Frustration Windows Are Generic: Certified Necessity Tests and the Almost-Quantum Excess of the Tilted I4422 Family

Two companion papers established that the almost-quantum excess of the tilted $I_{3322}$ family is a frustration-window phenomenon and conjectured the structure generic. Here we test the conjecture in both directions, with exact rational certificates for every headline claim. Necessity: we certify closures (gap $\leq 4\times10^{-7}$ in each case) for three deformations that remove one ingredient each -- tilted CHSH families in the 2222 scenario (three tilt patterns, swept; certified at $w=1$), one-wing anchoring in 3322, and the (3,2) hybrid scenario -- so cross-wing opposition and three settings per wing are each necessary, and no tilted two-setting functional in the families tested registers the almost-quantum excess (the 1+AB/level-2 gap) at all. Genericity: for the tilted $I_{4422}$ family (Collins-Gisin staircase, anchor weights (2,2,1,0)), we map the window and certify: the $I_{4422}$ almost-quantum landmark at $w=1$ to width $5\times10^{-7}$; the excess at $w=1$, $2.30033\times10^{-2} \leq \mathrm{gap} \leq 2.30047\times10^{-2}$ -- in comparable normalization $10.8\times$ the textbook $I_{3322}$ separation; a peak lower bound $\mathrm{gap}(w{=}7/5) \geq 3.888\times10^{-2}$; and the upper-edge closure $\mathrm{gap}(w{=}17/10) \leq 7.1\times10^{-7}$. The anchor law generalizes quantitatively and is certified: the matched words on the heavy anchored pair $\{1,2\}$ recover $f \in [0.69260, 0.69267]$ of the gap, every other single matched pair is inert, and the three matched pairs within the anchored triple -- exactly the marginal support -- jointly recover $f \in [0.98813, 0.98825]$. One structural novelty refines the conjecture: the $I_{4422}$ window opens at $w = 0^+$ rather than at an interior branch crossing, so the finite upper edge appears generic while the lower edge admits two types. Three further certified results: the quantum value is pinned at the peak by an exact rational strategy in dimension six meeting a rational dual of an intermediate relaxation ($9.522944386 \leq \max_Q J_{7/5} \leq 9.525780588$), giving a certified experimental corridor $\max_{\tilde{Q}} - \max_Q \geq 4.618\times10^{-2}$; grid points become intervals via convexity and dual chords against exact deterministic lower lines, pinning entire branches within $\sim 10^{-6}$; and the same machinery run in reverse certifies the window open on all of $[1/2, 8/5]$ with margin $\geq 1.36\times10^{-3}$. All certificates are single rational Bareiss-verifiable objects checkable with the Python standard library; certified zeros carry the necessity claims just as certified values carry the excess. Files in this record: the paper (PDF); certificate files with full rational proof objects (genericity_certificates.json, extended_certificates.json); standalone verifiers requiring only the Python 3 standard library (verify_genericity_certificates.py, verify_extended.py); and generating/exploration scripts (certlib.py, run_certs.py, ext_certs.py, cert_l2b.py, pin_maxq.py, quantum_cert.py, genericity_battery.py, polish.py). Companion records: the certified separation (doi:10.5281/zenodo.21534675) and the tilted I3322 window (doi:10.5281/zenodo.21603712).