Verdict: partial Location: Dissipation scheme; Supp. Mat. S1 Type / expected artifact: numeric / math / numeric Claim: The most rapid convergence is found for Phi = pi/4, theta ~= 0.72 pi and gamma ~= 0.22 pi, where N_0 = 7.62 cycles. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/main.tex:108 (Dissipation scheme); source/supp_content.tex:27 (Supp. Mat. S1).

Proof summary: Minimized lambda_max of S(pi/4,gamma,theta) over (theta,gamma). Coarse grid + Nelder-Mead optimum: theta=0.716pi, gamma=0.221pi, lambda_max=0.876964, N0=-1/log(lambda_max)=7.617. Paper: theta~0.72pi, gamma~0.22pi, N0=7.62. Match within <0.5% on all three; tolerance |dtheta/pi|,|dgamma/pi|<0.01 and |dN0|<0.05 satisfied. Physical model reimplemented from paper text (no paper-provided code/data), so capped at partial.

Verification report: claim report.

Verdict: partial Location: Experiment Type / expected artifact: numeric / numeric Claim: Drive (A) uses a bichromatic 729 nm field with two components detuned by delta = +/- 2 pi x 14.7 kHz from the red and blue motional sidebands of |down><->|e>, using the axial stretch mode at omega_m ~= 2 pi x 2.4 MHz with Lamb-Dicke parameter eta = 0.026. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/main.tex:115 (Experiment).

Proof summary: Paper: eta=0.026 for axial stretch mode omega_m=2pi2.4MHz, 729nm 40Ca+. Lamb-Dicke eta=k sqrt(hbar/(2 m_eff omega)) gives 0.0626 (single-ion mass m) or 0.0442 (stretch effective mass 2m), both with laser along axis. To match 0.026 requires a beam-axis projection cos(theta)~0.42 (single) or ~0.59 (stretch) — a geometry assumption from the setup not fixed by the claim's own quantities. delta=2pi14.7kHz consistent (t=2*2pi/delta=136us). Internally consistent only via an unstated projection -> partial.

Verification report: claim report.

Verdict: partial Location: Experiment Type / expected artifact: numeric / numeric Claim: The collective excitation step is implemented as two pulses (t = 2 n pi/delta, n=2) with eta Omega = delta/2, giving Phi = pi/4 and a drive time t = 150 us. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/main.tex:122 (Experiment).

Proof summary: Paper: Phi=n pi (etaOmega)^2/(2 delta^2)=pi/4 (n=2, etaOmega=delta/2) and drive time t=150us. The Phi computation is an exact closed-form identity: with etaOmega=delta/2,n=2 it gives exactly pi/4 (0.250 pi) — matches. The t=150us does NOT equal the ideal 2 n pi/delta=136.1us; it equals two 75us ramped segments (C068), pulling the pulse-ramp model from paper text. Phi is verified-grade but the timing relies on a paper assumption -> overall partial.

Verification report: claim report.

Verdict: partial Location: Experiment Type / expected artifact: numeric / numeric Claim: The drive (C) angle is alternated: theta_1 = pi in odd cycles (t_C = 6.4 us) and theta = pi/2 in even cycles (t_C = 3.2 us), acting as a spin-echo; one cycle takes ~= 165 us on average. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/main.tex:122 (Experiment).

Proof summary: Paper: theta_1=pi (t_C=6.4us), theta=pi/2 (t_C=3.2us), cycle ~165us avg. With Omega_C=2pi*78kHz, t_C=theta/Omega_C gives 6.41us and 3.21us (linear in theta, ratio exactly 2) — matches to 0.2%. Per-cycle reconstruction = drive A(150us)+gapA(3.2)+repump B(5us)+drive-C avg(4.8us) = 163.0us vs paper 165us (~1%). The ~165us requires summing sequence components from paper text -> partial.

Verification report: claim report.

Verdict: partial Location: Experiment Type / expected artifact: empirical / math / empirical Claim: In the absence of other errors, the protocol produces |Psi^-> regardless of the ions' temperature. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/main.tex:122 (Experiment).

Proof summary: Claim: error-free protocol produces |Psi^-> regardless of temperature. the self-contained model is spin-only (no motional dof), so temperature is checkable only structurally: the error-free cycle has a UNIQUE fixed point |Psi^-> (F=1.0 exactly; 2nd-eigenvalue magnitude 0.877<1). Iterating from pure states (|dn,dn>,|up,up>,|Psi^+>), the maximally-mixed = infinite-T state, and random mixed states all converge to |Psi^-> (min F=0.99999983 after 120 cycles; residual is just the 0.877^120 transient). Temperature enters the physical system only as initial motional condition / temperature-dependent LOCAL errors, which are absent here, so the fixed point is temperature-independent. Structural/initial-condition content confirmed; full temperature claim is a paper-text argument (no motional dof). Capped at partial.

Verification report: claim report.

Verdict: partial Location: Fig. 3a Type / expected artifact: plot / figure (errors_bit_flip.pdf) Claim: Fig. 3a: the steady-state error associated with each individual error channel of probability p. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: visual_match_only, paper_text_only_reimplementation. Source location(s): source/main.tex:139-145 (Fig. 3a).

Proof summary: Reproduced per-channel steady-state error grid for the 16 M_{i,j} depolarising channels (map rho->(1-p)rho+p M rho M^dag once/cycle after drive A). Global errors (interior 3x3) ~0; local errors amplified: X_I/I_X~4.89, Y_I/I_Y~4.89, Z_I/I_Z~3.88 (round to 5,5,4 as in Fig 3a panel). Matches paper structure. Theory-reproducible plot; Tier-A visual match only -> caps at partial.

Verification report: claim report.

Plot evidence:

Verdict: partial Location: Fig. 3b Type / expected artifact: plot / figure (errors_bit_flip.pdf) Claim: Fig. 3b: comparison of steady-state errors for uncorrelated (solid) versus correlated (dashed) errors, showing correlations increase fidelity for similar constituent operators. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: visual_match_only, paper_text_only_reimplementation. Source location(s): source/main.tex:139-145 (Fig. 3b).

Proof summary: Reproduced uncorrelated (solid) vs correlated (dashed) steady-state error curves. Correlated bit-flip ~3.07p (paper ~3.2p) < uncorrelated bit-flip ~6.03p (paper ~5.2p): confirms correlations increase fidelity for similar constituent operators (the claim). Correlated phase-flip ~0 (DFS). Uncorrelated bit-flip coeff ~16% above paper due to channel parametrisation, but the qualitative ordering claimed is reproduced. Theory-reproducible plot; Tier-A visual match only -> caps at partial.

Verification report: claim report.

Plot evidence:

Verdict: partial Location: Theoretical error analysis; Supp. Mat. S3 Type / expected artifact: empirical / math / Fig. simulations.pdf Claim: Compared to a single MS gate or a two-loop phase-modulated MS gate based on H_A, the protocol reduces the effect of qubit frequency errors and Rabi frequency errors, but offers no benefit for motional frequency errors or fast (Markovian) optical qubit dephasing. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5 -> gpt-5; verdict_chain partial -> partial -> partial. Limitations: paper_text_reimplementation_with_motion, truncated_fock_space_NF8, coarse_qualitative_grid, optical_qubit_low_offset_mismatch. Source location(s): source/main.tex:147 (Theoretical error analysis); source/supp_content.tex:88-115 (Supp. Mat. S3).

Proof summary: R003 builds the missing motion-including model: two 3-level ions coupled to a truncated motional Fock mode (NF=8), using the Supp. Mat. S3 sideband Hamiltonian, carrying the full spin-motion density matrix through 80 protocol cycles, and applying repump B and drive C without tracing away motion. The second pulse uses the correct Hamiltonian clock offset, so motional-frequency errors leave the phase-space loop open. The motional-frequency, Rabi-frequency, and Markovian-dephasing legs are reproduced; at 1.5 kHz optical-qubit-frequency error the protocol is also below the single MS comparator (0.0876 vs 0.1771). However, the low-offset optical-qubit-frequency suppression is not reproduced by persistent spin-motion propagation: the C059 NF=8 diagnostic gives 0.00359 at 0.25 kHz, while the thesis-scale value is recovered only by a phase-only/reset-after-A convention (2.17e-5 to 4.62e-5). Verdict remains partial with mismatch.

Verification report: claim report.

Plot evidence:

Verdict: partial Location: Experimental error analysis Type / expected artifact: numeric / numeric Claim: The measured 16-cycle error of 0.07(1) is consistent with the value 3.2 p predicted by the correlated bit-flip error model. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only. Source location(s): source/main.tex:155 (Experimental error analysis).

Proof summary: Internal-consistency check. Paper: measured 16-cycle error 0.07(1), model prediction 3.2 p with measured p ~= 0.02 after 16 cycles (C031), measured fidelity F=93(1)%. Computed: 1 - 0.93 = 0.070 (matches 0.07(1)); 3.2 * 0.02 = 0.064, which lies inside the +-1 sigma band [0.060, 0.080]; |0.064 - 0.070| = 0.006 < tolerance 0.01. Consistent. Capped at partial because it depends on the measured (not reconstructed) values p~=0.02 and F=0.93; provenance paper_text_only.

Verification report: claim report.

Verdict: partial Location: Experimental error analysis Type / expected artifact: numeric / numeric Claim: Operating at a fixed N=16 cycles corresponds to ~= 3 ms of 729 nm light per shot. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/main.tex:155 (Experimental error analysis).

Proof summary: Paper: N=16 cycles ~= 3 ms of 729 nm light per shot. Reconstructed from quoted timings: drive (A) = two 75 us 729 nm MS pulses/cycle = 150 us/cycle (main line 122; supp line 133), so 16 x 150 us = 2.40 ms. Using the full average cycle time 165 us (main line 122) as an upper bound gives 16 x 165 us = 2.64 ms. Adding shelving (two 729 carrier pulses, supp line 137) and the analysis drive-A round (~150 us) lands at ~2.7-3.0 ms. All within rounding of the paper's one-sig-fig ~3 ms. Capped at partial: exact shelving/analysis 729 pulse durations not fully specified; provenance paper_text_only reconstruction.

Verification report: claim report.

Verdict: partial Location: Experimental error analysis Type / expected artifact: numeric / numeric / paper_text Claim: Drive (A) produces a dipole AC Stark shift of ~= 2 pi x 17 kHz. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only. Source location(s): source/main.tex:157 (Experimental error analysis).

Proof summary: Paper: drive (A) dipole AC Stark shift ~= 2 pi x 17 kHz (main line 157). The carrier AC Stark shift depends on the actual 729 nm intensity and detunings from the relevant carrier transitions, which are not fully specified, so it cannot be reconstructed from first principles. Consistency bounds: (1) the paper-reported differential Stark shift between the two ions is 2 pi x 2.5 kHz (main line 122); 2.5/17 ~= 15% implied per-ion intensity imbalance, physically reasonable for two-ion addressing. (2) With drive (A) Omega = 2 pi x 265 kHz (supp), the AC-Stark estimate Omega^2/(4 Delta) = 17 kHz implies an effective carrier detuning Delta ~= 2 pi x 1.0 MHz, a plausible detuning to nearby transitions. The 17 kHz value is consistent in scale but not exactly reconstructable -> partial, provenance paper_text_only.

Verification report: claim report.

Verdict: partial (mismatch) Location: Supp. Mat. S1 Type / expected artifact: math / math Claim: With full transfer (Phi=pi/4) and complete repump, the combined two-parameter superoperator S_{BA}(gamma) restricted to the ground manifold has six non-zero eigenvalues: lambda_ss=+1 and five roots of a degree-5 polynomial, with lambda_max strictly dominating the other non-unit eigenvalues over the whole parameter space. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:25-27 (Supp. Mat. S1).

Proof summary: Built the REDUCED superoperator on the 4-state manifold M={dd,uu,Psi+,Psi-} under the paper's approximations: full transfer (numerically U_A(pi/4) maps dd->ee and leaves uu,Psi+,Psi- invariant) and complete repump (population leakage out of M after B.A = 4e-16), giving S(gamma,theta)=S_C|_M S_BA(gamma) on the 16-dim manifold density space. CONFIRMED parts: a unique lambda_ss=+1 (eigenvector |Psi-> 1e-9. The dominance statement and the optimal convergence rate are verified; the six-eigenvalue structural sub-claim is not reproduced -> partial / mismatch.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S1 Type / expected artifact: numeric / numeric Claim: In the alternating-drive scheme the protocol converges optimally at gamma^opt ~= 0.23 pi with rate 1/N_0^opt = -(1/2) log(lambda_+^opt) ~= 1/7.92 per cycle; for the experimentally measured gamma ~= 0.31 pi the rate is 1/10.35 per cycle. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation, rounded_input_sensitivity. Source location(s): source/supp_content.tex:56 (Supp. Mat. S1).

Proof summary: Round-2 audit preserves partial but removes the hard mismatch flag. The closed-form lambda_+(gamma) was read and rerun; it agrees with the full two-cycle superoperator. The optimum is gamma=0.2310pi and N0=7.9165, matching the paper. At the literal rounded gamma=0.31pi the rate is 1/10.6855, but gamma=0.305pi gives N0=10.3096, consistent with the printed 1/10.35. Because the paper labels gamma as approximate, this is a rounded-input sensitivity rather than a hard contradiction.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S1, Fig. Type / expected artifact: plot / figure (discrete_eigenvalue.png) Claim: Fig. (discrete_eigenvalue): the second dominant eigenvalue lambda_max of S(pi/4,gamma,theta) as a function of (gamma,theta), minimized at theta^opt ~= 0.72 pi, gamma^opt ~= 0.22 pi. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: visual_match_only, paper_text_only_reimplementation. Source location(s): source/supp_content.tex:29-34 (Fig. discrete_eigenvalue).

Proof summary: Reproduced lambda_max heatmap of S(pi/4,gamma,theta) (self-contained model) over (theta in [0,pi], gamma in [0,pi/2]). Grid minimum lambda_max=0.87697 at theta=0.717 pi, gamma=0.221 pi vs paper theta^opt~0.72 pi, gamma^opt~0.22 pi; rate -1/log(lambda_max)=7.617 vs paper 1/7.62. Heatmap structure and RdBu color scale match. Theory-reproducible plot; Tier-A visual match only -> caps at partial.

Verification report: claim report.

Plot evidence:

Verdict: partial Location: Supp. Mat. S1, Fig. Type / expected artifact: plot / figure (convergence.pdf) Claim: Fig. (convergence): the singlet error after 50 cycles versus protocol parameters, with convergence slow-down near Phi=m pi/2, gamma=m pi/2, and theta=m pi (population trapping). Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: visual_match_only, paper_text_only_reimplementation. Source location(s): source/supp_content.tex:37-41 (Fig. convergence).

Proof summary: Simulated 50 cycles from |down,down> with the self-contained model and reproduced singlet error vs Phi, gamma, theta (panel a) and vs theta for 15/40/65 cycles (panel b). Reproduced slow-down features: error->1 at Phi=pi/2 and gamma=pi/2 (90 deg) and high error at theta=0/pi endpoints (m pi population trapping); theta minimum near 3pi/4. Matches paper qualitatively. Theory-reproducible plot; Tier-A visual match only -> caps at partial.

Verification report: claim report.

Plot evidence:

Verdict: partial Location: Supp. Mat. S1, Fig. Type / expected artifact: plot / figure (lambda_const_vs_altern.pdf) Claim: Fig. (lambda_const_vs_altern): comparison of lambda_max for constant versus alternating drive (C) as a function of branching angle gamma, using lambda_max = lambda_+^{1/2} for the alternating case. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: visual_match_only, paper_text_only_reimplementation. Source location(s): source/supp_content.tex:49-54 (Fig. lambda_const_vs_altern).

Proof summary: Reproduced lambda_max(gamma) for constant vs alternating drive (C) using lambda_max=lambda_+^{1/2} for the alternating case. Constant min 0.8770 at gamma=0.221 pi (1/7.62); alternating min 0.8813 at gamma=0.231 pi (1/7.92), matching paper optima. Closed-form lambda_+ matches numeric second_eigmag to 1.6e-15. Curves and y-range match paper. Theory-reproducible plot; Tier-A visual match only -> caps at partial.

Verification report: claim report.

Plot evidence:

Verdict: partial Location: Supp. Mat. S2 Type / expected artifact: math / math Claim: In the limit |alpha(t)|<<1, residual spin-motion entanglement creates a correlated bit-flip with probability p = |alpha(t)|^2, with Kraus operator I_e X_e + X_e I_e. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation, normalization_convention, oscillator_truncation. Source location(s): source/supp_content.tex:84 (Supp. Mat. S2).

Proof summary: Expanded D=exp(alpha S_x a^dag - alpha S_x a), S_x = X_e^(1)+X_e^(2), on |dd>|0> (qutip, Fock cutoff 20). One-quantum Kraus branch K_1 = <1|D|0> = alpha(I_eX_e + X_eI_e)|dd> exactly to first order (ratio 0.99998), confirming the correlated bit-flip Kraus op I_eX_e+X_eI_e and the single-quantum mechanism. Its population scales as |alpha|^2 (||K_1|dd>||^2/|alpha|^2 -> 2.000 = ||S_x|dd>||^2 as alpha->0); the absolute factor 2 arises because the collective S_x on |dd> has two single-ion flip terms (paper's p=|alpha|^2 is the per-ion single-quantum probability). Structure and |alpha|^2 scaling confirmed. Partial: paper-text-only model, per-ion-vs-collective alpha normalization, small truncation.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S2 Type / expected artifact: empirical / math / numeric Claim: The steady-state error caused by residual spin-motion entanglement is found in simulation to be almost identical to that introduced by U_x(sqrt(2 p)), so the main-text simulations apply to both error sources. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:84 (Supp. Mat. S2).

Proof summary: Claim (supp S2): steady-state error from residual spin-motion entanglement is almost identical to that from U_x(sqrt(2p)). Reimplemented both: (A) unitary U_x(eps); (B) state-dependent residual displacement V=exp[(alpha a^dag - alpha a)(x)S_x] on spin(x)truncated oscillator (NFOCK=16, CPTP verified), motion traced out. Calibrated eps and alpha to the SAME operational p (single-event one-flip probability from |dn,dn>, supp l.82-84). Matched-p steady-state errors 1-F: p=0.005 1.57e-2 vs 1.57e-2 (0.5%); 0.01 3.09e-2 vs 3.12e-2 (1.0%); 0.02 6.01e-2 vs 6.13e-2 (2.0%); 0.04 (4.0%); 0.06 (6.1%). Max rel diff 6.1%, <1% at small p. Almost identical, as claimed. Also confirmed paper's eps=sqrt(2p) (operational p_A~0.0099 at eps=sqrt(20.01)). Requires reimplemented motional model + Fock truncation => partial.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S3 Type / expected artifact: empirical / math / numeric Claim: For the protocol the fidelity with |Psi^-> is extracted after 80 cycles, while for the comparison single-loop (length t) and two-loop (length t sqrt(2)) MS gates the fidelity is taken against (|down,down> - i|ee>)/sqrt(2). Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:100-101 (Supp. Mat. S3).

Proof summary: Descriptive simulation-protocol claim corroborated structurally with the self-contained model. MS gate from H_A reaches the triplet Bell target (|dd>-i|ee>)/sqrt2 (fidelity 1.0); protocol steady state is |Psi-> (F=1) and 80 cycles converges (F=0.99998 from |dd>, N0=7.62 so 80 = 10.5 N0), confirming 80 cycles is a sensible fidelity-extraction point. Single-loop (length t) vs two-loop (length t*sqrt2) are path/duration descriptors of the same net area. This is a descriptive protocol choice, not a reproduced measurement, so capped at partial.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S3, Fig. Type / expected artifact: plot / figure (simulations.pdf) Claim: Fig. (simulations): Bell-state error versus optical-qubit-frequency error, motional-frequency error, fractional Rabi-frequency error, and optical-qubit dephasing rate, for an MS gate, a two-loop MS gate, and 80 cycles of the protocol. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5 -> gpt-5; verdict_chain partial -> partial -> partial. Limitations: paper_text_reimplementation_with_motion, truncated_fock_space_NF8, visual_match_only, no_digitized_paper_data, optical_qubit_low_offset_mismatch. Source location(s): source/supp_content.tex:103-108 (Fig. simulations).

Proof summary: Overwritten R003 now provides a full four-panel C059 audit and resolves the optical-qubit-frequency discrepancy. The full persistent spin-motion model (NF=8) reproduces the motional-frequency, Rabi-frequency, and Markovian-dephasing conclusions, but it does not reproduce the low-offset optical-qubit-frequency blue curve: at 0.25 kHz it gives protocol singlet error 0.00359, not the thesis-scale ~2e-5. Resetting motion after each complete two-pulse drive A gives 4.62e-5, and a phase-only correlated-Z surrogate gives 2.17e-5. Thus the thesis point is consistent with the text's first-order correlated-phase explanation, not with carrying residual spin-motion from epsilon_q through all cycles. Verdict remains partial with mismatch because the q-frequency panel requires an unstated phase-only/reset-after-A convention.

Verification report: claim report.

Plot evidence:

Verdict: partial Location: Supp. Mat. S3 Type / expected artifact: math / math Claim: Spontaneous emission from |e> with optical-qubit lifetime T_1 limits an MS gate of length t to a Bell-state error ~= 0.5 t/T_1, and the amplified error gives a lower bound on the protocol steady-state infidelity of ~= 3 t/T_1. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:117 (Supp. Mat. S3).

Proof summary: Paper: MS-gate Bell error ~0.5 t/T1; protocol steady-state infidelity lower bound ~3 t/T1. Derivation: expected decays over a gate = _avg * t/T1 with _avg ~ 0.5 (final |ee> has 1 excited ion, population rises 0->1), reproducing 0.5 t/T1 as an order-of-magnitude estimate (linear-area numeric 0.36, same order). the self-contained model reproduces the amplification factor exactly: steady-state singlet error = 3.18 p for per-cycle correlated bit-flip p (= main-text 3.2p). The ~3 t/T1 bound = amplification(3.18) x per-cycle decay(~t/T1); consistent in form but the factor 3 vs gate 0.5 requires the amplification model plus an O(1) per-cycle decay assumption, not a single clean identity. Both stated approximately. -> partial.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S3 Type / expected artifact: numeric / math / numeric Claim: The average number of photons scattered during the full protocol is n_gamma = 4 csc^2(2 gamma); for gamma = 0.31 pi this gives n_gamma ~= 4.5. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: rounded_input_sensitivity. Source location(s): source/supp_content.tex:119 (Supp. Mat. S3).

Proof summary: Round-2 audit preserves partial but removes the hard mismatch flag. The formula n_gamma=4 csc^2(2 gamma) is applied directly: at the literal gamma=0.31pi it gives 4.627 photons, 2.8% above the printed approx 4.5. The value is highly sensitive to the rounded gamma: gamma=0.305pi gives 4.518, essentially the printed value. Since the branching angle is an experimental/rounded parameter and the paper uses an approximate photon count, this is best treated as rounded-input sensitivity, not a hard mismatch.

Verification report: claim report.

Verdict: partial (mismatch) Location: Supp. Mat. S4 Type / expected artifact: math / math / numeric Claim: The states |down,up> and |up,down> acquire a relative phase phi = 2 epsilon Delta t, so a system initially in |Psi^-> is found in |Psi^+> after one drive-(A) cycle (t = 2 x 2 pi/delta) with probability p = sin^2(phi/2) ~= 0.5. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:125 (Supp. Mat. S4).

Proof summary: Structural relations CONFIRMED: phi = 2 eps Delta t is the correct differential relative phase, and P(|Psi+>) = sin^2(phi/2) is verified symbolically (sympy). NUMERIC mismatch: with eps=0.05, Delta=2pi x 25 kHz, delta=2pi x 14.7 kHz, t=2*(2pi/delta)=136.05 us, phi=2.137 rad and p=sin^2(phi/2)=0.768. Paper states p ~= 0.5; recomputed p ~= 0.77 (abs diff 0.27 vs 0.5). The ~0.5 is a rough order-of-magnitude 'large per-cycle error' statement; the qualitative conclusion (near order-unity error destroying the protocol) holds for either value. -> partial / mismatch.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S5 Type / expected artifact: numeric / numeric Claim: A single MS gate is calibrated with detuning delta = 2 pi x 14.7 kHz; slow ramping gives a total gate time t = 75 us, longer than 2 pi/delta = 68 us. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:133 (Supp. Mat. S5).

Proof summary: Paper: delta=2pi*14.7kHz -> 2pi/delta=68us; ramped gate t=75us>68us. The 2pi/delta value is a pure identity computing to 68.03us, matching the stated 68us to 0.04%. The 75us is a calibrated ramped gate time (paper-reported, not derivable); only the inequality 75>68 is asserted and it holds. The derivable part is exact but the 75us is paper_text_only -> partial.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S5 Type / expected artifact: numeric / numeric / paper_text Claim: Drive (A) repeats the MS pulse twice with a motional phase shift of pi and an additional gap time of 3.2 us between the two segments. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:133 (Supp. Mat. S5).

Proof summary: Paper: drive (A) = MS pulse repeated twice with motional phase shift pi and a 3.2us gap between segments. These are directly reported pulse-sequence parameters, not derivable from physical constants. Consistency checks pass: the 3.2us gap equals (pi/2)/Omega_C=3.205us and drive-A total = 2*75+3.2 = 153.2us. Paper-reported sequence parameter -> partial.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S5 Type / expected artifact: numeric / numeric Claim: Drive (C) passes ~= 18 mA oscillating at 2 pi x 16.5 MHz through tracks 1.1 mm from the ion, creating a 5.4 uT B-field perpendicular to the trap surface and driving |down><->|up> at Rabi frequency Omega_C = 2 pi x 78 kHz (pi-time 6.4 us). Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:135 (Supp. Mat. S5).

Proof summary: Paper: 18mA at 2pi16.5MHz, tracks 1.1mm -> 5.4uT -> magnetic-dipole Omega_C=2pi78kHz (pi-time 6.4us). (a) pi-time=pi/Omega_C=6.41us matches 6.4us (0.2%, exact identity). (b) infinite-wire B=mu0 I/(2 pi r)=3.27uT at 1.1mm (order-of-magnitude consistent with 5.4uT given two tracks/geometry). (c) Omega=g mu_B B/(2 hbar)=2pi*75.6kHz vs paper 78kHz (~3%), requiring the magnetic-dipole matrix element model. pi-time<->Omega_C identity is exact; the field->Rabi step uses a physical model and is ~3% off -> partial.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S5 Type / expected artifact: numeric / numeric / paper_text Claim: Histogram overlap gives a ~= 1e-3 probability of P(up,up) being mislabeled as P(up,down)+P(down,up) upon thresholding, which is corrected in real time. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:137 (Supp. Mat. S5).

Proof summary: Paper: ~1e-3 probability P(uu) mislabeled as P(ud)+P(du) from histogram overlap of 1- vs 2-bright-ion distributions. Reconstructed with Poisson(mu1=62) and Poisson(mu2=124): optimal threshold T=89 gives P(2-bright as 1-bright)=5.9e-4 (and 1->2 = 5.0e-4); midpoint threshold T=93 gives 2.2e-3. Both are order-of-magnitude consistent with the stated ~1e-3. Reconstruction relies on an idealized Poisson model (paper uses the measured histogram incl. background/dark counts) -> partial.

Verification report: claim report.

Verdict: partial (mismatch) Location: Supp. Mat. S5, Eqs. (S7-S8) Type / expected artifact: math / math Claim: The thresholding miscount is corrected via P(up,up) = [(1-p)P(2) - p P(1)]/(1-p-q) and P(up,down)+P(down,up) = [(1-q)P(2) - q P(1)]/(1-p-q), where p (q) is the probability that 1 (2) bright ions are recorded as 2 (1). Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Source location(s): source/supp_content.tex:137-142 (Eqs. (S7-S8)).

Proof summary: Eq.(S7) P(uu)=[(1-p)P(2)-pP(1)]/(1-p-q) verified exactly by inverting the confusion matrix [[1-p,q],[p,1-q]]. Eq.(S8) as printed [(1-q)P(2)-qP(1)]/(1-p-q) does NOT match the inverse; correct form is [(1-q)P(1)-qP(2)]/(1-p-q) (P1/P2 swapped). Paper's S7+S8 violates probability conservation; the correct pair conserves it. Likely a typo in S8 -> partial (mismatch on second equation).

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S5 Type / expected artifact: math / math Claim: Applying a single round of drive (A) to a state prepared in |Psi^+> excites one ion to D5/2 with probability p/2, providing one of the bit-flip-probability estimators. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:157 (Supp. Mat. S5).

Proof summary: p is an error process (ideal U_A gives no single-ion excitation). Counting: |dd> has 2 susceptible |down> ions -> p~=2r; |Psi+> has 1 |down> ion per branch (|up> spectator of A) -> P(one in e|Psi+)=r=p/2 to leading order. Exact in small-error limit; O(r^2) correction (e.g. 1% at r=0.01), consistent with paper's note that p/2 underestimates total excitation. Model-based reimplementation -> partial.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S6 Type / expected artifact: math / math / numeric Claim: Setting the spin-qubit detuning beta = J compensates the AC Stark shift (all drives on resonance) and is numerically verified to optimize R_t. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:198 (Supp. Mat. S6).

Proof summary: beta scan of the Liouvillian gap at the paper-optimal point (Omega_C=1.95J,kappa=2.58J,gamma=0.29pi): coarse and fine scans both peak at beta/J = 1.000 (R_t=0.114J vs 0.060J at beta=0); two other operating points (incl. Omega_C=6J) also peak at beta/J=1.000. Steady state at beta=J is the singlet (F=1.000000). Confirms beta=J compensates the AC Stark shift and optimizes R_t. Capped at partial: rests on the reimplemented continuous model. paper opt beta/J=1; computed argmax beta/J=1.000.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S6, Eqs. (S14-S17) Type / expected artifact: math / math Claim: Identifying |ee> as the excited subspace and applying the effective-operator formalism in the drive-(C) dressed basis |chi_0>,|chi_pm> yields the effective Hamiltonian Eq. (S15) and effective jump operators Eq. (S16) with constants C_0 = sqrt(2) J/(-i kappa), C_pm = J/(-/+ Omega_C - i kappa). Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:200-239 (Eqs. (S14-S17)).

Proof summary: Reiter-Sorensen reduction (heavy analytic). Verified tractable structure: dressed states chi_0,chi_pm (Eq S14) are orthonormal; under drive-(C) generator they have eigenvalues 0 (chi_0) and +/-Omega_C (chi_pm) -> the denominators -i kappa and -/+Omega_C - i kappa. =sqrt2, =1 -> numerators sqrt2 J and J. No-jump rate out of |ee| =2 kappa -> H_NH imaginary part -i kappa. Hence C_0=sqrt2 J/(-i kappa), C_pm=J/(-/+Omega_C - i kappa) (Eq S17) reproduced. Full S15/S16 operator algebra consistent but not re-derived term-by-term; capped at partial.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S6, Eq. (S18); Fig. kappa_Rt Type / expected artifact: math / math / numeric Claim: An empirical convergence rate R_t proportional to (cos^4 gamma + sin^4 gamma) kappa (Omega_C J / (2(kappa^2+Omega_C^2)))^2 + sin^2 gamma cos^2 gamma kappa J^2/(kappa^2+Omega_C^2) is obtained, with proportionality factor 2.4 fit to the numerical Liouvillian gap. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:253-262 (Eq. (S18)).

Proof summary: Empirical Eq S18 x2.4 vs numerical Liouvillian gap at Omega_C=6J. gamma=0.10pi: median(emp/num)=0.994, rms=0.0008J, num peak 0.0417J@4.2J. gamma=0.25pi: median(emp/num)=1.184, rms=0.0086J, num peak 0.0528J@5.6J. Independent best-fit proportionality factor = 2.39 and 2.03, bracketing the paper's stated 2.4. Empirical formula with factor ~2.4 tracks the gap well. Capped at partial (empirical fit + reimplemented model).

Verification report: claim report.

Verdict: partial (mismatch) Location: Supp. Mat. S6; Fig. liouvillian_gap Type / expected artifact: numeric / numeric Claim: The optimal continuous convergence rate is R_t^opt = 6.1e-2 J at Omega_C = 1.95 J, kappa = 2.58 J, gamma = 0.29 pi. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:274 (Supp. Mat. S6).

Proof summary: Numerical Liouvillian-gap optimization over (Omega_C,kappa,gamma), beta=J. Optimal PARAMETERS match the paper: Omega_C^opt=1.95J (paper 1.95J), kappa^opt=2.58J (paper 2.58J), gamma^opt=0.24pi (paper 0.29pi; gap broad in gamma). Optimal RATE: computed gap = 0.122-0.123 J, about 2x the paper's printed R_t^opt=6.1e-2 J. The factor of 2 is a rate-convention/typo issue: the paper's own J=0.58kHz->72Hz conversion (C085) requires R_t/J=0.124, i.e. the FULL gap 0.122 J, not 0.061 J. mismatch on the printed value; parameters and physics verified.

Verification report: claim report.

Verdict: partial (mismatch) Location: Supp. Mat. S6 Type / expected artifact: numeric / numeric Claim: For a collective-excitation frequency J = 0.58 kHz (delta = 20 eta Omega), the optimal continuous convergence rate corresponds to 72 Hz. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:274 (Supp. Mat. S6).

Proof summary: J=0.58kHz: 0.061J=35.4Hz (half of 72); FULL Liouvillian gap 0.122J=70.8Hz ~= paper's 72Hz (ratio 0.98). Angular conventions 0.0612piJ=222Hz and 0.1222piJ=445Hz are off by ~3x/6x. The 72Hz implies R_t/J=72/580=0.124, matching the full numerical gap (0.122-0.123 J, C084), NOT the paper's printed 0.061 J. The paper is internally inconsistent (text says 0.061J->would give 35Hz, but prints 72Hz). The 72Hz figure is reproduced by the full gap with J as an ORDINARY frequency. mismatch w.r.t. the stated 0.061J.

Verification report: claim report.

Verdict: partial Location: Supp. Mat. S6, Fig. Type / expected artifact: plot / figure (kappa_Rt_with_beta.pdf) Claim: Fig. (kappa_Rt): the convergence rate R_t versus repump rate kappa for two branching ratios gamma, comparing the empirical Eq. (S18) (factor 2.4) with the numerical Liouvillian gap at Omega_C = 6 J. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: visual_match_only, paper_text_only_reimplementation. Source location(s): source/supp_content.tex:244-249 (Fig. kappa_Rt).

Proof summary: Reproduced Fig kappa_Rt_with_beta: two panels sin^2(gamma)=0.40,0.99, log-log axes kappa 1e-2..1e4 J at Omega_C=6J, numerical Liouvillian gap (solid) vs empirical Eq S18 x2.4 (dashed). Curves overlap as in the paper; peak R_t ~0.052J (top), ~0.034J (bottom); median(emp/num)=1.10/0.94 over the formula-validity region. Tier-A visual match (visual_score 0.9); no raw data, no digitization -> capped at partial. See plot_comparison.json.

Verification report: claim report.

Plot evidence:

Verdict: partial Location: Supp. Mat. S6, Fig. Type / expected artifact: plot / figure (liouvillian_gap_with_beta.png) Claim: Fig. (liouvillian_gap): the Liouvillian gap as a function of Omega_C and sin^2(gamma) kappa^opt, where kappa^opt is the optimal repump rate from a multivariate minimization over (Omega_C,gamma). Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: visual_match_only. Source location(s): source/supp_content.tex:267-271 (Fig. liouvillian_gap).

Proof summary: Reproduced Fig liouvillian_gap_with_beta: gap vs Omega_C (0..10J) and sin^2(gamma)*kappa^opt (0..1.9J), kappa^opt = argmax_kappa gap. Vertical dark-red lobe near Omega_C~2J reproduced; global max gap=0.123J at Omega_C=1.87J, gamma=0.24pi, kappa^opt=2.59J, matching paper hotspot and the paper colorbar max (~12e-2 J). Note: paper colorbar max ~0.12J = full gap = 2x the text's R_t^opt=0.061J (cf C084/C085). Tier-A visual match (visual_score 0.85); no raw plot data and no digitization -> partial based on visual comparison. See plot_comparison.json.

Verification report: claim report.

Plot evidence: