59 / C6-J: Log-Scale Renormalized Defect Flow, Telescoping Potentials, and Critical-Cycle Closure Tests

C6-I proved that a fixed critical cost cannot by itself force a finite-time contradiction — it only corrected the scaling type, without providing any cross-scale directionality. C6-J formally promotes $s=-\log r$ (equivalently, the standard backward Leray time $s=-\log(T^\ast-t)$) to a genuine dynamical time, reconnecting to the Navier–Stokes equations themselves. Using the standard backward Leray coordinates ($y=(x-x^\ast)/\sqrt{T^\ast-t}$, $U=\sqrt{T^\ast-t}\,u$, $P=(T^\ast-t)p$), C6-J.1 gives the exact autonomous equation $\partial_sU+\frac12U+\frac12(y\cdot\nabla)U+(U\cdot\nabla)U+\nabla P=\nu\Delta U$ — the finite-time blow-up horizon $t\uparrow T^\ast$ becomes exactly $s\to+\infty$, so the finite-time Zeno problem in physical coordinates turns into an infinite-time dynamical-systems problem in renormalized-scale time. A fixed point $U(y,s)=U_\ast(y)$ corresponds exactly to a backward self-similar blow-up profile, while a periodic orbit corresponds to a backward discretely self-similar (DSS) scenario (scaling factor $\lambda=e^{L/2}$). Already-published Liouville-type theorems (Seregin, Chae, Chae–Wolf, Nečas–Růžička–Šverák, Tsai) do rule out a broad class of nontrivial backward self-similar profiles and locally asymptotic DSS blow-up — but the document draws a clear line: these theorems require field-level profile hypotheses, and do not automatically apply to a periodic orbit that carries only C6 defect metadata; it likewise flags explicitly that forward DSS solutions genuinely exist, so "periodic in renormalized time" is not, by itself, intrinsically impossible within N–S — ruling it out genuinely requires using the boundary conditions of backward blow-up and critical-regularity constraints. The round's first key result comes from the exact $L^2$ balance: C6-J.3 gives $\frac12E'+\nu D-\frac14E=0$ ($E=\|U\|_2^2$) — the scaling-stretch term contributes an anti-dissipative drift $+\frac12E$, so $E$ itself is not a monotone quantity, and the bare $L^2$ balance by itself does not rule out a periodic renormalized orbit. C6-J.4/5 work out the exact identity for the weighted family $V_\alpha=e^{-\alpha s}E$, proving that universal monotonicity only starts to hold from $\alpha\ge1/2$ on, and that $\alpha=1/2$ reduces exactly to the physical kinetic energy $V_{1/2}=e^{-s/2}E=\|u(t)\|_2^2$ — this is the Criticality–Monotonicity Tradeoff: within this natural exponential $L^2$ family, scale-criticality and universal monotonicity cannot coexist, and the one weight capable of monotonicity, $e^{-s/2}=r$, is exactly the weight that appeared in C6-I's Zeno summation — this formally explains why the physical energy genuinely does admit a telescoping sum, but that summation weight decays with scale, so it cannot attach a fixed positive price to a scale-invariant recurrent event; it is not that "there is no telescoping quantity," but that "the only universally available telescoping-quantity weight decays." The genuine field-level elimination instead comes from another route: proving that $\|U\|_{L^3}$ and $\|U\|_{\dot H^{1/2}}$ are exactly invariant under the backward rescaling (a critical field topology of degree zero), while the already-published necessary conditions for a hypothetical blow-up require $\|u(t)\|_{L^3}\to\infty$ and $\|u(t)\|_{\dot H^{1/2}}\to\infty$ — C6-J.2 Critical Field-Compact Recurrence No-Go therefore cleanly proves: no tail of the renormalized orbit can possibly be compact in $L^3$ or $\dot H^{1/2}$ (compactness implies boundedness, but a hypothetical blow-up forces both norms to diverge — a contradiction) — fixed points and periodic orbits of finite critical norm both cannot be blow-up orbits, consistent with the existing Liouville literature. But this does not kill the C6 defect cycle, because compact defect recurrence is not the same as compact field recurrence: what C6 has actually been studying ever since C4/C5 is the image $\pi(U(s))$ of the full field state $\mathcal X_{crit}$ under the projection $\pi$ onto the compactified defect-state space $\mathcal K_{def}$ ($TS,GP,HF$ plus boundary faces), and the fiber $\pi^{-1}(\theta)$ of this projection can perfectly well fail to be compact. Defining the critical fiber radius $\mathfrak R_{fiber}(K)=\sup\{\mathfrak F_{crit}(U):\pi(U)\in K\}$, C6-J.6 proves that if a compact defect set $K$ has finite fiber radius, it cannot support the late-time recurrence of a hypothetical blow-up. C6-J.7 Critical Fiber Escape Theorem is the round's genuine core: for any compact defect set visited infinitely often by a hypothetical blow-up orbit, its critical fiber radius must be infinite — "every surviving compact defect recurrence requires critical non-compactness in the fiber," formally named Critical Fiber Escape. This is precisely the underlying reason why the already-published field-level Liouville theorems do not automatically kill the C6 defect cycle: periodic defect metadata can perfectly well correspond to field dynamics that are aperiodic and drift persistently to infinity in the fiber direction. The remaining problem is therefore formally reframed as a skew-product dynamical system $(\theta(s),\kappa(s))$ — $\theta\in\mathcal K_{def}$ is the compact recurrence base, $\kappa=\log(1+\mathfrak F_{crit})$ is the unbounded critical fiber coordinate; a hypothetical blow-up requires $\kappa(s)\to\infty$, while $\theta(s)$ may perfectly well keep recurring, even exactly periodically. Tying this concretely back to the candidates C6 has already established: at present no theorem proves that uniform $GP^\circ$ or uniform $HF^\circ$ recurrence implies $\mathfrak F_{crit}$ is bounded — GP's geometric/source-heredity reserve and HF's Duhamel/sign-coherence reserve each control only local combinatorial conditions, not the global critical norm, so the survival of either is exactly equivalent to whether the fiber direction can genuinely escape. C6-J.9 Critical-Cycle Closure Test gives a reusable four-step diagnostic procedure: has the candidate cycle already been typed and certified? does uniform recurrence imply critical-field compactness/boundedness? if so, it is incompatible with blow-up and is directly excluded; if not, precisely identify which fiber coordinate is escaping; does that escape trigger an external regularity gate, a capacity incompatibility, a telescoping potential, or a profile-decomposition contradiction? — only completing this fourth step genuinely eliminates a surviving cycle. The round finally repositions C5's defect-compactification work as not wasted: it precisely isolated the remaining degrees of freedom into the fiber, converting an unstructured infinite-dimensional flow problem into "a compact finite defect base plus a non-compact critical fiber requiring classification" — a clearer target better suited to concentration-compactness/profile-decomposition methods — and lists six candidate fiber-escape mechanisms (amplitude escape, multiplicity escape, sub-scale escape, translation/tail escape, frequency escape, profile splitting) for the next round to classify. Formally hands off to C6-K — Critical Fiber Escape, Defect-Fiber Compactness, and Profile-Splitting Closure, with ten constituent obligations (K1, selecting the critical field topology, through K10, updating the cycle).