$ at every generation (forcing $\sum b_n=\infty$), that divergence is itself not a contradiction unless $\sum b_n$ carries some independent finite upper bound — so "critical-barrier accumulation" alone is not enough to eliminate an infinite Zeno cycle; Miller's own "must diverge" criterion is, in essence, compatible with a hypothetical blow-up, not a refutation of it. The document further splits critical coordinates by logical role into three types (Type $K$ killer barriers, Type $D$ must-diverge currency, Type $C$ combinatorial efficiency), explicitly stating that neither type of barrier alone constitutes a contradiction — genuine cycle elimination needs either cross-coordinate incompatibility or a finite/telescoping critical budget — and it lists four candidate "genuine currencies" that might work (a finite log-scale measure, a cross-generation telescoping potential, a monotone critical flow, a finite-step barrier-to-external-gate theorem), explicitly admitting that none of them exist yet. The round finally reframes the whole problem as a genuine dynamical-systems question (C6-I.9 Renormalized-Cycle Reframing): introducing the log-scale variable $s=-\log r$, the UV limit $r\downarrow0$ becomes $s\to+\infty$, and the geometric ladder becomes an arithmetic sequence $s_n=s_0+n\log a$ — the genuine recurrence question is therefore rewritten as: can the criticalized state, in this log-scale time, form a fixed point, a periodic orbit, or a compact recurrent set, or must it necessarily drift toward the $\mathrm{REG}$/legality boundary — "this is more faithful than summing raw event energies." Formally hands off to C6-J — Log-Scale Renormalized Defect Flow, Telescoping Potentials, and Critical-Cycle Closure Tests, with eight constituent obligations (J1, defining log-scale generations, through J8, the cycle-closure verdict).