# Data dictionary

The 11 CSV tables in `research/data/` contain **532 data rows** in total, excluding headers. They are theoretical calculations, numerical verification fixtures, or synthetic sampling summaries generated by `research/scripts/run_research.py`. They contain **no laboratory measurements, human records, or trained-model examples**. No train/validation/test partition is supplied or claimed. A held-out control in the finite-shot example is an experimental decision check, not a machine-learning test split.

## Shared conventions and reproducibility

- Files are comma-separated text with one header row. Counts and indices are integers; probabilities, coherences, errors, and bounds are floating-point values.
- `n` and `bins` count discrete time bins. No bin duration in seconds is assigned by these tables. `pulses` counts ideal instantaneous refocusing pulses.
- Phase magnitudes (`alpha`, `common_alpha`, `slope_optimal_alpha`) are in **radians**. Other reported quantities are dimensionless or integer counts.
- Coherence means the real part of the characteristic-function average `W = E[exp(i sum_j alpha_j y_j xi_j)]`, where binary signs `xi_j` belong to `{-1,+1}` and the control signs `y_j` are `-1`, `0`, or `+1`. Physical coherence lies in `[-1,1]`.
- `degree` is a bound on the degree of **even Walsh moments**, not cumulant order or Ising interaction degree. Central symmetry and degree restrictions apply only where the generating experiment declares them.
- The generator initializes one NumPy random-number stream with seed **2026101602**. The inverse-roundtrip, finite-shot, and unitary tables consume that stream sequentially. Reproducing an isolated block with a newly reset stream need not reproduce its values; run the complete script with the recorded dependencies. The remaining tables are deterministic calculations.
- Floating-point residuals report numerical agreement; they are not, by themselves, formal proofs or experimental accuracy measurements. Modular ranks are computed modulo the prime **1,000,003**; a deficient modular rank alone is not a real-rank upper certificate.

## `research/data/rank_law.csv`

**49 rows.** Deterministic rank fixtures for `n = 2..8`, even `degree <= min(n,6)`, and pulse caps from zero through `min(degree,n-1)-1`. Each row compares canonical and full prefix-control feature matrices with the analytical rank formula.

| Column | Meaning |
|---|---|
| `n` | Number of bins. |
| `degree` | Maximum retained even Walsh-moment degree. |
| `pulses` | Maximum permitted refocusing pulses per setting. |
| `parameters` | Number of nonempty even moment coordinates through `degree`. |
| `canonical_settings` | Number of rows in the canonical acquisition bank; known one-bin data and terminal-bin-pulse settings are omitted. |
| `full_settings` | Number of rows in the full acquisition bank, including those redundant settings. |
| `modular_rank` | Canonical feature-matrix rank modulo 1,000,003. |
| `full_modular_rank` | Full feature-matrix rank modulo the same prime. |
| `theoretical_rank` | Analytical value `sum_{j=1}^{min(degree,pulses+1,n-1)} binom(n-1,j)`. |

## `research/data/setting_counts.csv`

**29 rows.** Deterministic parameter/acquisition counts for `n = 2..30`; counts are not shot totals or hardware runtimes.

| Column | Meaning |
|---|---|
| `n` | Number of bins. |
| `full_symmetric_parameters` | Number of free nonconstant even moments without a degree cutoff: `2^(n-1)-1`. |
| `degree_2` | Number of even moment coordinates of sizes 2 through `min(n,2)`. |
| `degree_4` | Number of even moment coordinates of sizes 2 through `min(n,4)`. |
| `degree_6` | Number of even moment coordinates of sizes 2 through `min(n,6)`. |

## `research/data/inverse_roundtrips.csv`

**180 rows.** Seeded, positive, centrally symmetric finite sign laws with `n = 4..9`, even degree at most 6, and binwise phase magnitudes uniformly drawn from `[0.45,0.85)` radians. Random moment coefficients are scaled to have absolute sum 0.6. Exact finite-law averages are reconstructed by dense and stencil inverses; no shot noise is added.

| Column | Meaning |
|---|---|
| `case` | Zero-based synthetic case identifier, 0..179. |
| `n` | Number of bins. |
| `degree` | Maximum retained even Walsh-moment degree. |
| `settings` | Number of canonical settings with pulse cap `degree-1`. |
| `minimum_probability` | Smallest atom probability among all `2^n` binary sign strings. |
| `dense_error` | Maximum absolute recovered-moment error of the dense inverse. |
| `stencil_error` | Maximum absolute recovered-moment error of the local mixed-difference inverse. |
| `cross_inverse_error` | Maximum absolute difference between moments recovered by the two inverses. |

## `research/data/hidden_control_checks.csv`

**12 rows.** Deterministic direct checks of the `lambda = +1` hidden family for degrees 2, 4, and 6, with `bins = 2*degree-1`, and common amplitudes 0.25, 0.55, `pi/4`, and 1.05 radians. The full control bank has pulse cap `degree-2`. Degrees 2 and 4 are exhaustively checked; degree 6 uses a deterministic subsample.

| Column | Meaning |
|---|---|
| `degree` | Degree parameter of the explicit hidden-family construction. |
| `bins` | Number of bins. |
| `common_alpha` | Phase magnitude applied equally in all bins, in radians. |
| `total_available_controls` | Size of the full bank under the declared pulse cap. |
| `direct_controls_tested` | Number of controls actually evaluated by direct averaging. |
| `maximum_hidden_residual` | Largest absolute difference between computed real coherence and the independent-sign value `cos(common_alpha)^b`, using each control's readout endpoint `b`. |

## `research/data/global_amplitude_sweep.csv`

**121 rows.** Deterministic seven-bin, degree-4 hidden-family sweep over common amplitude 0.1..1.4 radians, inclusive. The revealing control has pulses after bins 1, 3, and 5.

| Column | Meaning |
|---|---|
| `alpha` | Common phase magnitude in radians. |
| `baseline` | Full-duration independent-sign coherence `cos(alpha)^7`; also the seven-bin shallow-control baseline for both hidden laws. Earlier readout endpoints have their own `cos(alpha)^b` baseline. |
| `lambda_plus` | Real coherence of the revealing control for hidden-family strength `lambda = +1`. |
| `lambda_minus` | Real coherence of that control for `lambda = -1`. |
| `revelation_slope` | Positive slope of the revealing response with respect to `lambda`: `sin(alpha)^4*cos(alpha)^3`. Thus `lambda_plus = baseline + revelation_slope` and `lambda_minus = baseline - revelation_slope`. |

## `research/data/capability_exchange.csv`

**16 rows.** Deterministic seven-bin, degree-4 construction. Starts with common amplitude `pi/4`; selected unrefocused adjacent pairs are assigned unequal amplitudes 0.3 and 1.1 radians. Pulses are after bins 1, 3, and 5 as the pulse count increases. Pair coverage is disjoint by construction.

| Column | Meaning |
|---|---|
| `pulses` | Number of applied refocusing pulses, 0..3. |
| `imbalanced_pair_budget` | Nominal budget of locally unequal adjacent pairs, 0..3; capped in actual use by the number of remaining uncovered pairs. |
| `coherence_contrast` | Absolute difference between the revealing response at `lambda = +1` and `lambda = -1`. |
| `predicted_revealed` | Integer indicator: 1 when `pulses + imbalanced_pair_budget >= 3`, otherwise 0. This prediction is for the declared pair-coverage construction. |

## `research/data/staircase_sharp_bounds.csv`

**88 rows.** Deterministic comparisons for `pairs = 1..5` at phase magnitude `pi/4`, on 17 evenly spaced uncertainty values in `[0,1]` plus applicable branch breakpoints. Duplicate uncertainty values are removed. Optimizes target echo coherence subject to `|w_j| <= epsilon` for each staircase calibration. Symmetry-reduced vertices are used in the numerical solve; the underlying pointwise theorem does not require a symmetric law.

| Column | Meaning |
|---|---|
| `pairs` | Pair count `k`; the model has `2*k` bins. |
| `epsilon` | Uniform absolute bound on each ideal calibration mean, dimensionless. |
| `theoretical_lower` | Exact formula `max(1-(2^k-1)*epsilon, -(2^(k-1)-1)*epsilon, -1)`. |
| `lp_lower` | Floating linear-program optimum for the minimum real echo coherence. |
| `constructive_value` | Real echo coherence of the explicitly constructed attaining probability law. |

## `research/data/calibration_deletion.csv`

**15 rows.** Deterministic ablation calculation for pair counts 1..5. Omits one staircase calibration, fixes all remaining ideal calibration means to zero, and minimizes target echo coherence at phase magnitude `pi/4`.

| Column | Meaning |
|---|---|
| `pairs` | Pair count `k`; there are `2*k` bins and initially `k` calibrations. |
| `omitted_prefix_index` | One-based index of the omitted calibration, 1..`k`; its readout endpoint is bin `2*index`. |
| `minimum_echo_coherence` | Floating linear-program minimum real target-echo coherence under the remaining constraints. |

## `research/data/finite_shot_coverage.csv`

**4 rows.** Seeded Monte Carlo summaries, 1,000 trials per shot count, for a six-bin matched-pair sign law with `k=3` and common phase `pi/4`. Each trial samples independent binary Pauli outcomes for three staircase calibrations plus a Ramsey baseline. Hoeffding intervals use nominal simultaneous confidence 95% across those four acquired settings.

| Column | Meaning |
|---|---|
| `shots_per_setting` | Repeated binary outcomes for each acquired setting: 200, 1,000, 5,000, or 20,000. |
| `trials` | Monte Carlo repetitions at that shot count, always 1,000. |
| `simultaneous_coverages` | Trials in which all four true acquired-setting means lie in their simultaneous intervals. |
| `positive_certificates` | Trials with a strictly positive certified target-minus-Ramsey real-coherence gain. |
| `max_bound_minus_true_gain` | Largest, over trials, difference between the certified lower gain bound and the exact true gain. Negative values indicate conservative bounds in those sampled trials; this is not a proof of coverage. |

The individual 50,000-shot held-out decision example is recorded separately in `research/results/finite_shot_decision.json`, not as rows of this table.

## `research/data/visibility_cost.csv`

**6 rows.** Deterministic hidden-family visibility estimates at even degrees 2..12 and `bins = 2*degree-1`. The shot column is a sufficient concentration bound for distinguishing a **known** mean shift; it is not an observed sample requirement, an optimality theorem, or an unknown-baseline two-sample bound.

| Column | Meaning |
|---|---|
| `degree` | Hidden-family even moment degree. |
| `bins` | Number of bins. |
| `quarterturn_slope` | Magnitude of the response slope at `alpha = pi/4`, equal to `2^(-bins/2)`; the column name is literal. |
| `slope_optimal_alpha` | Common phase magnitude maximizing the slope magnitude: `arcsin(sqrt(degree/bins))`, in radians. |
| `slope_optimal_value` | Maximum slope magnitude `(degree/bins)^(degree/2)*(1-degree/bins)^((bins-degree)/2)`. |
| `hoeffding_sufficient_shots_known_mean` | Integer `ceil(2*log(2/0.05)/quarterturn_slope^2)+1`; uses the `pi/4` slope, not the optimized slope. |

## `research/data/unitary_propagation.csv`

**12 rows.** Seeded independent 2-by-2 matrix propagation checks, with 3..6 bins, random finite sign-law probabilities, phase magnitudes uniformly drawn from `[0.2,0.9)` radians, and randomly selected pulse locations. No central-symmetry or moment-degree restriction is imposed on these random laws.

| Column | Meaning |
|---|---|
| `case` | Zero-based synthetic case identifier, 0..11. |
| `bins` | Number of bins. |
| `pulses` | Number of selected refocusing pulses. |
| `unitary_entanglement_fidelity` | Probability-weighted entanglement fidelity from explicit unitary propagation, relative to the known ideal refocusing operation. |
| `characteristic_entanglement_fidelity` | The same dimensionless fidelity computed as `(1 + Re(W))/2`. |
| `absolute_residual` | Absolute difference between those two fidelity calculations. |

## Related files

The `research/data/` directory also contains two NumPy matrix archives and a native-code fixture text file; these are verification inputs rather than additional CSV datasets. The claim ledger under `research/protocols/` is documentation and is excluded from the 11-table and 532-row totals above. Consult the manuscript and generating source for the full mathematical assumptions and derivations.