44 / C5-H: All-Order Effective-Volume Defects, Spectral–Multiplicity Ladders, and Asymptotic-Critical Compatibility

C5-G obtained the first theorem-ready fixed-order direct gate. C5-H asks: if a hypothetical survivor keeps $\mathfrak G_k^{dir}>1$ at every fixed $k$, can a purely volume-based argument push this all the way to $k\to\infty$ to force a contradiction? The answer is no — and not only does the fixed-order direct route fail, so does the asymptotically favorable chain route, on pure volume alone. First, an honest theorem audit: the spatial scale in Grujić–Xu's (2024) Theorem 3.5 carries a $2^{-k}$ factor, and the admissible later-time window carries a $4^{-k}$ factor — this is an exponential penalty built into the published theorem itself, not a product of C5's methodology. An explicit purely illustrative smooth single-scale analytic wave-packet model is constructed (explicitly stated to be not an N–S counterexample, only an illustrative no-go): **C5-H.1 All-Order Direct-Gate No-Go** proves that even for the most favorable smooth, single-scale data, the coarse volume certificate still forces $\mathfrak G_k^{dir}$ to diverge exponentially in $k$; §8 further proves the admissible time window itself undergoes exponential Zeno-type contraction, which cannot be made to vanish simply by raising the order. To study the all-order spectral ladder, the $L^2$ Fourier moment $M_k=\||\xi|^ku\|_2^2$ is introduced, and **C5-H.2** (Cauchy–Schwarz) proves $M_k^2\le M_{k-1}M_{k+1}$, forcing the spectral frequency $\Lambda_k$ to be monotonically increasing. But then comes the round's central no-go: using the Agmon inequality to decompose the effective volume exactly into "spectral-cell scale × physical multiplicity" $r_{eff,k}=\mathfrak N_k^{1/3}\Lambda_k^{-1}$, log-convexity controls only the monotonicity of $\Lambda_k$ and gives no control at all over the multiplicity $\mathfrak N_k$ — high spectral frequency can perfectly well coexist with unbounded multiplicity, so spectral cascade does not imply physical concentration, which is the core no-go of the all-order effective-volume route. Decomposing the direct-gate ratio into an order penalty $2^k$, a physical-multiplicity factor $\mathfrak N_k^{1/3}$, and a spectral-peak mismatch, it is shown that even under perfect single-cell concentration ($\mathfrak N_k\sim1$), the bare $2^k$ factor alone can still make the fixed-order gate fail at high order — direct failure at high order does not equate to packet multiplicity. Next, the more favorable chain scale from Theorem 3.14 is tested: **C5-H.3 Volume-Only Chain No-Go** uses the same wave-packet model to prove that — even for a chain theorem whose asymptotic-critical exponent genuinely tends to zero — closing it at high order via a coarse global volume certificate generally still fails (the theorem constant itself grows like $k^2$, overwhelming any gain from pure volume alone). §24–25 explain the geometric reason: what Theorem 3.14 actually exploits is component/sign one-dimensional micro-geometry (rapid sign alternation, filamentary geometry, directional oscillation), which global volume simply cannot see — small volume is a sufficient certificate of 1D sparseness, not the full geometry the theorem actually uses. The genuine positive fact that survives (**C5-H.4**): the gap between the chain and a priori exponents, $1/[(k+1)(2k+3)]\sim1/(2k^2)\to0$, does genuinely tend to zero — if a survivor could maintain any fixed positive concentration-gain exponent, and the theorem-constant burden were negligible, the chain spatial condition would eventually be crossed — but (**C5-H.5** and what follows) if the theorem constant or the multiplicity $\mathfrak N_k$ grows fast enough in step, it can completely overwhelm this vanishing exponential gap — a vanishing exponent gap does not imply automatic regularity. A multi-packet realization example (§36) further shows that $N$ mutually distant, approximately disjoint copies of the same wave packet let $\mathfrak N_k$ simply multiply by $N$ without changing the derivative spectral scale, continuing the "carrier multiplicity" theme that has run throughout C3/C4. **C5-H.6 Static All-Order Volume Closure No-Go** formally declares: a static all-order effective-volume closure scheme is a methodological dead end; the direct gate should remain a fixed-order "kill switch," not the main engine for automatic order escalation; the correct high-order asymptotic object must instead turn to what Grujić–Xu's paper genuinely relies on and C5 has not yet made rigorous: component/sign one-dimensional micro-geometry + Type-A/B derivative chains + harmonic-measure compatibility. Formally hands off to C5-I — Derivative Sign-Geometry Defects, Chain Sections, and Harmonic-Measure Compatibility, with 8 targets left to prove.