# The `(m,2,d)` gap: first pass at the one-dimensional case

Reference: Glenn Bruda, Joseph Cooper, Kareem Jaber, Raul Marquez, and Steven J. Miller, **“Variants of Conway Checkers and k-nacci Jumping,”** arXiv:2408.08856v3, December 2025.  I use their notation and cite Theorem 1.1/equation (1.2), §4.2, §4.3, and §4.4 below.

## Question checked

The tempting narrow question was to characterize the set of `m >= 2` for which the one-dimensional Conway `(m,2,1)` game attains the upper side of the gap in Bruda--Cooper--Jaber--Marquez--Miller, Theorem 1.1/equation (1.2):

```text
floor(log_phi m) + 3d - 2 <= n_M <= floor(log_phi m) + 3d - 1.
```

For `d = 1` this displayed corollary reads

```text
floor(log_phi m) + 1 <= n_M <= floor(log_phi m) + 2.
```

The useful correction is that the paper's one-dimensional section is already sharper than the displayed `d`-dimensional corollary.  In §4.2, for `k=2,d=1,m>1`, they prove exactly

```text
n_M(m,2,1) = floor(log_phi(m) + 2) = floor(log_phi(m)) + 2.
```

In §4.3, Theorem 4.3 generalizes this to `k >= 2`:

```text
n_M(m,k,1) = floor(log_{phi_k}(m) - log_{phi_k}(phi_k - 1) + 1),  m > 1.
```

So the one-dimensional characterization is closed, but not newly open:

```text
S_1 = { m in N : m >= 2 }
L_1 = emptyset
```

where `S_1` means “hits `floor(log_phi m)+2`, the upper side of the `d=1` gap-of-1 corollary.”  If instead `S_1` is defined as “hits `floor(log_phi m)+1`,” then the naming is inverted relative to equation (1.2), and that set is empty for `m >= 2`.

## Why this settles `d=1`

### Upper bound

For a target on row `n` in the one-dimensional board, set `alpha = 1/phi`.  Bruda et al. §4.2 use the standard pagoda function; the initial energy of the infinite half-line is

```text
E_0 = m alpha^n / (1 - alpha) = m phi^{2-n}.
```

A checker on the target has energy `1`, so reachability requires `E_0 >= 1`, hence

```text
n <= log_phi(m) + 2.
```

Since `m` is an integer and `phi` is irrational, equality `m phi^{2-n} = 1` cannot occur for `m > 1`; taking floors gives the upper bound `n_M <= floor(log_phi(m)+2)`.

### Lower bound / construction

Section 4.2 then supplies the matching Fibonacci-jumping construction.  To reach row `n`, it aims to build `F(n+1)` checkers in row `0` and `F(n)` in row `-1`.  The number of imported checkers needed at row `-i` is equation (4.5):

```text
D_i(n,m) = F(n+i+1) - (F(i+3) - 1)m.
```

If two consecutive demands are non-positive, the finite construction can start there and work upward.  As `i -> infinity`,

```text
D_i(n,m) ~ c phi^{i+1} (phi^n - m phi/(phi-1)).
```

The right side is eventually negative exactly when `n < log_phi(m) + 2`; because the equality case cannot happen for integer `m > 1`, the construction reaches every `n <= floor(log_phi(m)+2)`.  This matches the pagoda upper bound.

This is the key obstruction to a fractional-part characterization in `d=1`: there is no nontrivial boundary interval in `{log_phi m}`.  Every fractional part occurring from an integer `m >= 2` is on the attainable side.

## Computational reproduction

Artifacts:

- Script: [`research/scripts/mkd_1d_table.py`](scripts/mkd_1d_table.py)
- Script README: [`research/scripts/README.md`](scripts/README.md)
- CSV table: [`research/data/m-k-d-table.csv`](data/m-k-d-table.csv)

Run:

```sh
python3 research/scripts/mkd_1d_table.py --min 2 --max 100 --out research/data/m-k-d-table.csv
```

Output from this run:

```text
wrote 99 rows to research/data/m-k-d-table.csv
floor(log_phi m)+2 hits: 99; floor(log_phi m)+1 hits: 0
```

The script is deliberately not a blind search over all legal one-dimensional moves.  The paper proves in §4.2 that upward moves suffice in `Z`, then gives the exact row-demand recurrence above.  The script therefore records the exact pagoda upper bound and verifies the constructive side by finding a finite depth where the recurrence has two consecutive non-positive demands.

Columns:

- `m`: initial checkers per cell.
- `floor_log_phi_m`: `floor(log_phi m)`.
- `frac_log_phi_m`: fractional part of `log_phi m`.
- `n_M_d1_k2`: exact value from §4.2.
- `pagoda_upper`: the matching pagoda upper bound.
- `construction_depth`: first depth where the finite Fibonacci-jumping construction has a two-row non-positive base.
- `eq_floor_log_plus_1`: whether `n_M = floor(log_phi m)+1`.
- `eq_floor_log_plus_2`: whether `n_M = floor(log_phi m)+2`.

| m | floor log | frac log | n_M | construction depth | hits upper |
|---:|---:|---:|---:|---:|---:|
| 2 | 1 | 0.440420090413 | 3 | 2 | 1 |
| 3 | 2 | 0.283011828589 | 4 | 3 | 1 |
| 4 | 2 | 0.880840180825 | 4 | 1 | 1 |
| 5 | 3 | 0.344551876369 | 5 | 3 | 1 |
| 6 | 3 | 0.723431919002 | 5 | 2 | 1 |
| 7 | 4 | 0.043770430650 | 6 | 7 | 1 |
| 8 | 4 | 0.321260271238 | 6 | 3 | 1 |
| 9 | 4 | 0.566023657179 | 6 | 2 | 1 |
| 10 | 4 | 0.784971966782 | 6 | 2 | 1 |
| 11 | 4 | 0.983034804894 | 6 | 1 | 1 |
| 12 | 5 | 0.163852009414 | 7 | 4 | 1 |
| 13 | 5 | 0.330187713371 | 7 | 3 | 1 |
| 14 | 5 | 0.484190521063 | 7 | 2 | 1 |
| 15 | 5 | 0.627563704958 | 7 | 2 | 1 |
| 16 | 5 | 0.761680361650 | 7 | 2 | 1 |
| 17 | 5 | 0.887663595352 | 7 | 1 | 1 |
| 18 | 6 | 0.006443747591 | 8 | 11 | 1 |
| 19 | 6 | 0.118800132980 | 8 | 5 | 1 |
| 20 | 6 | 0.225392057194 | 8 | 4 | 1 |
| 21 | 6 | 0.326782259240 | 8 | 3 | 1 |
| 22 | 6 | 0.423454895306 | 8 | 3 | 1 |
| 23 | 6 | 0.515829521730 | 8 | 2 | 1 |
| 24 | 6 | 0.604272099827 | 8 | 2 | 1 |
| 25 | 6 | 0.689103752738 | 8 | 2 | 1 |
| 26 | 6 | 0.770607803784 | 8 | 2 | 1 |
| 27 | 6 | 0.849035485768 | 8 | 2 | 1 |
| 28 | 6 | 0.924610611475 | 8 | 1 | 1 |
| 29 | 6 | 0.997533424224 | 8 | 1 | 1 |
| 30 | 7 | 0.067983795371 | 9 | 6 | 1 |
| 31 | 7 | 0.136123897329 | 9 | 5 | 1 |
| 32 | 7 | 0.202100452063 | 9 | 4 | 1 |
| 33 | 7 | 0.266046633483 | 9 | 4 | 1 |
| 34 | 7 | 0.328083685764 | 9 | 3 | 1 |
| 35 | 7 | 0.388322307019 | 9 | 3 | 1 |
| 36 | 7 | 0.446863838004 | 9 | 2 | 1 |
| 37 | 7 | 0.503801287919 | 9 | 2 | 1 |
| 38 | 7 | 0.559220223393 | 9 | 2 | 1 |
| 39 | 7 | 0.613199541960 | 9 | 2 | 1 |
| 40 | 7 | 0.665812147607 | 9 | 2 | 1 |
| 41 | 7 | 0.717125542882 | 9 | 2 | 1 |
| 42 | 7 | 0.767202349652 | 9 | 2 | 1 |
| 43 | 7 | 0.816100768570 | 9 | 2 | 1 |
| 44 | 7 | 0.863874985719 | 9 | 2 | 1 |
| 45 | 7 | 0.910575533548 | 9 | 1 | 1 |
| 46 | 7 | 0.956249612143 | 9 | 1 | 1 |
| 47 | 8 | 0.000941375938 | 10 | 15 | 1 |
| 48 | 8 | 0.044692190240 | 10 | 7 | 1 |
| 49 | 8 | 0.087540861301 | 10 | 6 | 1 |
| 50 | 8 | 0.129523843151 | 10 | 5 | 1 |
| 51 | 8 | 0.170675423941 | 10 | 4 | 1 |
| 52 | 8 | 0.211027894196 | 10 | 4 | 1 |
| 53 | 8 | 0.250611699038 | 10 | 4 | 1 |
| 54 | 8 | 0.289455576180 | 10 | 3 | 1 |
| 55 | 8 | 0.327586681263 | 10 | 3 | 1 |
| 56 | 8 | 0.365030701888 | 10 | 3 | 1 |
| 57 | 8 | 0.401811961570 | 10 | 3 | 1 |
| 58 | 8 | 0.437953514637 | 10 | 3 | 1 |
| 59 | 8 | 0.473477233027 | 10 | 2 | 1 |
| 60 | 8 | 0.508403885784 | 10 | 2 | 1 |
| 61 | 8 | 0.542753211985 | 10 | 2 | 1 |
| 62 | 8 | 0.576543987742 | 10 | 2 | 1 |
| 63 | 8 | 0.609794087829 | 10 | 2 | 1 |
| 64 | 8 | 0.642520542475 | 10 | 2 | 1 |
| 65 | 8 | 0.674739589740 | 10 | 2 | 1 |
| 66 | 8 | 0.706466723895 | 10 | 2 | 1 |
| 67 | 8 | 0.737716740170 | 10 | 2 | 1 |
| 68 | 8 | 0.768503776177 | 10 | 2 | 1 |
| 69 | 8 | 0.798841350320 | 10 | 2 | 1 |
| 70 | 8 | 0.828742397432 | 10 | 2 | 1 |
| 71 | 8 | 0.858219301891 | 10 | 2 | 1 |
| 72 | 8 | 0.887283928416 | 10 | 1 | 1 |
| 73 | 8 | 0.915947650740 | 10 | 1 | 1 |
| 74 | 8 | 0.944221378332 | 10 | 1 | 1 |
| 75 | 8 | 0.972115581327 | 10 | 1 | 1 |
| 76 | 8 | 0.999640313806 | 10 | 1 | 1 |
| 77 | 9 | 0.026805235544 | 11 | 8 | 1 |
| 78 | 9 | 0.053619632373 | 11 | 7 | 1 |
| 79 | 9 | 0.080092435230 | 11 | 6 | 1 |
| 80 | 9 | 0.106232238019 | 11 | 5 | 1 |
| 81 | 9 | 0.132047314357 | 11 | 5 | 1 |
| 82 | 9 | 0.157545633295 | 11 | 5 | 1 |
| 83 | 9 | 0.182734874084 | 11 | 4 | 1 |
| 84 | 9 | 0.207622440065 | 11 | 4 | 1 |
| 85 | 9 | 0.232215471721 | 11 | 4 | 1 |
| 86 | 9 | 0.256520858983 | 11 | 4 | 1 |
| 87 | 9 | 0.280545252813 | 11 | 4 | 1 |
| 88 | 9 | 0.304295076131 | 11 | 3 | 1 |
| 89 | 9 | 0.327776534120 | 11 | 3 | 1 |
| 90 | 9 | 0.350995623960 | 11 | 3 | 1 |
| 91 | 9 | 0.373958144021 | 11 | 3 | 1 |
| 92 | 9 | 0.396669702556 | 11 | 3 | 1 |
| 93 | 9 | 0.419135725918 | 11 | 3 | 1 |
| 94 | 9 | 0.441361466351 | 11 | 3 | 1 |
| 95 | 9 | 0.463352009350 | 11 | 2 | 1 |
| 96 | 9 | 0.485112280652 | 11 | 2 | 1 |
| 97 | 9 | 0.506647052858 | 11 | 2 | 1 |
| 98 | 9 | 0.527960951713 | 11 | 2 | 1 |
| 99 | 9 | 0.549058462072 | 11 | 2 | 1 |
| 100 | 9 | 0.569943933563 | 11 | 2 | 1 |

## Pattern hypothesis after the table

There is no nontrivial `d=1` pattern to find: the upper side is attained for all `m >= 2`.  In fractional-part language,

```text
m hits the d=1 upper side iff {log_phi m} in [0,1).
```

The more interesting unresolved characterization remains the higher-dimensional projection step behind Theorem 1.1/equation (1.2).  Section 4.4 proves an exact one-dimensional row-1 capacity,

```text
floor(m/(phi_k - 1))
```

checkers, and §4.5 uses repeated projections into lower-dimensional games.  The gap-of-1 in the full `(m,2,d)` theorem appears when this one-dimensional exact machinery is composed across dimensions and then rounded.  A next useful target is therefore not `d=1`, but `d=2`: compute exact small `m` values for the projected construction versus the pagoda upper bound and isolate which rounding step loses the row.

## Outcome shape

This attempt lands as a **documented correction/obstruction** rather than a new open-result proof:

1. The proposed `d=1` subproblem is already exactly solved in the reference, §4.2--§4.3.
2. The correct one-dimensional upper-attainment set is all integers `m >= 2`.
3. The generated table for `m = 2..100` reproduces that fact and gives finite construction depths.
4. The genuine open work should move to `d >= 2`, where the paper's “almost all `m`” statement is not accompanied by an explicit classification.