The Almost-Quantum Window of the Tilted I3322 Family

For the tilted family $J_w = w(a_1+a_2-b_1-b_2) + \sum_{ij} s_{ij}E_{ij}$ containing the $I_{3322}$ Bell functional at $w=1$, we map where and why the almost-quantum set (NPA level $1{+}AB$) exceeds NPA level 2, using a subset-relaxation diagnostic: adjoin the twelve level-2 words to the $1{+}AB$ basis in subsets and measure each subset's share of the gap. The excess is sharply localized and structurally rigid. (i) At $w=1$, the four matched words $A_1A_2, A_2A_1, B_1B_2, B_2B_1$ on the marginal-anchored settings recover 92% of the gap, while either party's complete pair structure, all eight third-setting words, and every mismatched cross-party pairing recover exactly none; at the level of the certified separation witness $T^*$, infeasibility appears already under a single cross-party pair of anchored words, one per wing, while every single-party and every third-setting subset leaves $T^*$ feasible. (ii) The mechanism is algebraic: reversal-symmetrized ordering differences obey $\langle A_1A_2B_jB_3\rangle - \langle A_1A_2B_3B_j\rangle = \frac{1}{2}\langle[A_1,A_2][B_j,B_3]\rangle$, an expectation of commutator products across the wings, and sharp sequential measurement reads only anticommutators, so the excess lives in a sector unreadable in principle on either wing alone. (iii) The gap requires the functional's marginal anchor to oppose and co-locate with the aligned correlator block: moving either, aligning their signs, or deleting the marginals annihilates the gap. (iv) In the anchor weight the gap opens only on a finite window, $w \in (0.90, 0.95)$ to $(2.25, 2.50)$, with a two-lobed interior peaking near $w=2$ at $4.07\times10^{-3}$ -- 7.7 times the separation of standard $I_{3322}$, which sits just inside the lower edge. Experiments probing the almost-quantum boundary should therefore run the tilted functional near $w=2$, not the textbook one. The headline values are certified in exact rational arithmetic: the two-word infeasibility of $T^*$, the matched-four value, the peak, and one closure; remaining grid values are floating-point with two solver families agreeing to five digits. Files in this record: the paper (PDF); exact rational proof objects for the four headline certificates (window_paper_certificates.json); a standalone verifier requiring only the Python 3 standard library (verify_window_certificates.py); analysis scripts (why_level2.py, anchor_tracking.py); and certificate generators (certify_window.py, refreeze_window.py). See README_window.txt. Companion record with the certified separation and its certificates: doi:10.5281/zenodo.21534675.