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).
Three companion papers established that the excess of the almost-quantum set over NPA level 2 on tilted Imm22 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]×10−2 at m=5 and [4.274637,4.274741]×10−2 at m=6 -- with the I5522 and I6622 classical bounds (11 and 16) re-enumerated exactly -- so the growth ratio decelerates (10.8→3.54→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 ≤8×10−6 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 ≤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 I4422 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≤maxQ≤9.524082942 (width 1.14×10−3) and the certified experimental corridor widens to 4.788×10−2. The I5522 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).
Two companion papers established that the almost-quantum excess of the tilted I3322 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 ≤4×10−7 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 I4422 family (Collins-Gisin staircase, anchor weights (2,2,1,0)), we map the window and certify: the I4422 almost-quantum landmark at w=1 to width 5×10−7; the excess at w=1, 2.30033×10−2≤gap≤2.30047×10−2 -- in comparable normalization 10.8× the textbook I3322 separation; a peak lower bound gap(w=7/5)≥3.888×10−2; and the upper-edge closure gap(w=17/10)≤7.1×10−7. The anchor law generalizes quantitatively and is certified: the matched words on the heavy anchored pair {1,2} recover f∈[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∈[0.98813,0.98825]. One structural novelty refines the conjecture: the I4422 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≤maxQJ7/5≤9.525780588), giving a certified experimental corridor maxQ~−maxQ≥4.618×10−2; grid points become intervals via convexity and dual chords against exact deterministic lower lines, pinning entire branches within ∼10−6; and the same machinery run in reverse certifies the window open on all of [1/2,8/5] with margin ≥1.36×10−3. 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).
For the tilted family Jw=w(a1+a2−b1−b2)+∑ijsijEij containing the I3322 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 A1A2,A2A1,B1B2,B2B1 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 ⟨A1A2BjB3⟩−⟨A1A2B3Bj⟩=21⟨[A1,A2][Bj,B3]⟩, 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∈(0.90,0.95) to (2.25,2.50), with a two-lobed interior peaking near w=2 at 4.07×10−3 -- 7.7 times the separation of standard I3322, 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.
Gigena et al. [npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional Bαβ=α⟨A0⟩+β⟨B0⟩+CHSH and observed that the NPA level required to reach it grows without evident bound toward the critical line α+β=2. We quantify the mechanism on the symmetric slice s=2−α−β: (i) the quantum value leaves the local bound cubically, cQ=4−s+s3/6−s4/36+O(s5); (ii) each NPA level overshoots quadratically, ck(s)=4−s+aks2+O(s3), with the almost-quantum coefficient computed exactly, a1+AB=3/64; (iii) the divergence of the required exact level is equivalent to positivity of the single sequence (ak) - proven for every k in the companion paper. We prove the supercritical side completely: for all α,β≥1 and every level the hierarchy is exact, via three explicit rational certificates realizing an affine identity. The hierarchy's exactness thus undergoes a phase transition at the critical line. On the subcritical side we certify the first four levels in exact arithmetic (rational pseudo-moments beating cQ, confirmed by Sturm's theorem). We identify the exact mechanism: rescaled to the critical corner, the limiting obstruction is the Motzkin polynomial, the classical nonnegative-but-not-sum-of-squares form, so the finite-level failure sits in the restricted-certificate regime. The phase boundary has a precise geometric reading via Nie's finite-convergence theorem and Marshall's boundary Hessian condition: a self-tested optimum is finitely NPA-certifiable whenever its boundary Hessian is nondegenerate (contact order two), which holds for the single tilt and fails exactly at the doubly-tilted cubic touch. Three verified errata in the published polynomial system of Gigena et al. are documented.
Four companion papers located, dissected, generalized, and laddered the excess of the almost-quantum set over NPA level 2 on tilted Imm22 functionals. This paper asks why the numbers are what they are, and answers for the bottom of the tilted I3322 window: the constants are algebraic all the way down. Three thresholds are proven exactly. The no-signaling-line visibility threshold of I3322 is the golden ratio conjugate, λ3∗=(5−1)/2 (explaining a numerical coincidence between two cost conventions, since 1/λ−1=λ precisely at φ; the same constant appears elsewhere in Bell physics as a critical detection efficiency, with no known derivation connecting the two). The quantum maximum of the I3322 correlator part is exactly 5, attained by a hexagonal configuration -- a non-facet functional whose value appears to be new. The I4422 threshold is λ4∗=210−6, the positive root of x4+12x2−4, by a symmetrization argument in which two feasibility intervals kiss exactly at the quartic root. Below the window, the flat branch is a theorem: an affine family of zero-slack rational duals proves the gap vanishes identically on [0,471/500], certified closure gives gap≤2.3×10−8 at w=2357/2500, and certified opening gives gap≥1.10×10−6 at w=1889/2000, so the lower edge lies in [0.942,0.9445]. Inside, ten dual-solver points spanning five orders of magnitude follow the onset law gap=21(w−22/3)2 to 1--2%, down to gaps of 5×10−10; the coefficient follows from the envelope theorem given the measured unit marginal susceptibility (1.00(2)), and the edge is a staggered pair of bifurcations, level 1+AB at we=22/3 and level 2 at weL2=0.955934(3). The edge constant is derived at first order: the flat branch's 1+AB completion obeys a Clifford--Wick pairing and carries an eight-dimensional relation ideal (the dual hexagon plus five operator sum rules); the marginal mode must break relations to turn on, and the cheapest tilt breaks exactly three while preserving five, at a price certified in both directions by exact arithmetic over Q(2) to be 82/3 per 4 units of anchor -- hence 22/3. The certificates' entries lie in 61Z[2] and 181Z[2], extending the series' exact-certificate method beyond the rationals for the first time. A failed derivation route -- a homogenized fractional SDP whose optimum sits at infinity, so that every finite iterate overestimates the edge by 1.7×10−4 -- is documented alongside the successes, in the company of the field's own precedent that at this boundary only exact objects adjudicate. Files in this record: the paper (The_Algebraic_Edge.pdf and .tex); the Q(2) certificate pair for the marginal-price theorem with its standalone verifier (q2_edge_certificates.json, verify_q2_edge.py); the lower-edge theorem's proof objects -- the zero-slack flat-dual family, the rational level-2 flat completion, and the closure/opening pair -- with verifier (edge_bracket_certificates.json, verify_edge_bracket.py); the onset-law data table (onset_table.json); generating scripts (gen_edge_bracket.py, gen_onset_table.py); and a README. Both verifiers require only the Python 3 standard library. Companion records: the certified separation (doi:10.5281/zenodo.21534675), the tilted I3322 window (doi:10.5281/zenodo.21603712), the genericity and necessity tests (doi:10.5281/zenodo.21616031), and the ladder anatomy (doi:10.5281/zenodo.21625241).
We give exact rational certificates for two results in the bipartite Bell scenario with three two-outcome measurements per party. First, the maximum of the I3322 (Froissart) functional over the almost-quantum set -- the 1+AB level of the Navascues-Pironio-Acin (NPA) hierarchy -- is certified to lie between 0.2514708838 and 0.2514709016 (width <2×10−8), in the normalization with classical bound 0 and quantum maximum ≈0.2508754; to our knowledge no value for this landmark had been published. Second, we exhibit a single exact rational behavior T∗ carrying certificates in both directions: a rational moment matrix proving T∗ almost-quantum, and a rational dual object proving that every NPA level-2 completion of T∗ requires slack at least q=4.5934×10−4, so that T∗ is not quantum. The strict inclusion Q⊊Q~ in the minimal three-setting scenario is thereby established end-to-end with no reliance on floating-point computation. Anatomically, the witness is locally quantum: all nine of its 2×2 correlator subtables pass the Tsirelson-Landau-Masanes criterion, so its non-quantumness is irreducibly a three-setting phenomenon, concentrated in ordering-dependent pseudo-moments coupling each party's third setting to the other's first two; the level-2 certificate is equivalently an exact sum-of-squares identity that any quantum state values nonnegatively and the witness is forced to value at −q. The certificates follow the rational rounding-projection-lifting programme recently developed for noncommutative optimization; for NPA moment matrices in canonical form the required projection reduces to a per-moment uniform shift, making certification essentially free at these sizes, and we work the programme in both the feasibility and infeasibility directions on one witness. Machine-checkable certificates and verifiers requiring only the Python standard library accompany the paper. Files in this record: the paper (PDF); exact rational certificates for both theorems (i3322_1AB_certificate.json, i3322_level2_infeasibility_certificate.json); standalone verifiers requiring only the Python 3 standard library (verify_i3322_certificate.py, verify_level2_certificate.py); and the generating scripts (certify_landmark.py, certify_level2.py). See README_ancillary.txt for which artifact proves which theorem and how to run the verifiers. v2: corrects the mechanism attribution in Sec. 5 (anatomy) and adds the archive reference to Data Availability; theorems, certificates, and verifiers unchanged.
Version 4 (2026-08-25). Two additions from the correction campaign, carried as dated in-body correction records. The (2,5) entry, printed as a prediction in v3, is now census-exact -- and it was the wall law's first divergence test. At the constructed maximal-wall stratum point the stiff transverse count is exactly 10, so alpha_stratum(2,5) = 5; a fresh 60k ensemble's calibrated read is consistent with 5, with alpha = 10 (the superseded single-wall count's prediction) excluded at p = 0.000 and demonstrated power ~ 1; and a stratum-conditional paired ladder tracks the exact rank-10 Hessian form to <= 0.002 while a rank-20 comparator separates by >= 2.7 on identical draws. The law m_C(2,d) = 2(d-1) is census-exact at d = 3, 4, 5, now including the first point where it diverges (by 2x) from the superseded count. The (3,4) entry, printed open in v3, is now a PAIR of statements, and neither alone is alpha(3,4). The posted product-wall counting landed census-exact (alpha_stratum(3,4) = 5, so the asymptotic exponent satisfies alpha(3,4) <= 5), while the accessible window at n = 3e4 is steeper (calibrated 90 percent set [5.4, 6.9], with the previously read 4.0-4.1 excluded at power ~ 1): the codim-10 stratum's prefactor has not engaged at reachable depth, and window and asymptote have visibly decoupled. This is the scope lesson of the (d >= 3) x (d' > d) regime -- census plus conditional ladder measure the law, global windows measure the pre-asymptotic shoulder -- and it retro-explains the earlier fresh-seed ladders' instability as shoulder reads. Version 3 (2026-08-20). One SUPERSESSION: the qutrit-qutrit exponent closes. Versions 1 and 2 left alpha(3,3) open, v1 by a small-sample fit reading 3.25 and v2 by the bracket [2.3, 3.0] read off a descending window ladder. Both readings were artifacts of the estimator problem for which the pair-type table was already corrected in v2. The descent is attenuation tracking sample count rather than tail depth: an ensemble drawn from an EXACT alpha = 3.0 law descends by 0.62 across the ladder against the real data's 0.34, so a descending ladder is evidence FOR a constant exponent, not against one. Replacing the window fit with a calibrated Monte-Carlo test and inverting it over candidate exponents gives alpha(3,3) = 3.0 at p = 0.517 with a 90 percent confidence set [2.75, 3.10]; v1's 3.25 is excluded at p = 0.017. Power against alternatives is 0.990 / 0.590 / 0.120 / 0.575 / 0.990 / 1.000 at true alpha = 2.50 / 2.75 / 3.00 / 3.25 / 3.50 / 4.00. An independent re-audit at n = 10^4 reads flat near three on two seeds. Disclosed defect, carried in the paper: the test's own calibration row missed at the low edge (rejection rate 0.207 at a nominal 0.10 when the truth is alpha = 2.50, 4.4 standard errors high), so the lower end of the confidence set is softer than printed. This ensemble separates 3.0 from 3.25 with about 58 percent power and has none to separate 3.0 from 2.95. The (3,3) CARRIER family remains unattributed: the exponent is settled, the stratum that carries the accessible tail there is not. Version 2 (2026-08). Changes from v1: the pair-type exponent table is corrected (v1's alpha(2,4) = 2.08 was a 50-sample order-statistics fluctuation, placed at the 0.15th percentile of its own estimator's sampling distribution; three independent codebases now agree on the wall law alpha(2,d) = d, census-exact at d = 3 and 4); the transverse constant 1/12 is derived in closed form (diagonal cost 1/6, halved by the congruent partner's equal-cost correction); the (2,3) tail's carrier stratum is fully attributed (an isotropic-intertwiner family sharing the exponent 3); the qutrit-qutrit exponent is honestly bracketed open at [2.3, 3.0]; and the dispersion rule (tail-exponent fits print bootstrap dispersion or do not print) is adopted. The v2 ledger appendix carries the correction anatomy. Why is the multipartite certification constant heavy-tailed, where does the tail live, and what kills it? Localization: the credential pencil is exactly block-diagonal over party-subset supports; the tail is the worst pair; geometry statistics reconstruct from pair statistics (14/14 factor-band comparisons). The law: a frame-congruence theorem: whenever the two parties' operation frames subtend equal Bloch angles, the rotation intertwining them is an exact common kernel direction; the zero set is codimension one at generic angles, transverse growth is universally quadratic (mu = (Delta theta)^2/12), and the qubit-pair tail exponent is exactly 1/2 (the previously measured 0.39 is the finite-window slope of this law). The dichotomy: pairs containing a d >= 3 party measure exponents 2.08 to 3.69 because higher-dimensional operation pairs carry many local invariants that must all match: freedom helps the certifier. Engineering: heterogeneous dimension profiles tame the tail; qubit-pair presence splits all measured geometries into two clean classes; screening is a five-minute two-party calculation. Pre-registered; two entry gates that caught wrong conventions and all deaths ledgered.