The Clique Law: Certified Anatomy of the Almost-Quantum Excess along the Imm22 Ladder

Three companion papers established that the excess of the almost-quantum set over NPA level 2 on tilted $I_{mm22}$ functionals is a frustration phenomenon: it exists only where a marginal anchor opposes a co-located correlator block, it lives on a finite window in the anchor weight, and it is policed by matched cross-party level-2 words at the anchored settings. This paper climbs the ladder and dissects the mechanism, with exact rational certificates for every headline value. The ladder grows: the normalized excess at $w=1$ is certified to $[2.038580, 2.038617]\times10^{-2}$ at $m=5$ and $[4.274637, 4.274741]\times10^{-2}$ at $m=6$ -- with the $I_{5522}$ and $I_{6622}$ classical bounds (11 and 16) re-enumerated exactly -- so the growth ratio decelerates ($10.8 \to 3.54 \to 2.10$), roughly tracking $(m-3)^2$. The anchor law of the previous paper fails upward and is replaced: at $m=5$ every single matched pair is inert (the heaviest, certified to $\leq 8\times10^{-6}$ of the gap) while the triangle of matched pairs on the three heaviest settings carries $[0.74590, 0.74595]$; at $m=6$ the triangle in turn drops to $\leq 0.2596$ while the 4-clique carries $[0.81660, 0.81665]$; at $m=7$ the pattern continues to the 5-clique. A chordless 4-cycle recovers only 0.20 where the 4-clique recovers 0.82: the policing unit is the clique, not the cycle. In every certified instance the minimal adequate clique is the complete graph on the marginal support minus its lightest (sacrificial) setting -- the same setting the anchor sweep shows being driven anti-aligned and then released, so the static law and the dynamic cascade are one structure. Genuinely triadic words become load-bearing: at the tilted $I_{4422}$ peak, same-wing length-3 words (the real, Bargmann-type 3-cycle sector) police a sector mixed words cannot reach, with a certified unique share of $[0.18845, 0.18884]$ -- the completion refusal retreats into sectors of higher cyclic order as the functional's tension increases. As a corollary the quantum value at the peak is pinned to $9.522944386 \leq \max_Q \leq 9.524082942$ (width $1.14\times10^{-3}$) and the certified experimental corridor widens to $4.788\times10^{-2}$. The $I_{5522}$ upper edge is certified onto deterministic branches $12w-1$ and $18w-11$ crossing at exactly $w=5/3$. Three of our own published predictions failed this round and are reported with the same standard as the confirmations. Every certificate is a rational Bareiss-verifiable object checkable with the Python standard library. Files in this record: the paper (PDF and source); the certificate file with full rational proof objects (paper4_certificates.json); a standalone verifier requiring only the Python 3 standard library (verify_paper4.py, which also re-enumerates the classical bounds 11, 16, 22); generating and exploration scripts (certlib.py, p4_certs.py, i5522.py, i6622.py, m7.py, m7scan.py, syn.py, lobes.py, lean.py, raylib.py); and the interior-anatomy figure (lobe_anatomy.png). The improved quantum pin is archived in the extended_certificates.json of the genericity record. Companion records: the certified separation (doi:10.5281/zenodo.21534675), the tilted I3322 window (doi:10.5281/zenodo.21603712), and the genericity and necessity tests (doi:10.5281/zenodo.21616031).