46 / C5-J: Line-Section Sign Processes, Order-Sandwich Coupling, and Harmonic Critical Saturation

C5-I preserved only the chord occupancy rate $b_k([\nu])$, without knowing whether the sign-thick set is actually one long interval, a few islands, or heavily fragmented. C5-J's question: can fragmentation itself create a new harmonic escape? The answer is no. **C5-J.1 Fragmentation-Neutral Harmonic Lower Bound** proves that Solynin's extremal theorem needs only the total measure of the chord complement — whether the active set is one interval, finitely many pieces, or a highly fragmented measurable set, as long as the complement measure stays $\ge2(1-\beta)$, the harmonic-measure lower bound $h(\beta)$ holds — fragmentation cannot make the harmonic-measure lower bound any worse than the pure-occupancy bound. **C5-J.2** likewise proves that C5-I's descent estimate depends only on the total length of the same-sign high-value set, so fragmentation position does not affect the descent cost. But fragmentation is not free: it forces repeated threshold crossings along the chord, and such crossings are exactly the variation of the next-order derivative. Using dual-threshold hysteresis counting ($\lambda_0<\lambda_1$, to avoid threshold-noise artifacts) to define a robust high-value island count $N_k$, each interior gap costs at least a fixed total variation $2(\lambda_1-\lambda_0)A_k$, and since total variation is bounded above by $|f_k'|\le C_DA_{k+1}$, **C5-J.3** forces $A_{k+1}\ge\frac{(\lambda_1-\lambda_0)(N_k-1)}{C_Dr_k}A_k$ — fragmentation genuinely pays upward to the next order. The round's most elegant result is multiplying this upper-order cost by C5-I's lower-order cost $A_{k-1}\ge\kappa_{\lambda,\delta}r_kA_k$: the chain-scale radius $r_k$ cancels exactly, giving a dimensionless three-order derivative sandwich inequality $A_{k-1}A_{k+1}/A_k^2\ge c_{\lambda_0,\lambda_1,\delta}(N_k-1)$ (**C5-J.4**), which converts into Grujić–Xu's own normalized chain-root notation (the normalization constants likewise cancel exactly, **C5-J.5**). Defining the log chain curvature $Y_k=(k+1)\log\mathcal R_k$, **C5-J.6** proves that if the fragmentation count $N_k\to\infty$, the discrete second difference $\Delta^2Y_k\to+\infty$ — unbounded fragmentation cannot coexist with a locally flat/affine chain-root log profile, and must force a sharp order-space convexity event. Separately, defining the dimensionless line roughness $\mathfrak U_k=r_kA_{k+1}/A_k$, it is shown that the fragmentation count is bounded by the roughness bound: **C5-J.7**, if $\sup_k\mathfrak U_k<\infty$, the normalized line profile $\psi_k$ is equi-Lipschitz, and Arzelà–Ascoli gives a genuinely continuous limiting profile, generalized to the full angular chord process $\Psi_k(\nu,s)$ compactifying likewise on $S^2\times[-1,1]$ (**C5-J.9**). C5-J.10 Bad-Core Line-Process Dichotomy closes the round: every recurrent chain-scale sign-thick bad core, after passing to a subsequence, can only be one of a compact hysteretic sign core (bounded roughness) or upper-order-derivative roughness ($\mathfrak U_k\to\infty$) — fragmentation is thus completely absorbed into the derivative-chain metadata, and is formally removed from C5's survivor list; it is no longer an independent category. Line micro-geometry ultimately compresses to just four states: harmonic passage, compact critical core, upper-order roughness, and strong sign-thickness. The most important honest boundary (§49–55) continues and sharpens C5-I's rule: all the new inequalities (descent, fragmentation's upper-order cost, the three-order sandwich) are same-time, but the published theorem uses different admissible later times at different derivative orders, so same-time sandwiches cannot be directly stitched across orders — explicitly defining a "root-transfer factor" $\mathfrak T_{k\to k+1}$ measuring the cross-time chain-root ratio; if this factor diverges, it constitutes by itself a newly named derivative-root time-reversal defect, belonging to the Type-A/B dynamic-interpolation layer, which this round does not resolve. Formally hands off to C5-K — Chain-Time Stitching, Type-Switch Defects, and Dynamic-Interpolation Closure Audit, with 8 targets left to prove.