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The Algebraic Edge: Exact Constants and the Onset Law of the Almost-Quantum Window

Caleb Weiner

Published Jul 29, 2026• 0 citations
Publisher

Abstract

Four companion papers located, dissected, generalized, and laddered the excess of the almost-quantum set over NPA level 2 on tilted $I_{mm22}$ functionals. This paper asks why the numbers are what they are, and answers for the bottom of the tilted $I_{3322}$ window: the constants are algebraic all the way down. Three thresholds are proven exactly. The no-signaling-line visibility threshold of $I_{3322}$ is the golden ratio conjugate, $\lambda^*_3 = (\sqrt{5}-1)/2$ (explaining a numerical coincidence between two cost conventions, since $1/\lambda - 1 = \lambda$ precisely at $\varphi$; 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 $I_{3322}$ correlator part is exactly $5$, attained by a hexagonal configuration -- a non-facet functional whose value appears to be new. The $I_{4422}$ threshold is $\lambda^*_4 = \sqrt{2\sqrt{10}-6}$, the positive root of $x^4 + 12x^2 - 4$, 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 $\mathrm{gap} \leq 2.3\times10^{-8}$ at $w = 2357/2500$, and certified opening gives $\mathrm{gap} \geq 1.10\times10^{-6}$ 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 $\mathrm{gap} = \tfrac{1}{2}(w - 2\sqrt{2}/3)^2$ to 1--2%, down to gaps of $5\times10^{-10}$; 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 $w_e = 2\sqrt{2}/3$ and level 2 at $w_e^{L2} = 0.955934(3)$. 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 $\mathbb{Q}(\sqrt{2})$ to be $8\sqrt{2}/3$ per 4 units of anchor -- hence $2\sqrt{2}/3$. The certificates' entries lie in $\tfrac{1}{6}\mathbb{Z}[\sqrt{2}]$ and $\tfrac{1}{18}\mathbb{Z}[\sqrt{2}]$, 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 $1.7\times10^{-4}$ -- 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 $\mathbb{Q}(\sqrt{2})$ 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).

DOI
10.5281/zenodo.21658228

Related papers

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

Caleb Weiner

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).

•Jul 27, 2026•2 citations

The Almost-Quantum Window of the Tilted I3322 Family

Caleb Weiner

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.

•Jul 26, 2026•3 citations

A phase transition in the exactness of the NPA hierarchy at the critical doubly-tilted CHSH functional

Anton Pakhunov

Gigena et al. [npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional $B_{αβ}=α\langle A_0\rangle+β\langle B_0\rangle+\mathrm{CHSH}$ 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, $c_Q=4-s+s^3/6-s^4/36+O(s^5)$; (ii) each NPA level overshoots quadratically, $c_k(s)=4-s+a_k s^2+O(s^3)$, with the almost-quantum coefficient computed exactly, $a_{1+AB}=3/64$; (iii) the divergence of the required exact level is equivalent to positivity of the single sequence $(a_k)$ - proven for every $k$ in the companion paper. We prove the supercritical side completely: for all $α,β\ge 1$ 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 $c_Q$, 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.

quant-pharXiv:2607.13774•Jul 16, 2026•0 citations

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

Caleb Weiner

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).

•Jul 27, 2026•1 citation

No finite level of the NPA hierarchy is exact for the doubly-tilted CHSH functional near the critical tilt

Anton Pakhunov

Gigena, Panwar, Scala, Araujo, Farkas and Chaturvedi [npj Quantum Inf. 11, 82 (2025)] determined the quantum maximum of the doubly-tilted CHSH functionals, observed that the Navascues-Pironio-Acin level needed for exactness grows without evident bound toward the critical tilt, and asked whether any finite level suffices. We answer this in the negative. For the symmetric critical family $B_s=(1-s/2)(\langle A_0\rangle+\langle B_0\rangle)+\mathrm{CHSH}$ we prove: for every NPA level $k\ge 2$ there are an explicit rational $g_k>0$ and an $s^*_k>0$ with $c_k(s)\ge 4-s+g_k s^2$ on $(0,s^*_k]$; since the quantum value leaves the local bound only cubically, every finite level strictly overshoots on an interval: no finite level is exact on any neighbourhood of the critical point. Unconditionally $a_2>1/39$, $a_3>1/188$, $a_4>1/641$. The proof is a primal construction: an exactly feasible moment curve at each level, built from level-uniform structural laws and one level-independent signed witness -- a closed-form class function $y^*$ with $N_k^TΓ(y^*)N_k=u_k u_k^T$ at every level. The mechanism forces the sign: for $k\ge 3$ no quantum state and no smooth curve of quantum models can realize the gain direction, so the overshoot lives strictly in the non-quantum part of the NPA tangent cone. The proof is computer-assisted in the strict sense: finite exact-integer verifications with proven degree bounds are constituent parts of the argument; the chain has been re-verified against independent implementations, including a symbolic per-regime proof of the witness identity and a clean-room implementation written from the paper text alone. The one external input is the published quantum value of Gigena et al., cross-checked to twelve digits.

quant-pharXiv:2607.13762•Jul 16, 2026•0 citations

Certified exact identification of the I3322 quantum value

Artus Krohn-Grimberghe

We determine the quantum value of the three-setting, two-outcome Bell functional I3322. The value is the unique intersection \(q_\dagger\) of an analytic A-high stable curve of the exact Jacobi characteristic map with a central reverser curve. A computer-assisted proof isolates this intersection in a rational interval of width \(10^{-55}\), beginning 0.2508753845139765356173366109459512989013219739714764755. We prove that the supremum over finite-dimensional tensor-product strategies, the unrestricted spatial value and the commuting-operator value all equal \(q_\dagger\). No finite-dimensional strategy attains the value, whereas a normal infinite-dimensional spatial strategy and a commuting GNS strategy do. The proof combines an exact Bellman/path variational reduction, extraction of a positive bilateral calibrated Jacobi fibre from a maximizing weak-star state, a reflected-fibre central-incidence theorem and interval certificates for the unique stable/central intersection. A separate operator-positive three-remainder decomposition gives the commuting upper bound. The complete verification record is archived at DOI 10.5281/zenodo.22160086.

quant-pharXiv:2609.29693•Sep 1, 2026•0 citations
Four companion papers located, dissected, generalized, and laddered the excess of the almost-quantum set over NPA level 2 on tilted Imm22I_{mm22}Imm22​ functionals. This paper asks why the numbers are what they are, and answers for the bottom of the tilted I3322I_{3322}I3322​ window: the constants are algebraic all the way down. Three thresholds are proven exactly. The no-signaling-line visibility threshold of I3322I_{3322}I3322​ is the golden ratio conjugate, λ3∗=(5−1)/2\lambda^*_3 = (\sqrt{5}-1)/2λ3∗​=(5​−1)/2 (explaining a numerical coincidence between two cost conventions, since 1/λ−1=λ1/\lambda - 1 = \lambda1/λ−1=λ precisely at φ\varphiφ; 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 I3322I_{3322}I3322​ correlator part is exactly 555, attained by a hexagonal configuration -- a non-facet functional whose value appears to be new. The I4422I_{4422}I4422​ threshold is λ4∗=210−6\lambda^*_4 = \sqrt{2\sqrt{10}-6}λ4∗​=210​−6​, the positive root of x4+12x2−4x^4 + 12x^2 - 4x4+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][0, 471/500][0,471/500], certified closure gives gap≤2.3×10−8\mathrm{gap} \leq 2.3\times10^{-8}gap≤2.3×10−8 at w=2357/2500w = 2357/2500w=2357/2500, and certified opening gives gap≥1.10×10−6\mathrm{gap} \geq 1.10\times10^{-6}gap≥1.10×10−6 at w=1889/2000w = 1889/2000w=1889/2000, so the lower edge lies in [0.942,0.9445][0.942, 0.9445][0.942,0.9445]. Inside, ten dual-solver points spanning five orders of magnitude follow the onset law gap=12(w−22/3)2\mathrm{gap} = \tfrac{1}{2}(w - 2\sqrt{2}/3)^2gap=21​(w−22​/3)2 to 1--2%, down to gaps of 5×10−105\times10^{-10}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+AB1{+}AB1+AB at we=22/3w_e = 2\sqrt{2}/3we​=22​/3 and level 2 at weL2=0.955934(3)w_e^{L2} = 0.955934(3)weL2​=0.955934(3). The edge constant is derived at first order: the flat branch's 1+AB1{+}AB1+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)\mathbb{Q}(\sqrt{2})Q(2​) to be 82/38\sqrt{2}/382​/3 per 4 units of anchor -- hence 22/32\sqrt{2}/322​/3. The certificates' entries lie in 16Z[2]\tfrac{1}{6}\mathbb{Z}[\sqrt{2}]61​Z[2​] and 118Z[2]\tfrac{1}{18}\mathbb{Z}[\sqrt{2}]181​Z[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 1.7×10−41.7\times10^{-4}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)\mathbb{Q}(\sqrt{2})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).
Two companion papers established that the almost-quantum excess of the tilted I3322I_{3322}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\leq 4\times10^{-7}≤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=1w=1w=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 I4422I_{4422}I4422​ family (Collins-Gisin staircase, anchor weights (2,2,1,0)), we map the window and certify: the I4422I_{4422}I4422​ almost-quantum landmark at w=1w=1w=1 to width 5×10−75\times10^{-7}5×10−7; the excess at w=1w=1w=1, 2.30033×10−2≤gap≤2.30047×10−22.30033\times10^{-2} \leq \mathrm{gap} \leq 2.30047\times10^{-2}2.30033×10−2≤gap≤2.30047×10−2 -- in comparable normalization 10.8×10.8\times10.8× the textbook I3322I_{3322}I3322​ separation; a peak lower bound gap(w=7/5)≥3.888×10−2\mathrm{gap}(w{=}7/5) \geq 3.888\times10^{-2}gap(w=7/5)≥3.888×10−2; and the upper-edge closure gap(w=17/10)≤7.1×10−7\mathrm{gap}(w{=}17/10) \leq 7.1\times10^{-7}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}\{1,2\}{1,2} recover f∈[0.69260,0.69267]f \in [0.69260, 0.69267]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]f \in [0.98813, 0.98825]f∈[0.98813,0.98825]. One structural novelty refines the conjecture: the I4422I_{4422}I4422​ window opens at w=0+w = 0^+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≤max⁡QJ7/5≤9.5257805889.522944386 \leq \max_Q J_{7/5} \leq 9.5257805889.522944386≤maxQ​J7/5​≤9.525780588), giving a certified experimental corridor max⁡Q~−max⁡Q≥4.618×10−2\max_{\tilde{Q}} - \max_Q \geq 4.618\times10^{-2}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\sim 10^{-6}∼10−6; and the same machinery run in reverse certifies the window open on all of [1/2,8/5][1/2, 8/5][1/2,8/5] with margin ≥1.36×10−3\geq 1.36\times10^{-3}≥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)+∑ijsijEijJ_w = w(a_1+a_2-b_1-b_2) + \sum_{ij} s_{ij}E_{ij}Jw​=w(a1​+a2​−b1​−b2​)+∑ij​sij​Eij​ containing the I3322I_{3322}I3322​ Bell functional at w=1w=1w=1, we map where and why the almost-quantum set (NPA level 1+AB1{+}AB1+AB) exceeds NPA level 2, using a subset-relaxation diagnostic: adjoin the twelve level-2 words to the 1+AB1{+}AB1+AB basis in subsets and measure each subset's share of the gap. The excess is sharply localized and structurally rigid. (i) At w=1w=1w=1, the four matched words A1A2,A2A1,B1B2,B2B1A_1A_2, A_2A_1, B_1B_2, B_2B_1A1​A2​,A2​A1​,B1​B2​,B2​B1​ 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∗T^*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∗T^*T∗ feasible. (ii) The mechanism is algebraic: reversal-symmetrized ordering differences obey ⟨A1A2BjB3⟩−⟨A1A2B3Bj⟩=12⟨[A1,A2][Bj,B3]⟩\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⟨A1​A2​Bj​B3​⟩−⟨A1​A2​B3​Bj​⟩=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)w \in (0.90, 0.95)w∈(0.90,0.95) to (2.25,2.50)(2.25, 2.50)(2.25,2.50), with a two-lobed interior peaking near w=2w=2w=2 at 4.07×10−34.07\times10^{-3}4.07×10−3 -- 7.7 times the separation of standard I3322I_{3322}I3322​, which sits just inside the lower edge. Experiments probing the almost-quantum boundary should therefore run the tilted functional near w=2w=2w=2, not the textbook one. The headline values are certified in exact rational arithmetic: the two-word infeasibility of T∗T^*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⟩+CHSHB_{αβ}=α\langle A_0\rangle+β\langle B_0\rangle+\mathrm{CHSH}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=2α+β=2α+β=2. We quantify the mechanism on the symmetric slice s=2−α−βs=2-α-βs=2−α−β: (i) the quantum value leaves the local bound cubically, cQ=4−s+s3/6−s4/36+O(s5)c_Q=4-s+s^3/6-s^4/36+O(s^5)cQ​=4−s+s3/6−s4/36+O(s5); (ii) each NPA level overshoots quadratically, ck(s)=4−s+aks2+O(s3)c_k(s)=4-s+a_k s^2+O(s^3)ck​(s)=4−s+ak​s2+O(s3), with the almost-quantum coefficient computed exactly, a1+AB=3/64a_{1+AB}=3/64a1+AB​=3/64; (iii) the divergence of the required exact level is equivalent to positivity of the single sequence (ak)(a_k)(ak​) - proven for every kkk in the companion paper. We prove the supercritical side completely: for all α,β≥1α,β\ge 1α,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 cQc_QcQ​, 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.
Three companion papers established that the excess of the almost-quantum set over NPA level 2 on tilted Imm22I_{mm22}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=1w=1w=1 is certified to [2.038580,2.038617]×10−2[2.038580, 2.038617]\times10^{-2}[2.038580,2.038617]×10−2 at m=5m=5m=5 and [4.274637,4.274741]×10−2[4.274637, 4.274741]\times10^{-2}[4.274637,4.274741]×10−2 at m=6m=6m=6 -- with the I5522I_{5522}I5522​ and I6622I_{6622}I6622​ classical bounds (11 and 16) re-enumerated exactly -- so the growth ratio decelerates (10.8→3.54→2.1010.8 \to 3.54 \to 2.1010.8→3.54→2.10), roughly tracking (m−3)2(m-3)^2(m−3)2. The anchor law of the previous paper fails upward and is replaced: at m=5m=5m=5 every single matched pair is inert (the heaviest, certified to ≤8×10−6\leq 8\times10^{-6}≤8×10−6 of the gap) while the triangle of matched pairs on the three heaviest settings carries [0.74590,0.74595][0.74590, 0.74595][0.74590,0.74595]; at m=6m=6m=6 the triangle in turn drops to ≤0.2596\leq 0.2596≤0.2596 while the 4-clique carries [0.81660,0.81665][0.81660, 0.81665][0.81660,0.81665]; at m=7m=7m=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 I4422I_{4422}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][0.18845, 0.18884][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≤max⁡Q≤9.5240829429.522944386 \leq \max_Q \leq 9.5240829429.522944386≤maxQ​≤9.524082942 (width 1.14×10−31.14\times10^{-3}1.14×10−3) and the certified experimental corridor widens to 4.788×10−24.788\times10^{-2}4.788×10−2. The I5522I_{5522}I5522​ upper edge is certified onto deterministic branches 12w−112w-112w−1 and 18w−1118w-1118w−11 crossing at exactly w=5/3w=5/3w=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).
Gigena, Panwar, Scala, Araujo, Farkas and Chaturvedi [npj Quantum Inf. 11, 82 (2025)] determined the quantum maximum of the doubly-tilted CHSH functionals, observed that the Navascues-Pironio-Acin level needed for exactness grows without evident bound toward the critical tilt, and asked whether any finite level suffices. We answer this in the negative. For the symmetric critical family Bs=(1−s/2)(⟨A0⟩+⟨B0⟩)+CHSHB_s=(1-s/2)(\langle A_0\rangle+\langle B_0\rangle)+\mathrm{CHSH}Bs​=(1−s/2)(⟨A0​⟩+⟨B0​⟩)+CHSH we prove: for every NPA level k≥2k\ge 2k≥2 there are an explicit rational gk>0g_k>0gk​>0 and an sk∗>0s^*_k>0sk∗​>0 with ck(s)≥4−s+gks2c_k(s)\ge 4-s+g_k s^2ck​(s)≥4−s+gk​s2 on (0,sk∗](0,s^*_k](0,sk∗​]; since the quantum value leaves the local bound only cubically, every finite level strictly overshoots on an interval: no finite level is exact on any neighbourhood of the critical point. Unconditionally a2>1/39a_2>1/39a2​>1/39, a3>1/188a_3>1/188a3​>1/188, a4>1/641a_4>1/641a4​>1/641. The proof is a primal construction: an exactly feasible moment curve at each level, built from level-uniform structural laws and one level-independent signed witness -- a closed-form class function y∗y^*y∗ with NkTΓ(y∗)Nk=ukukTN_k^TΓ(y^*)N_k=u_k u_k^TNkT​Γ(y∗)Nk​=uk​ukT​ at every level. The mechanism forces the sign: for k≥3k\ge 3k≥3 no quantum state and no smooth curve of quantum models can realize the gain direction, so the overshoot lives strictly in the non-quantum part of the NPA tangent cone. The proof is computer-assisted in the strict sense: finite exact-integer verifications with proven degree bounds are constituent parts of the argument; the chain has been re-verified against independent implementations, including a symbolic per-regime proof of the witness identity and a clean-room implementation written from the paper text alone. The one external input is the published quantum value of Gigena et al., cross-checked to twelve digits.
We determine the quantum value of the three-setting, two-outcome Bell functional I3322. The value is the unique intersection q†q_\daggerq†​ of an analytic A-high stable curve of the exact Jacobi characteristic map with a central reverser curve. A computer-assisted proof isolates this intersection in a rational interval of width 10−5510^{-55}10−55, beginning 0.2508753845139765356173366109459512989013219739714764755. We prove that the supremum over finite-dimensional tensor-product strategies, the unrestricted spatial value and the commuting-operator value all equal q†q_\daggerq†​. No finite-dimensional strategy attains the value, whereas a normal infinite-dimensional spatial strategy and a commuting GNS strategy do. The proof combines an exact Bellman/path variational reduction, extraction of a positive bilateral calibrated Jacobi fibre from a maximizing weak-star state, a reflected-fibre central-incidence theorem and interval certificates for the unique stable/central intersection. A separate operator-positive three-remainder decomposition gives the commuting upper bound. The complete verification record is archived at DOI 10.5281/zenodo.22160086.