Appendix B – Reproducibility

Complete documentation of the computational program behind this paper: software environment, seed conventions, sample sizes per data point, and the file manifest. All numerical results in the main text are reproducible from the artifacts documented here.

B.1 Software environment

All computations use the following package versions, pinned on a single Linux machine (Ubuntu 24.04 LTS, Python 3.12.3):

Every pip install used --break-system-packages as required by Ubuntu 24 system Python; all results are reproducible under any equivalent install of the above versions.

B.2 Seed conventions

All Monte Carlo draws route through either numpy.random.default_rng(seed) or the project’s SeededRNG wrapper (which threads a single default_rng through both numpy and scipy.stats.unitary_group for a unified stream):

class SeededRNG:
    def __init__(self, seed: int = 0xBADC0DE):
        self.rng = np.random.default_rng(int(seed))
    def haar_unitary(self, n):
        return unitary_group.rvs(n, random_state=self.rng)
    def haar_state(self, d):
        # complex Gaussian then normalize
        ...
    def aehpv_map(self, d_eff, d_fund):
        # first d_fund rows of haar_unitary(d_eff)
        ...

Each phase uses a distinct BASE_SEED constant, with per-point seeds derived deterministically from BASE_SEED and the relevant dimension parameters so that every data point’s sample stream is fully determined by its nominal coordinates:

The seed-to-data correspondence is bit-identical across reruns under numpy >= 2.0: the same seed yields the same state vector, unitary, and statistics down to the last bit.

B.3 Sample sizes per data point

Note on file names in §§B.3–B.4. The phase-prefixed scripts and CSVs named below (e.g. phase5_boost_point.py, phase5_extended_scan.csv, phase5_money_plot.py) are historical development artifacts documenting how the data were produced; they are not part of the shipped reproducibility/ bundle, which contains the consolidated regenerators and data listed in §B.5. They are referenced here only to record the original sampling and wall-time methodology.

Phase 5 extended scan (Haar bulk)

Nine points with cumulative sample counts from merged batches:

Batch-merge metadata preserved in phase5_extended_scan.csv. Batch offsets use independent seed streams (BASE_SEED + d_B*10000 + batch_index) to guarantee statistical independence across batches.

Phase 6 state-class scan

Haar and Product classes, six primary points plus Product extension:

Phase 7 theorem verification

Per-figure sample sizes:

Figure 3 (Theorem 1, Product):

• Panel (a) Dirichlet variance: $N \in \{10^5, 10^5, 10^5, 10^5, 5 \cdot 10^4, 5 \cdot 10^4, 2.5 \cdot 10^4, 1.5 \cdot 10^4, 10^4\}$ for $d \in \{4, 8, 16, 32, 64, 128, 256, 512, 1024\}$.
• Panel (b) prefactor: $N \in \{10^5, 5 \cdot 10^4, 5 \cdot 10^4, 3 \cdot 10^4, 2.5 \cdot 10^4, 1.5 \cdot 10^4, 1.5 \cdot 10^4, 8 \cdot 10^3\}$.
• Panel (c) Gaussian ratio: same as (b).
• Panel (d) Phase-6 theory: $N = 30\,000$ per theory point, paired with Phase 6 data.

Figure 4 (Theorem 2, Haar):

• Panels (a, b) large-$d$ scan: $N \in \{3 \cdot 10^4, 1.5 \cdot 10^4, 10^4, 6 \cdot 10^3, 4 \cdot 10^3, 2 \cdot 10^3\}$ for $d \in \{16, 24, 32, 48, 64, 96\}$.
• Panel (c) Phase-5 comparison: matched to Phase 5 $N$ values in §B.3 above.
• Panel (d) out-of-sample: $d = 128$ at $N = 3000$; $d_B = 18$ at $N = 60$ (full HUZ + $V$ pipeline).

Structural identity verification (Figure 2)

Haar-$V$ averages at $d_B \in \{4, 6, 8\}$ with $N_V \in \{500, 300, 200\}$ Haar-$V$ draws per point, two bulk state classes. 18 diagonal entries compared per class (6 at $d_B = 4$, 6 at $d_B = 6$, 6 at $d_B = 8$).

B.4 Per-batch wall-time budget

The execution environment imposed a wall-time ceiling of approximately 5 minutes per bash invocation. This ruled out single-shot runs at large $d_B$ (where per-sample cost scales as $d_{\rm eff}^3 = (d_B^2 d_M)^3$ for the full two-observer HUZ + $V$ simulation). Three workarounds enabled the large-$d_B$ scan:

1. Checkpoint-every-N-samples (phase5_boost_point.py): pickle partial results to phase5_cache/point_dB{N}.pkl every 25–50 samples so a timeout at sample $k$ loses at most 50 samples, not the full run.
2. Independent batch seeds: each invocation uses BASE_SEED + d_B*10000 + batch_index so running the same command twice extends the sample count rather than duplicating samples.
3. Efficient two-observer entropy computation (two_observer_entropies_fast in phase5_money_plot.py): replaces the naïve formation of the full joint density matrix (which would require $\sim 270$ GB at $d_B = 16$) with direct einsum contractions on the state tensor, reducing memory to $\sim 2$ MB at $d_B = 16$. Bit-identical to the naïve computation on five cross-checked test cases.

All Phase 5, 6, and 7 data was collected under these three mechanisms. A single fresh machine with no wall-time limits could reproduce the full scan in approximately 3–4 CPU-days on a single core.

B.5 File manifest (bundled reproducibility/ directory)

The accompanying package ships a consolidated, self-contained artifact set sufficient to regenerate every figure and Table 1 from the documented seeds. It does not ship the full phase-by-phase development tree; the files below are exactly what is included.

Backend

• bh_lab_backend.py – SeededRNG, aehpv_map, and the two-observer HUZ reduced-state machinery used by all drivers.

Figure generators (canonical seed 707070)

• make_fig2.py – Figure 2 (structural identity, Lemma 1): diagonal agreement and off-diagonal suppression.
• make_figs_1345.py – Figures 3 and 5 (product Dirichlet model; the Table 1 log-log landscape and exponent-gap ratio). Loads Table 1 from table1_full_scan.csv.
• make_figs_41.py – Figures 1 and 4 (finite-$d_B$ vs asymptotic; the Haar prefactor test, subleading structure, HUZ+$V$ residuals, and out-of-sample $d=128$). Loads Table 1 from table1_full_scan.csv and the prefactor scan from fig4_haar_prefactor.csv.

Proof-verification scripts

• scratch_fourth_moment.py – the fourth-moment projector estimate (Lemma C.5) and assembled off-diagonal bound.
• scratch_Fdiag.py – direct Monte-Carlo verification of $F_{\rm diag}$ and of the diagonal-to-bulk error $F_{\rm diag}$ (Lemma C.6).
• scratch_grouped_dirichlet.py – verification of the grouped-Dirichlet moments of Lemma 3 (Var$(T_A)$, Cov$(T_A,T_B)$, Var$(T_A-T_B)$, and the negative off-diagonal / vanishing $p$–$q$ covariances).
• scratch_centered_C6.py – confirms the centered-operator representation of Lemma C.6 ($L_A-L_B = -(d/\rho)\mathrm{Tr}[(P-\rho I)(G_A-G_B)]$, correlation $>0.999$ with the direct linear error) and the resulting $O(d^{-4}d_M^{-2})$ scaling.
• scratch_Mdom.py – diagnostic only (not used by any proof): inspects the conditional diagonal-covariance matrix; retained for completeness.

Data (one row per figure/table point)

• table1_full_scan.csv – the single source of truth for Table 1 (Haar + product, measured / sem / theory / $z$); consumed by both figure generators.
• fig1_full_V.csv, fig1_no_V.csv – Figure 1 curves.
• fig2_struct_identity.csv, fig2_offdiag.csv – Figure 2 panels.
• fig3_dirichlet_var.csv, fig3_prefactor.csv, fig3_gaussian_ratio.csv, fig3_panel_d.csv – Figure 3 panels.
• fig4_haar_prefactor.csv, fig4_full_V.csv, fig4_out_of_sample.csv – Figure 4 panels.
• fig5_class_ratio.csv – Figure 5 ratio panel.
• entropy_replacement.csv – $F_{AB}$ end-to-end check (§C.7).
• phase2_dM_scan.csv, phase2_exponents.csv – $d_M$-dependence and fitted exponents.

B.6 Reproducibility verification

Two classes of checks were run. The first concerns the artifacts shipped in reproducibility/: each figure and Table 1 regenerates from the bundled CSVs and generators under the documented environment and seed, and the proof-verification scripts reproduce the quoted numbers (e.g. scratch_grouped_dirichlet.py reproduces the Lemma 3 moments to $<1\%$, scratch_centered_C6.py the Lemma C.6 representation and scaling, and scratch_Fdiag.py the $F_{\rm diag}$ ratios; scratch_Mdom.py is a diagnostic and supports no proof step).

The second class is a record of the broader development program (the historical phase scripts, which are not part of the shipped bundle): all non-trivial data points were spot-checked there for bit-identical reproducibility – integer rank predictions; error-scaling and EGH verifications to $\le 10^{-12}$; the extended Haar scan within a batch (merged-batch means reproducible to $\le 10^{-10}$); and the Dirichlet-variance check (Lemma 4) to $\le 10^{-13}$. The one caveat there: merged-batch means are reproducible in aggregate but not strictly bit-identical across batch-execution orderings, as summation order shifts the accumulated mean by $\le 10^{-14}$.

B.7 Data and code availability

The reproducibility artifacts are provided as supplementary material with this submission, in the reproducibility/ directory of the accompanying package: the unified backend bh_lab_backend.py (containing SeededRNG, aehpv_map, and the two-observer HUZ machinery), the figure-generation scripts, the verification scripts referenced in §B.5, and the CSV data files underlying every figure and table. A public archival deposit (Zenodo or equivalent) with a DOI is planned at time of journal submission; until then the bundled reproducibility/ directory is the authoritative artifact set. The phase-by-phase script names in §B.5 reflect the development history; the consolidated, self-contained scripts in reproducibility/ are sufficient to regenerate all figures and the Table 1 scan from the documented seeds.