57 / C6-H: Critical Boundary-Face Transition Graph, Debt-Coercivity Audit, and Boundary-Cycle Elimination

C6-G compressed C6's interior graph to $\{TS,GP,HF\}$ and proved there is no certified interior SCC at all; any infinite survivor must belong to one of uniform GP, uniform HF, approaching some boundary superclass $B_\ast\in\mathfrak B$ (C6-G roughly listed ten), or the legality exit. C6-H's original task was to treat these ten boundary faces as nodes, search for $B_i\to B_j$ edges and a global finite debt, and try to eliminate a boundary SCC — but it begins with two key corrections. First: not every "reserve tending to zero" is entitled to become a physical boundary node — the paper sorts boundaries into four types: Type $P$ (a physical-state boundary, an observable limiting property of the PDE state itself, which can legitimately become a compactification node), Type $E$ (an edge failure, meaning only "the attempted edge did not compose successfully," which does not determine a unique physical state and cannot be promoted to an SCC node), Type $N$ (a normalization boundary, e.g. $LOAD$ — a cost that shrinks relative to the chosen normalization scale), and Type $\infty$ (an infinite boundary, e.g. $CAP^\infty$ — a normalized capacity ratio that diverges, but becomes legitimate after compactifying via $\widehat C=C/(1+C)\to1$). C6-H.1 Edge-Boundary Node No-Go therefore removes $B_{FIELD}$ and $B_{HER}$ from the physical boundary alphabet — they are merely "metadata of a failed transition attempt"; C6-H.2 Setup Quotient proves $B_{SETUP}\equiv\mathsf A$, likewise removed, first compressing the alphabet from ten down to seven. The second correction is the round's genuine mathematical core: under the standard Navier–Stokes parabolic rescaling $u_\lambda(x,t)=\lambda u(\lambda x,\lambda^2t)$, kinetic-energy dissipation is not scale-invariant — an exact computation gives $D_E[u_\lambda;I_\lambda]=\lambda^{-1}D_E[u;I]$, while most of C6's boundary reserves (Duhamel coherence, middle-gap ratio, axis angle, pressure signature, source-capture fraction, etc.) are dimensionless/scale-normalized, and stay exactly invariant under the corresponding rescaling. C6-H.3 Scaling Obstruction to Uniform Energy Coercivity therefore proves a clean counterexample theorem: for any event class $\mathcal E$ defined solely by scale-invariant metadata and closed under N–S rescaling, membership alone can never force a scale-independent positive lower bound $D_E\ge\varepsilon_0>0$ — taking an event of finite positive dissipation and letting $\lambda\to\infty$, the scale-invariant metadata stays inside $\mathcal E$, but $D_E(\lambda)=\lambda^{-1}D_E\to0$, a contradiction. This formally explains why so many individual "per-event debt" arguments throughout the C6 series have never been able to become coercive in the high-scale limit — not an oversight specific to any one round, but a structural scale-type mismatch. What genuinely suits UV recurrence is not a finite budget but a scale-critical barrier: if a scale-invariant quantity $b_n$ carries a regularity threshold ($b_nthis does not imply additivity. Cheskidov–Dai's high-frequency vorticity tolerance $\int\|\Delta_q\omega\|_\infty dt$ is a typical model of this (vorticity's own scaling and the dyadic shell-index shift cancel exactly, letting the threshold stay invariant across arbitrarily small scales), and Grujić–Xu's harmonic/sign sparseness is the geometric version of a critical barrier. C6-H.4 formally names this distinction — finite global budget vs. critical barrier — as the fundamental dichotomy C6 has been implicitly relying on all along. With this tool in hand, two clean dichotomy theorems further compress the alphabet: C6-H.5 Coherence Boundary Dichotomy proves that any $\Gamma_n=R_n/C_n\to0$ must, after passing to a subsequence, be either $R_n\to0$ (i.e. $LOAD$) or $C_n/R_n\to\infty$ (i.e. $CAP^\infty$) — so $B_{COH}\Rightarrow B_{LOAD}\vee B_{CAP^\infty}$, and $COH$ is no longer an independent terminal node; C6-H.6 Middle-Gap Boundary Dichotomy uses C5-E's exact inequality $\int_{\vartheta\le\delta}|S|^3\ge M_\delta/(\sqrt6\delta)$ to prove that middle-gap collapse likewise routes precisely into $LOAD\vee CAP^\infty$. After this second compression, the physical boundary alphabet has only six members left: $\mathfrak B_{phys}^{(2)}=\{LOAD,SEG,GEOM^{res},MEAN,PROV,CAP^\infty\}$ — this is the round's genuine structural achievement, compressing ten coarse superclasses down to six. But C6-H.7 Energy-Coercivity vs Critical-Coercivity Theorem honestly draws a boundary: all six remaining classes are, on their face, scale-invariant, and a finite kinetic-energy budget alone cannot supply any of them with a uniform positive per-event cost — a future successful proof eliminating boundary cycles must instead use a scale-critical global finite quantity, a monotone scale-normalized quantity, a cross-generation telescoping potential, or a proof that critical-barrier cost forces the external $\mathrm{REG}$ gate after finitely many steps — it can no longer simply "count total dissipation events." The round also identifies six genuinely certified boundary transitions (H-B1 through H-B6: $COH\to LOAD\vee CAP^\infty$, $GAP\to LOAD\vee CAP^\infty$, $SETUP\to\mathsf A$, plus three external closures given by already-published theorems: the harmonic-favorable side, the high-frequency critically-small side, and the pressure-favorable side each going to $\mathrm{REG}$ under their own theorem hypotheses), but explicitly states that $SEG\to GEOM^{res}$, $GEOM^{res}\to MEAN$, $MEAN\to PROV$, $PROV\to CAP^\infty$, or their reverses, are all not certified, so even among the six compressed physical faces, there is still no certified nontrivial boundary SCC (explicitly flagged: this is the current state of the research graph, not a PDE theorem). Formally hands off to C6-I — Scale-Normalized Critical Debt, Capacity-at-Infinity Compactification, and Barrier-Accumulation Cycles, with eight composition obligations (I1 computing the critical-scale degree of every surviving debt through I8 recomputing the boundary graph in critical coordinates).