# Data dictionary

## Core notation

- `N` is the total number of tokens.
- `k` is the OEIS A048833 index. For even total size, `N = 2k`.
- `a_N` is the number of partitions of `N` whose parts have bitwise XOR zero.
- `p_N` is the ordinary partition number `p(N)`.
- `t_N` is the exact unrestricted-partition saddle.
- `L_N_star = log(1/t_N)/t_N`.
- `J_N_star` is the nearest power of two to `L_N_star` in multiplicative distance.
- `Lambda_N_star = exp(-t_N J_N_star)/t_N`.
- `B_N` is the explicit scale based on `tau_N = pi/sqrt(6N)`.

## Count files

### `nim_losing_counts_N0_N20000.tsv`

| Column | Meaning |
|---|---|
| `total_size_N` | Total size `N` |
| `oeis_index_k` | `N/2` for even `N`, blank for odd `N` |
| `parity` | `even` or `odd` |
| `losing_count_a_N` | Exact count `a_N` |
| `partition_number_p_N` | Exact partition number `p(N)` |
| `density_a_N_over_p_N` | Exact density, written to 24 decimal digits |

### `oeis_A048833_k0_k10000.txt`

Compact two-column sequence file. Comment lines begin with `#`. Data rows have the form `k A048833(k)`, with `A048833(k) = a_{2k}`.

### `oeis_A048833_k0_k10000.tsv`

The same exact sequence indexed by `k`, together with `N = 2k`, `p(2k)`, and the exact density.

### `oeis_A048833_public_k0_k750.txt`

Frozen snapshot of the preceding public OEIS b-file through index `750`. It is used only to verify that the released extension preserves all previously public terms.

### `exact_count_verification.txt`

Summary of the count range, CRT uniqueness bound, independent-prime check recorded for the original run, public b-file comparison, and endpoint value.

### `knapsack_check_N1500.txt`

Output of the algorithmically independent check at `N = 1500`. The check computes distinct-part XOR counts by a direct knapsack modulo four primes, reconstructs them by CRT, and applies the odd-core convolution.

## Table files

### `table_01_first_order_milestones.tsv`

The exact density and first-order formula at the six values printed in Table 1. It includes both exact-saddle quantities and the explicit scale `B_N`.

### `table_02_error_decomposition.tsv`

- `first_order_density` is the one-block first-order prediction.
- `exact_boltzmann_fixed_size_2R` is `2R(exp(-t_N))`.
- `two_level_fixed_size_approximation` is the two-level approximation after multiplying by two.
- The error columns are signed percentages.

### `table_03_two_level_error_budget.tsv`

- `observed_two_level_absolute_relative_error_percent` is the absolute relative error of the two-level Boltzmann formula.
- `low_block_scale_beta_percent` is `100 beta_J(t_N)`.
- `upper_tail_scale_percent` is `100 exp(-4 t_N J_N_star)/t_N`.

### `table_04_second_order_fixed_size.tsv`

- `curvature_term_percent` is the term involving `F''`.
- `skew_term_percent` is the term involving the third cumulant and `F'`.
- `second_order_prediction_percent` is their sum.

### `table_05_major_arc_validation.tsv`

Raw 5-sigma and 10-sigma quadrature values and their signed relative errors. Values printed as below `1e-13` in the paper are retained numerically in this file.

## Figure data

### `fig_03_04_hierarchy_and_error_sources_N2000_N20000.tsv`

The grid is `N = 2000, 2050, ..., 20000`.

- `one_level_absolute_relative_error_percent` supports Figure 3.
- `two_level_absolute_relative_error_percent` supports Figures 3 and 4.
- `low_block_scale_beta_percent` and `upper_tail_scale_percent` support Figure 4.

### `fig_05_fixed_size_correction_N2000_N20000.tsv`

The same grid is used. The file records the observed fixed-size error and the formal second-order curvature and skew terms.

### `fig_06_exact_density_N100_N20000.tsv`

The grid contains every even `N` from `100` through `20000`. It records the exact normalized density, the first-order prediction, and the limiting dyadic phase.
