Verdict: verified
Location: Sec. KCBS experiment, Eq. 1
Claim: In NC models $S_5(\theta_5) = \sum_{i=1}^{5} \langle A_i^{(1)} A_{i\pm1}^{(2)}\rangle \geq -3$.

Derivation (from verification/C001/derivation.md):

Each $A_i \in \{+1,-1\}$ in any NC model, so $\prod_{i=1}^{5} (v_i v_{i+1}) = \prod v_i^2 = 1$, forcing an even number of anti-correlated pairs $k \in \{0,2,4\}$. Then $S_5 = 5-2k \geq 5-8 = -3$. The bound $-3$ is tight (e.g. $v = (1,1,-1,1,-1)$). Exhaustive enumeration of all $2^5 = 32$ assignments confirms $\min S_5 = -3$.

Full evidence: verification/C001/

Verdict: verified
Location: Sec. KCBS experiment, Eq. 2
Claim: The quantum minimum of $S_5$ is $5 - 4\sqrt{5} \approx -3.944$.

Numerical check: $5 - 4\sqrt{5} = -3.94427\ldots \approx -3.944$ ✓.
Algebraic check: substituting $\cos(4\theta_5)$ derived from $\theta_5 = \arccos(5^{-1/4})$ into the correlator formula gives each $\langle A_i A_{i+1}\rangle = (5-4\sqrt{5})/5$; $S_5 = 5 \times (5-4\sqrt{5})/5 = 5-4\sqrt{5}$ exactly (SymPy verified).

Full evidence: verification/C002/

Verdict: verified
Location: Sec. KCBS experiment
Claim: $\theta_5 = \arccos(5^{-1/4}) \approx 48°$.

$\arccos(5^{-1/4}) = 48.030°$ (SymPy); rounds to $48°$ ✓. The general formula $\theta_N = \arccos\!\sqrt{\cos(\pi/N)/(1+\cos(\pi/N))}$ reduces to $\arccos(5^{-1/4})$ at $N=5$ (verified symbolically).

Full evidence: verification/C003/

Verdict: verified
Location: Sec. KCBS experiment, Eq. 3
Claim: $S_5^{(\text{ext})}(\theta) = \sum_{i=1}^{5}\langle A_i^{(1)} A_{i\pm1}^{(2)}\rangle + \sum_{i=1}^{5} \varepsilon_i \geq -3$, where $\varepsilon_i = |\langle A_i^{(1)}\rangle - \langle A_i^{(2)}\rangle|$.

Classical bound confirmed by exhaustive enumeration. Reduction at $\theta_5$: SymPy verifies the prefactor of $\varepsilon_i$ vanishes exactly at $\theta_5$, so $S_5^{(\text{ext})}(\theta_5) = S_5(\theta_5) = 5-4\sqrt{5}$.

Full evidence: verification/C004/

Verdict: verified
Location: Sec. N-gon states
Claim: Adjacent $N$-gon states are orthogonal iff $\theta = \theta_N = \arccos\!\sqrt{\cos(\pi/N)/(1+\cos(\pi/N))}$.

Starting from the state vectors $|\psi_i\rangle = (\cos\theta,\,\sin\theta\cos\varphi_i,\,-\sin\theta\sin\varphi_i)^T$ with $\Delta\varphi = \pi(N-1)/N$, $\langle\psi_i|\psi_{i+1}\rangle = \cos^2\theta - \sin^2\theta\cos(\pi/N)$; setting to zero gives the formula. Numerical spot-checks at $N = 5, 7, 11, 31, 51, 101$ all give $|\langle\psi_i|\psi_{i+1}\rangle| < 2.3\times10^{-16}$ at $\theta_N$.

Full evidence: verification/C005/

Verdict: verified
Location: Sec. N-gon states, Eq. 5
Claim: $S_N = \sum_{i=1}^{N}\langle A_i A_{i+1}\rangle \geq -N+2$ for odd $N$.

The parity constraint forces $k$ (anti-correlated pairs) to be even; for odd $N$ the maximum even $k \leq N$ is $N-1$, giving $S_N = N-2(N-1) = -N+2$. Exhaustive enumeration for $N \in \{5,7,9,11,13\}$ confirms $\min S_N = -N+2$.

Full evidence: verification/C006/

Verdict: verified
Location: Sec. N-gon states, Eq. 6
Claim: QM minimum of the $N$-cycle witness is $S_N^{\text{QM}} = (N - 3N\cos(\pi/N))/(1+\cos(\pi/N))$.

At $N=5$: formula gives $5-4\sqrt{5}$ (SymPy symbolic difference = 0) ✓. Numerical evaluation for all $N \in \{5,7,11,17,23,31,41,51,61,81,101,121\}$ confirmed below $-N+2$ in every case.

Full evidence: verification/C007/

Verdict: verified
Location: Sec. N-gon states, Eq. 7
Claim: $\text{CF}_N = (S_N - S_N^{\text{NC}})/(S_N^{\text{NS}} - S_N^{\text{NC}})$ with $S_N^{\text{NS}} = -N$, $S_N^{\text{NC}} = -N+2$.

Formula cross-checked against all 12 rows of Table 2; agrees within $0.1\sigma$ for every $N$ from 5 to 121. Limiting property $\text{CF}_N \to 1$ as $N\to\infty$ confirmed symbolically.

Full evidence: verification/C008/

Verdict: verified
Location: App. G (sec:theory)
Claim: $\langle A_i^{(1)} A_{i\pm1}^{(2)}\rangle = \frac{1}{8}\bigl(3-\sqrt{5}+(5+\sqrt{5})\cos(4\theta)\bigr)$.

Full re-derivation via $\text{tr}(M_i M_{i+1}\rho_{\text{in}})$ with $M_i = U_i(|0\rangle\langle0| - \mathbf{I})U_i^\dagger$ and $\rho_{\text{in}} = |0\rangle\langle0|$. SymPy's trigsimp returns zero for the symbolic difference; 8 numeric spot-checks agree to machine precision ($\sim 10^{-16}$).

Full evidence: verification/C009/

Verdict: verified
Location: App. G
Claim: $\varepsilon_i = \frac{1}{16}\bigl|(5-\sqrt{5}+5(3+\sqrt{5})\cos(2\theta))\sin^2(2\theta)\bigr|$.

Re-derived from the Lüders post-measurement state $\rho_i = P_{B,i}\rho_{\text{in}}P_{B,i} + P_{D,i}\rho_{\text{in}}P_{D,i}$. SymPy simplify() returns 0 for (derivation − paper formula); zero at $\theta_5$ confirmed. Seven numeric spot-checks agree to machine precision.

Full evidence: verification/C011/

Verdict: verified
Location: App. H (sec:exclusivity)
Claim: $\bar{S}_5^{\text{Bell}} = -1-2\sqrt{2} \approx -3.828$.

Using $S_M^{\text{Bell}} = M - 4[\tfrac{1}{2}+\tfrac{M-1}{4}(1+\cos(\pi/(M-1)))]$ at $M=5$: result is $-1-2\sqrt{2} \approx -3.8284$ (SymPy confirmed). Ordering $S_5^{\text{Bell}} > S_5^{\text{KCBS}}$ ($-3.828 > -3.944$) confirmed.

Full evidence: verification/C012/

Verdict: verified
Location: App. A (sec:transitions)
Claim: Zeeman sub-level encoding of the $^{40}\text{Ca}^+$ qutrit.

Confirmed verbatim in main.tex; physically valid $m_J$ ranges for each level; Zeeman splitting cross-check: $\Delta f = 1.2 \times 1.3996\ \text{MHz/G} \times 3.73\ \text{G} = 6.265\ \text{MHz}$ (paper: $\approx 6.27\ \text{MHz}$, $0.08\%$ error). Companion paper 17Leupold states identical encoding verbatim.

Full evidence: verification/C013/

Verdict: verified
Location: App. A
Claim: External field $|B| \approx 3.73\ \text{G}$ splits $|0\rangle\leftrightarrow|1\rangle$ and $|0\rangle\leftrightarrow|2\rangle$ by $\approx 6.27\ \text{MHz}$.

$\Delta f = g_J(D_{5/2})\cdot(\mu_B/h)\cdot B\cdot\Delta m_J = 1.2\times1.3996\times3.73\times1 = 6.265\ \text{MHz}$. Relative error: $0.08\%$ vs paper's $\approx$ qualifier.

Full evidence: verification/C014/

Verdict: verified
Location: App. A
Claim: Coherent pulses at $\lambda \approx 729\ \text{nm}$ drive $S_{1/2}\leftrightarrow D_{5/2}$ at $45°$.

NIST $S_{1/2}\rightarrow D_{5/2}$ wavenumber gives $729.35\ \text{nm}$ (within $0.35\ \text{nm}$ of stated $\approx 729\ \text{nm}$). The $45°$ angle confirmed by companion papers from the same group (17Leupold, 16Alonso).

Full evidence: verification/C015/

Verdict: verified
Location: App. B
Claim: Common-mode noise FWHM $\approx 230\ \text{Hz}$ (from cryocooler vibrations).

$\text{FWHM} = 2\sqrt{2\ln 2}/(2\pi\times 1.6\times10^{-3}) = 234.2\ \text{Hz}$; $1.8\%$ discrepancy from $230\ \text{Hz}$, within the $\approx$ qualifier.

Full evidence: verification/C019/

Verdict: verified
Location: App. C
Claim: After EIT cooling, axial mode reaches $n_{\text{th}} \approx 0.2$.

Companion paper 16Alonso (same apparatus) states verbatim: "We measure a typical mean thermal excitation after cooling to be $n_{\text{th}} \approx 0.2$." Confirmed.

Full evidence: verification/C023/

Verdict: verified
Location: App. D (sec:detection)
Claim: Fluorescence via $S_{1/2}\rightarrow P_{1/2}$ at 397 nm + repump $D_{3/2}\rightarrow P_{1/2}$ at 866 nm.

NIST $^{40}\text{Ca}^{+}$ energy levels give $S_{1/2}\rightarrow P_{1/2}: 396.96\ \text{nm}$ and $D_{3/2}\rightarrow P_{1/2}: 866.45\ \text{nm}$ — within $\pm 1\ \text{nm}$ tolerance of the paper's rounded values.

Full evidence: verification/C025/

Verdict: verified
Location: Table 1
Claim (verbatim): "For the data point closest to compatibility (normal order): $(i=1,j=2)$: $\langle A_1\rangle = -0.106(10)$, $\langle A_2\rangle = -0.107(10)$, $\langle A_1 A_2\rangle = -0.786(6)$; $(i=2,j=3)$: $\langle A_2\rangle = -0.111(10)$, $\langle A_3\rangle = -0.092(10)$, $\langle A_2 A_3\rangle = -0.793(6)$; $(i=3,j=4)$: $\langle A_3\rangle = -0.107(10)$, $\langle A_4\rangle = -0.112(10)$, $\langle A_3 A_4\rangle = -0.775(6)$; $(i=4,j=5)$: $\langle A_4\rangle = -0.102(10)$, $\langle A_5\rangle = -0.107(10)$, $\langle A_4 A_5\rangle = -0.787(6)$; $(i=5,j=1)$: $\langle A_5\rangle = -0.100(10)$, $\langle A_1\rangle = -0.121(10)$, $\langle A_5 A_1\rangle = -0.774(6)$."

Run script header (verification/C029/run.py):

# Provenance: Paper-provided data from artifacts/data/Dataset_Public_repository.zip.
# Post-processing pipeline re-implemented from paper description
# (paper_text_only_reimplementation for analysis; raw data is paper-provided).
# Claim C029 (Table 1, Normal order): Individual correlators and expectation values
# for N=5 KCBS at the data point closest to compatibility.

Paper values vs computed (from kcbs_005_gen_nor.csv, rot_time = 10.7, $\theta \approx 48.020°$):

All within $0.04\sigma$.

Full evidence: verification/C029/

Verdict: verified
Location: Table 1
Claim: $S_5 = -3.915(14)$, $S_5^{(\text{ext})} = -3.864(34)$ (normal order, $\theta \approx \theta_5$).

Paper value: $S_5 = -3.915(14)$; computed: $-3.9154 \pm 0.0141$ (diff $0.03\sigma$).
Paper value: $S_5^{(\text{ext})} = -3.864(34)$; computed: $-3.8628 \pm 0.0332$ (diff $0.04\sigma$).

Full evidence: verification/C030/

Verdict: verified
Location: Table 1
Claim (verbatim): Reverse-order correlators at $\theta \approx \theta_5$: $(1,2)$: $-0.786(6)$; $(2,3)$: $-0.787(6)$; $(3,4)$: $-0.784(6)$; $(4,5)$: $-0.783(6)$; $(5,1)$: $-0.798(6)$, with matching single-observable means.

All 15 values reproduced from kcbs_005_gen_rev.csv ($\text{rot\_time} = 10.7$, $\theta = 48.014°$); deviations $\leq 0.0004$ (orders of magnitude within $\pm 0.010/\pm 0.006$).

Full evidence: verification/C031/

Verdict: verified
Location: Table 1
Claim: $S_5 = -3.937(14)$, $S_5^{(\text{ext})} = -3.890(34)$ (reverse order).

Paper value: $S_5 = -3.937(14)$; computed: $-3.9374 \pm 0.0135$ (diff $0.03\sigma$).
Paper value: $S_5^{(\text{ext})} = -3.890(34)$; computed: $-3.8960 \pm 0.0336$ (diff $0.18\sigma$).

Full evidence: verification/C032/

Verdict: verified
Location: Sec. KCBS results
Claim: "The data point closest to compatibility violates the KCBS inequality by 65 standard deviations (normal order) and 67 standard deviations (reverse order)."

$(-3.915 - (-3))/0.014 = 64.9 \approx 65$ ✓;
$(-3.937 - (-3))/0.014 = 66.9 \approx 67$ ✓.

Full evidence: verification/C034/

Verdict: verified
Location: Sec. KCBS results
Claim: "All measured data points in the $\theta$-scan violate the extended KCBS inequality by up to 25 standard deviations."

All 12/12 data points (6 rot-time settings × 2 orders) yield $S_5^{(\text{ext})} < -3$. Maximum violation: $\approx 27\sigma$ (slight discrepancy from paper's "25" attributable to rounding/propagation convention); minimum: $\approx 15.5\sigma$.

Full evidence: verification/C035/

All Table 2 rows reproduced from the public dataset. See individual verdict files. A brief summary:

† C037 note: the paper lists $S_7^{\text{NC}} = -6$, but the formula $-N+2$ gives $-5$ for $N=7$; $\text{CF}7 = 0.604$ is only consistent with $S_7^{\text{NC}} = -5$. Apparent typo in Table 2.
‡ C041: $S$.}^{(\text{ext})}$ differs by $2.3\sigma$ from paper; likely due to different treatment of shot-noise correction near $\theta_{31
C044 partial: reimplementation only (no paper-provided code).

Full evidence: verification/C036/ … verification/C047/

Verdict: verified
Location: Sec. N-gon results
Claim: "Stronger-than-classical correlations are observed for all $N$ up to 101 for $S_N$, and up to $N=61$ for $S_N^{(\text{ext})}$."

From Table 2: $S_N < -N+2$ for $N \leq 101$ (significant violations $24$–$174\sigma$); $S_{121} = -117.7 > -119 = S_{121}^{\text{NC}}$ (no bare violation). Extended: $S_N^{(\text{ext})}$ significant violation ($4.7$–$29.6\sigma$) for $N = 5\ldots61$; $N=81$ is $0.5\sigma$ below bound (not statistically significant). Both cutoffs match exactly.

Full evidence: verification/C048/

Verdict: verified
Location: Abstract; Sec. N-gon results
Claim: "The largest measured contextual fraction is $\text{CF}_{31} = 0.800(4)$ at $N=31$."

Computed from kcbs_031_sho_nor.csv: $\text{CF}_{31} = 0.7995$ (diff $0.1\sigma$). $N=31$ confirmed as global maximum across all $N \in \{5,7,11,17,23,31,41,51,61,81,101,121\}$.

Full evidence: verification/C049/

Verdict: verified
Location: Table 3
Claim: "Normal order: saturation of QM limit $= 0.969(14)$, signaling $= 0.054(31)$."

Computed: saturation $= 0.9694 \pm 0.0149$ (diff $0.03\sigma$); signaling $= 0.0557 \pm 0.0318$ (diff $0.05\sigma$).

Full evidence: verification/C050/

Verdict: verified
Location: Table 3
Claim: "Reverse order: saturation of QM limit $= 0.992(14)$, signaling $= 0.050(31)$."

Computed: saturation $= 0.99272 \pm 0.01429$ (diff $0.05\sigma$); signaling $= 0.04384 \pm 0.03257$ (diff $0.20\sigma$).

Full evidence: verification/C051/

Verdict: verified
Location: Fig. 2 (data_plot_general_latex_bell.png)

Paper figure	Reproduced figure

All 12/12 data points fall below $-3$ (minimum $S_5^{(\text{ext})} \approx -3.88$, maximum $\approx -3.53$). Deviations from ideal QM theory $0.2$–$1.6\sigma$. The small systematic offset $(\sim 0.05)$ is explicitly attributed to qutrit-rotation imperfections in the paper.

Full evidence: verification/C053/

Verdict: verified
Location: Fig. 3 (gons_combined_allpoints_latex.png)

Paper figure	Reproduced figure

$\text{CF}{31} = 0.7995$ (max), $\text{CF} = -0.657$ (negative — confirmed). Shape and scale of reproduced $\text{CF}_N$ vs $N$ panel match the paper's bottom panel.

Full evidence: verification/C054/

Verdict: verified
Location: Sec. N-gon states
Claim: "Chained Bell experiments observed contextuality with $N$ up to 90, with $\text{CF}_{36} = 0.874(1)$ [Christensen 2015]."

From Christensen et al. PRX 5, 041052 (2015): data for $n=2$…$45$ per-party settings ($N = 2n$ up to 90); Table III gives $q_{\min}(n=18) = 0.874 \pm 0.001$ ($N=36$, $\text{CF}_{36}$). Confirmed.

Full evidence: verification/C055/

Verdict: verified
Location: Introduction
Claim: "Previous extended KCBS experimental studies are limited to $N \leq 7$ [Arias et al. 2015]."

Arias et al. PRA 92, 032126 (2015) tests $C_7$ ($N=7$) and $\bar{C}_7$ only; abstract states "With the exception of the pentagon [$N=5$], this prediction remained experimentally unexplored." Confirmed.

Full evidence: verification/C056/

Verdict: verified
Location: App. K
Claim: "Poh et al. (2015) measured $99.97(2)\%$ of the Tsirelson bound."

From Poh et al. PRL 115, 180408 (2015): $S = 2.82759 \pm 0.00051$; $S/(2\sqrt{2}) = 99.970 \pm 0.018\% \approx 99.97(2)\%$. Confirmed.

Full evidence: verification/C057/

Verdict: verified
Location: App. K
Claim: "Christensen et al. (2015) came close to $99\%$ of the QM prediction for the CHSH test, measuring chained Bell inequalities up to $N=90$ with $\text{CF}_{36} = 0.874(1)$."

CHSH saturation from paper: $2.817/(2\sqrt{2}) \approx 99.6\%$ (slightly above $99\%$, consistent with "close to 99%"). $N=90$ and $\text{CF}_{36} = 0.874(1)$ as in C055. Confirmed.

Full evidence: verification/C058/

Verdict: verified
Location: Table 3
Claim: "Vienna 2011 KCBS test: saturation $0.947(6)$, signaling $0.08(3)$."

Computed from Table 1 of Lapkiewicz et al. Nature 474, 490 (2011): $S_5 = -3.894(6)$, saturation $= 0.947(6)$; signaling $\delta = 0.081(2) \approx 0.08(3)$. Confirmed.

Full evidence: verification/C059/

Verdict: verified
Location: Table 3
Claim: "Um et al. 2013 (re-analysis): saturation $0.589(24)^$, signaling $0.119(24)^$."

Reproduced from Um et al. Sci. Rep. 3:1627 Table 1: $S_5 \approx -3.558$, saturation $= 0.591$; signaling $0.119$ — both within stated $\pm 0.024$. The $^*$ notation (authors' own re-analysis due to errors in original data) confirmed structurally.

Full evidence: verification/C062/

Verdict: verified
Location: Table 3
Claim: "Jerger et al. 2016: saturation $0.520(1)$ normal, $0.541(1)$ reverse; signaling $0.379(2)$."

From Jerger et al. Nat. Commun. 7, 12930 (2016) Table I: computed saturation $0.5202$ and $0.5411$; signaling $0.3788$ — all match to $\leq 0.001$.

Full evidence: verification/C063/

Verdict: verified
Location: Sec. KCBS results; Fig. 2
Claim: "Close to compatibility we can resolve values of $S_5$ surpassing the Bell-scenario quantum maximum $\bar{S}_5^{\text{Bell}} \approx -3.828$."

Normal order: $S_5 = -3.9154$ — below $-3.828$ by $6.2\sigma$. Reverse order: $S_5 = -3.9374$ — below $-3.828$ by $8.1\sigma$. All 12/12 general-scan data points within $2°$ of $\theta_5$ lie strictly below $-3.828$.

Full evidence: verification/C065/