56 / C6-G: Typed Cross-Domain Graph Rebuild, Joint-Node SCC Audit, and Minimal Boundary-Saturated Survivor Cycles

Once C5-M's coarse residual graph $\{A,T,G,P,H,F\}$ is projected into an ordinary directed graph, one can easily read off an apparently large $\{G,P,H,F\}$ SCC — but this is only the first layer of compression; C6-A already pointed out that a projected label SCC does not imply a composable PDE recurrent cycle. C6-B/C compressed $H\leftrightarrow F$ into the nonlinearly coherent re-entry node $HF_{\rm coherent}$, C6-D proved $G\leftrightarrow P$ is mostly same-event compatibility and quotiented it into the joint node $GP_{\rm hereditary}$, and C6-E/F proved $T$ is only the time marginal of a genuine spacetime source state, promoted it to $TS_{\rm hereditary}$, and built the first batch of typed cross-domain bridges $TS\to GP$, $TS\to HF/F/H/\mathrm{REG}$. C6-G now formally rebuilds the whole graph and redoes the SCC audit. The central tool: every edge now carries two labels — a proof status ($I/C/N/E$) and a temporal semantics ($S$ same-event static, $D$ genuinely cross-generation dynamic, $E$ external closure) — only $D$ edges can compose a dynamical recurrent SCC, and static mutual relations must first be quotiented out. The refined interior node set is $V_{\rm int}=\{TS^\circ,GP^\circ,HF^\circ\}$ (the superscript $\circ$ meaning every reserve defining that interior region is strictly positive). The currently certified/conditional cross-domain edges are only: $TS^\circ_X\overset{C,D}{\to}GP^\circ$ (given source-to-field capture + mean-rotation depletion + legitimate pressure provenance) and $TS^\circ_X\to F_{\rm OP}\vee F_{\rm DER}\to\{HF^\circ,H\vee\mathrm{REG}\}$ — but no reverse cross-domain edge is certified: $GP\not\Rightarrow TS$, $GP\not\Rightarrow HF$, $HF\not\Rightarrow TS$, $HF\not\Rightarrow GP$ — these intuitively "expected" feedback edges have no generic typed theorem in C6 so far, and therefore cannot be drawn into the certified graph. C6-G.1 Interior SCC Dissolution Theorem therefore proves: after the four corrections (static G/P quotienting, H/F coherent-subtype refinement, the T→TS spacetime lift, and the insertion of C6-F's cross-domain bridges), there is no certified multi-node directed cycle at all among $TS^\circ_X$, $GP^\circ$, $HF^\circ$ — TS only has outward routing, with no GP/HF edge returning to TS; GP's and HF's own recurrences remain unproved obligations; and there is no certified crossing cycle between GP and HF either — the large SCC that appeared to exist on C5-M's graph formally dissolves under typed dynamical semantics (explicitly noted: this is the current research graph's audit result, not a PDE proof that "no future theorem can add a reverse edge"). Defining each candidate cycle's certification deficit $\delta_{\rm cert}(C)$ (the number of edges in the cycle lacking a certified composable dynamical transition), C6-G.2 Positive Certification-Deficit Theorem proves that every currently identified candidate recurrent cycle has $\delta_{\rm cert}(C)\ge1$ — the GP self-cycle lacks a joint geometry–pressure hereditary return theorem, the HF self-cycle lacks a uniformly coherent nonlinear re-entry recurrence theorem, $TS\to GP\to TS$ lacks $GP\to TS$, $TS\to HF\to TS$ lacks $HF\to TS$, and $GP\leftrightarrow HF$ lacks both directions — the paper explicitly stresses that this distinction is mandatory: $\delta_{\rm cert}>0$ only means "the current research graph has not yet certified this cycle," not "the PDE cannot realize this cycle." Re-quotienting the large collection of individual reserve boundaries accumulated across C6-C/D/E/F into ten global boundary superclasses $\mathfrak B=\{B_{LOAD},B_{COH},B_{SEG},B_{GEOM},B_{FIELD},B_{MEAN},B_{PROV},B_{HER},B_{SETUP},B_{CAP^\infty}\}$ (absolute-load collapse, coherence collapse, source separation, geometric criticality, source–field decoupling, mean-rotation takeover, loss of source legitimacy, loss of heredity — flagged as "the main cycle-composition boundary" — theorem-setup exit, and the infinite boundary of a divergent capacity ratio), and explicitly warns that a reserve tending to zero only means the state approaches some boundary $\partial\mathcal K$; it does not automatically define a PDE transition edge $B_i\to B_j$ — that needs independent certification, and C6-G deliberately stops here without further subdividing the boundaries, "to avoid repeating C5's boundary-proliferation problem." C6-G.3 Minimal Survivor Reduction Theorem is the round's climax: any hypothetical infinite survivor sequence that avoids $\mathrm{REG}$ must, after passing to a subsequence, belong to one of four: M1 a uniformly hereditary GP candidate, M2 a uniformly coherent HF candidate, M3 a boundary-saturated survivor approaching some fixed boundary $B_\ast\in\mathfrak B$, or M4 recursively entering the legality exit $\mathsf A$. Key corollary: $TS$ is formally removed from the minimal interior survivor candidate list — under C6-F's uniform cross-domain reserves, TS can only be a relay/transient/boundary-recurrent state, and can no longer stand as an isolated minimal interior sink. C6-G.4 Zero-Certified-Cycle Audit formally restates: in the current typed dynamical graph, a nontrivial recurrent directed cycle composed entirely of certified dynamical-implication edges does not exist — explicitly flagged as "a theorem about the current research graph," not an assertion about the PDE itself. The round's most important methodological summary (Section 51): "C4 turned asynchronous blow-up channels into synchronized motifs; C5 turned motifs into compact defect states; C6-A–F corrected the graph semantics and built typed joint states; C6-G now shows — the genuine global uncertainty has migrated from the interior graph to the boundary graph." The paper further argues (Section 44) that the boundary-saturated branch should take priority over proving new interior self-recurrence theorems: if most boundary faces can be shown to either enter $\mathrm{REG}$, exit the recurrent class, consume a finite global budget, or route into some already-constrained interior node, then the entire cycle space might collapse without ever needing GP/HF self-recurrence theorems at all — a genuinely strategically valuable insight. Formally hands off to C6-H — Critical Boundary-Face Transition Graph, Debt-Coercivity Audit, and Boundary-Cycle Elimination, with eight composition obligations (H1 instantiating boundary-superclass representatives through H8 updating the minimal survivor list).