49 / C5-M: Unified Defect-State Closure, Compatibility Graph Audit, and C5 Phase Boundary

C5's task was never to find yet another magic inequality, but to turn C4's recurrent compensation motifs into compact recurrent states, and then route the apparently different escapes into debts. This round performs only a phase-closure audit and introduces no new estimates. Core conclusion: C5 should close as a research phase, but Navier–Stokes global regularity remains open. After listing the external theorem gates used throughout C5 (Miller's middle-eigenvalue gate, Miller's strain–vorticity operator gate $\langle-\Delta S,\omega\otimes\omega\rangle=0$, Grujić–Xu's fixed-order direct gate Theorem 3.5, Grujić–Xu's chain gate Theorem 3.14, and the Bradshaw–Tsai/Constantin pressure gate), all the recurrent survivor states encountered across C5-A through L are encoded into a six-letter residual alphabet $\mathfrak D_{C5}=\{\mathsf A,\mathsf T,\mathsf G,\mathsf P,\mathsf H,\mathsf F\}$: $\mathsf A$ legality/lineage/theorem-setup (explicitly stated to be a proof-entry legality defect, not a physical singular mechanism); $\mathsf T$ temporal-phase defects (Young oscillation, load concentration, separated scalar compensation periods); $\mathsf G$ field-geometry degeneracy (middle gap, compression-axis dispersion, strain-derivative fluctuation, vorticity leakage, cubic intermittency); $\mathsf P$ pressure compensation/provenance (mean rotation, pressure concentration, far-field signature, zero-determinant boundary, source splitting/flipping, axis locking); $\mathsf H$ high-order harmonic/theorem-window defects (fixed-order gate failure, window-persistent sign defect, harmonic–temporal critical saturation, persistent bad clusters); $\mathsf F$ forcing/order-variation debt (viscous/projected-nonlinear reversal, order curvature, chain-clock variation, theorem-constant drift). Nine pseudo-defects are formally struck from the list of independent nodes: free seven-point cancellation, generic line fragmentation, generic Type-A/B switching, generic root reversal, generic clock mismatch, amplitude-level carrier transfer, isolated large operator norm, isolated vorticity-constraint complement, and static all-order effective-volume escalation — all now routed into the six classes or into an external regularity gate. A certified compatibility graph is built, tagged (unconditional U, conditional C, external-theorem closure E), detailing the routing relations among the six classes (e.g. $\mathsf H\to\mathrm{REG}$ via the published theorem, $\mathsf F\to\mathsf H$ via viscous congestion pushing activity to higher order, $\mathsf G\to\mathsf P$ via strong middle coherence forcing pressure return). Finite Recurrence Principle: since the residual alphabet is finite, any infinite sequence of hypothetical survivor labels must have some class recur infinitely — a purely combinatorial but genuine pigeonhole argument, resting on the substantive result that the alphabet really is finite. If the compatibility graph is complete for some survivor path, its condensation graph of strongly connected components is finite and acyclic, so any infinite path must eventually fall into some sink SCC after a finite transient — C6's genuine object is the recurrent sink SCC/minimal recurrent defect cycle, not a new isolated defect node. Three unexcluded candidate recurrent cycles are flagged: the high-order forcing cycle $\mathsf H\leftrightarrow\mathsf F$ (persistent bad theorem window → descent/load cost → viscous/nonlinear/clock debt → activity pushed to higher derivative order → the new window fails again — C5 currently has no finite all-order budget to exclude this, possibly the hardest candidate sink); the geometry–pressure cycle $\mathsf G\leftrightarrow\mathsf P$; and a possibly isolated temporal cycle $\mathsf T$ (removing this candidate SCC would need a generic theorem for a common $\mathsf T\to\mathsf G/\mathsf P/\mathsf H$ source, currently open). The most important honest boundary: a finite defect graph does not imply global regularity — the graph may still contain directed cycles, compactness does not eliminate cycles, and debt routing without a finite-additivity theorem does not itself yield a contradiction. Six-Class Closure Theorem: under the current C3/C4/C5 rules and the conditional-lineage framework, every recurrent survivor state encountered in C5-A through L can be encoded into these six classes plus compact metadata within each class, with no mechanism requiring a seventh independent residual class — C5's state-space/motif-compactification task is structurally complete; this is the closure of a research phase, not the closure of a PDE proof. Formally declares C5 — Recurrent Motif Limits, Defect Measures, and Compensation Compactness status PHASE CLOSED, PDE status GLOBAL REGULARITY OPEN, and hands off to C6 — Minimal Recurrent Defect Cycles, Sink-SCC Extraction, and Cross-Domain Closure, opening with C6-A, whose task is to extract the genuine sink strongly connected components and determine whether their debt can be paid indefinitely.