45 / C5-I: Derivative Sign-Geometry Defects, Chain Sections, and Harmonic-Measure Compatibility

C5-H formally declared the static all-order effective-volume closure scheme dead; C5-I is the first round to treat component/sign one-dimensional micro-geometry as the primary object, rather than an appendage of volume geometry. It faithfully encodes the exponentially separated derivative sections of Grujić–Xu's Definition 3.15, the section maxima $m_i$, and Type-$\mathcal A$/Type-$\mathcal B$ strings. It defines the exact 1D chord occupancy rate $b_E(x_0,r,[\nu])$ and the best directional occupancy $\beta_E=\inf_{[\nu]}b_E$ — Theorem 3.14's spatial passage condition is precisely "every dangerous base point has a direction/scale with occupancy $\le\delta$." If the spatial condition genuinely fails, it produces an everywhere sign-thick bad core: at the maximal chain-scale radius, the occupancy in every projection direction is $>\delta$. **C5-I.1 Angular Sign-Core Compactness** proves that the recurrent bad core's angular profile $b_n$ weak-* compactifies in $L^\infty(\mathbb{RP}^2)$, with the limit $\ge\delta$ almost everywhere — a genuine mathematical object, not just a label; two limiting types result: strictly thick ($\beta_\ast>\delta$) or a harmonic critical saturation defect ($\beta_\ast=\delta$, a finite-order failure approaching the theorem threshold rather than violating it). A bad core is not a free escape: first, Solynin's harmonic-measure lower bound (§12) confirms that the external contraction mechanism falls within the published theorem's scope; then the contrapositive of "3D $\delta^3$ sparseness ⟹ 1D $\delta$ sparseness" (**C5-I.2**) forces a genuine local $L^2$ derivative concentration, which links directly to C5-H's spectral-cell multiplicity (the bad-core fraction $\Phi_k^{bad}\gtrsim(r_k\Lambda_k)^3/\mathfrak N_k$); the number of disjoint bad cores is likewise controlled by this same multiplicity. The round's central new bridge is C5-I.3 Sign-Thick Chord Descent Lemma: integrating a chosen derivative component along a coordinate line through the bad core, combined with the everywhere lower bound $\ge-A_k$, forces exactly a lower bound on the one-order-lower amplitude, $A_{k-1}\ge\kappa_{\lambda,\delta}r_kA_k$ (with $\kappa_{\lambda,\delta}=(1+\lambda)\delta-1>0$, coming directly from the theorem's own tuning-constant constraint), holding for both signs, which converts into Grujić–Xu's own normalized chain-root notation $\mathcal R(k-1,c,s)\ge d_k(c)\mathcal R(k,c,s)$. The headline result, C5-I.4 Harmonic-or-Descent Dichotomy: at every order, it is either harmonic passage or a descent-order cost. Conversely (§28, C5-I.5), if the adjacent root ascends steeply enough ($\mathcal R(k)>d_k^{-1}\mathcal R(k-1)$), descent is impossible, directly forcing spatial-geometry passage — the first time C5 forces sign geometry directly from amplitude-chain shape — generalized to a same-time-interval Type-A Puncture Criterion: a sufficiently strong same-time ascent gain implies the entire interval cannot be sign-thick throughout — at least one order genuinely passes. It also explicitly clarifies Type-B compatibility: descent cost is geometrically compatible with Type-B (descending) behavior, but the published Theorem 3.9/Corollary 3.12 are specifically designed for stable descending chains, so Type-B is not a free hypothetical singular survivor. The most important honest hard rule (§45): different derivative orders are evaluated in the theorem at different admissible later times, and a same-time descent inequality cannot be multiplied across orders unconditionally to form an all-order contradiction — this is exactly where the published paper's dynamic-interpolation mechanism (Lemma 3.16/3.17 tracking Type-A↔B switching) is genuinely needed, and C5-I explicitly does not claim to have solved this timing-stitching problem. The round also precisely clarifies that the sign-geometry defect is not an independent Boolean failure — it is simultaneously coupled to descent-order cost and local $L^2$ concentration, splitting into three distinct recurrent motifs: single/few cores, multi-core multiplicity, and small-core fraction. Formally hands off to C5-J — Line-Section Sign Processes, Order-to-Order Descent Coupling, and Harmonic Critical Saturation, with 8 targets left to prove.