Local Semisimple Givental--Teleman Reconstruction from Coordinate-Free Renormalized Tail Orbits:\ Isomonodromic Tau Functions, Quantized \(R\)-Matrices, and Finite-Window Descendant Recovery

The preceding paper of this series pushed coordinate-free renormalized-tail analysis all the way to semisimple miniversal ADE geometry. From finitely many asymptotic probes of the tail orbit one could recover, on every simply connected semisimple chamber, the Jacobi multiplication operators, the primitive metric, the Euler grading, the canonical coordinates, and the calibrated Stokes data of the Dubrovin connection. That achievement still left one decisive structural layer open. Semisimple singularity theory is not exhausted by the genus-zero Frobenius germ. The genuinely stronger package is the quantized higher-genus and descendant theory: the isomonodromic tau function, the genus-one potential, the symplectic \ (R\) -matrix, the Givental Lagrangian cone, and the local semisimple cohomological field theory reconstructed by the Givental--Teleman theorem. The present paper proves that the renormalized-tail orbit sees that layer too. We remain in the same sharply delimited regime as the previous paper: coordinate-free smooth simple-pole branch families, a finite bounded ADE candidate list, and simply connected semisimple chambers of the miniversal base. Within that regime, however, the orbit category determines not only the semisimple Frobenius manifold germ but also its full local semisimple Givental--Teleman quantization. The first main theorem is an orbit tau-function closure theorem. Let \ J_ (z, u) =_ (u) R_ (z, u) e^V_^{can (u) /z}, R_ (z, u) =I+R, ₁ (u) z+R, ₂ (u) z²+, \ be the calibrated packet matrix recovered from the orbit on a semisimple chamber. Writing \ (v, ₈\) for the canonical coordinates and \ (, ₈\) for the diagonal primitive-metric weights, we show that the one-form \ _: =12ᵢ (R, ₁) ₈₈\, dv, ₈ -148ᵢ d, ₈ \ is closed. Consequently the orbit recovers a local isomonodromic tau function \ (_\) and a genus-one potential \ (F₁, \), both unique up to additive or multiplicative constants compatible with primitive scalar gauge. The second theorem is a symplectic \ (R\) -matrix and Givental-cone theorem. The recovered calibration is shown to satisfy the symplectic identity \ R_ (-z, u) ^ R_ (z, u) =I, \ relative to the recovered primitive metric. Therefore its quantization is well defined on the standard Givental Fock space. We obtain a canonical local ancestor potential \, ₔ =_ (u) \, R_ (u) ₈=₁^_ _ (, ₈ (u), q^ (i) ), \ and an equivalent description of the corresponding Lagrangian cone as the image of the tensor product of point-theory cones under the recovered symplectic transformation. This gives a concrete quantized object in the orbit category rather than merely an abstract existence claim. The third theorem is the central reconstruction result: a local semisimple Givental--Teleman theorem for renormalized-tail orbits. Using the semisimple Frobenius manifold already reconstructed in the previous paper and the orbit-determined \ (R\) -matrix proved here, we show that the local semisimple cohomological field theory germ is determined uniquely up to Weyl relabeling of canonical idempotents, primitive scalar gauge, and the usual unstable translation ambiguity. Equivalently, the orbit determines the full local ancestor potential and, after the standard \ (S\) -calibration, the local descendant potential as well. The fourth theorem is quantitative. Fix a finite ADE candidate list, a compact semisimple class, and a truncation level \ ( (G, M) \) in genus and descendant degree. Then finitely many scalar probes taken on finitely many selected rays and on a bounded observation window \ (n=N, , N+J-1\) asymptotically recover the true ADE type together with the truncated ancestor and descendant jets up to \ ( (G, M) \), with error \ \! (N^-1/h_+N+N) \ under the natural spectral-gap, injectivity, model-accuracy, and noise-floor hypotheses. The theorem is stated explicitly as a local finite-horizon asymptotic recovery scheme; it is not claimed to be a globally conditioned numerical algorithm under arbitrary noise. One conceptual point deserves emphasis. In sparse quantized regimes, the recovered \ (R\) -matrix may collapse to the identity and the higher-genus package reduces to a simple tensor product of point theories. The genuine novelty of this paper begins exactly when the recovered off-diagonal coefficients of \ (R_ (z, u) \) survive: then the orbit detects higher-genus and descendant mixing that is invisible at the level of canonical coordinates and first metric weights alone. The scope is exact and intentionally limited. We do not claim non-semisimple coalescence, modal singularities, higher-order poles, arbitrary singular varieties, or globally conditioned arbitrary-noise algorithms. Within the semisimple ADE regime, however, the present paper upgrades the orbit category from Frobenius-germ reconstruction to quantized semisimple reconstruction: orbit tau functions, symplectic calibrations, Givental cones, Teleman-type higher-genus closure, and finite-window truncated descendant diagnosis.