In M-games, the elements of Sudoku carry weight $v$, and elements sharing a common boundary or vertex with exterior regions "influence" these regions $v/2$.

Infinite spaces $ℤ$ may be “split” by arbitrarily assigning any cell coordinate 0; the $(c < 0 , 0 < c)$ disjoints will have the same cardinality:

$c < 0 > c$

In order to maintain the common boundaries, make the common boundaries 0.

$> 0$ is $[1,2,…]$ and $[…, -2, -1]$ is $< 0$

$c > 0 < c$ is the boundary

$[c > 0, .., c < 0]$

$[c_{-i} > 0, .., c_{-i} < 0] , [c_{i} > 0, .., c_{i} < 0]$

Now the sets expand infinitely inward—the center of regions can never be reached but connections are maintained.

(i don't know if this breaks anything. naively, i feel like this is "twisting" the paradox and stowing it in the center to solve the problem of the common boundaries. i’m more of an engineer so I can’t help thinking of infinity procedurally—how high i can count in how much time i have. i see coordinates as scalars because the elements have weight, and here there are 2n vectors for any n dimensions. Each pair of opposite vectors points towards the other but can never intersect. My thought is they probably spiral toward infinity so I’m calling this spiral sudoku: "At the still point of the turning world, neither flesh nor fleshless; neither from nor towards; at the still point, there the dance is.";)

Spiral Sudoku can be expressed procedurally with an additional constraint that element expression $p_x$ must start in the greatest $c < 0$ or least $c > 0$, with the least unexpressed element $p > 0$ for any region, and all subsequent $p_x$ must share a cell boundary with a previous $p_x$

The toroidal surface is attractive because there is no center—every region and vertex is the same until the initial $p_x$

Where n is number of dimensions: $p_x \cap n$

Wherever that first $p_x$ is expressed becomes the center vertex of the sudoku.

This becomes a stacking game around a single vertex, or, because spaces can be virtual, players may “jump” across “hyperspace” to the closest uncolonized vertices to express new $p_x$

If new p must only share a vertex with $p_x$ sparser aggregations may be formed with empty spaces between $p_x$. Element weights $v$ may be balanced where $p_x$ can also be the greatest unexpressed $p < 0$.

In M-games, the elements of Sudoku carry weight $v$, and elements sharing a common boundary or vertex with exterior regions "influence" these regions $v/2$.

Infinite spaces $ℤ$ may be “split” by arbitrarily assigning any cell coordinate 0; the $(c < 0 , 0 < c)$ disjoints will have the same cardinality:

$c < 0 > c$

In order to maintain the common boundaries, make the common boundaries 0.

$> 0$ is $[1,2,…]$ and $[…, -2, -1]$ is $< 0$

$c > 0 < c$ is the boundary

$[c > 0, .., c < 0]$

$[c_{-i} > 0, .., c_{-i} < 0] , [c_{i} > 0, .., c_{i} < 0]$

Now the sets expand infinitely inward—the center of regions can never be reached but connections are maintained.

(i don't know if this breaks anything. naively, i feel like this is "twisting" the paradox and stowing it in the center to solve the problem of the common boundaries. i’m more of an engineer so I can’t help thinking of infinity procedurally—how high i can count in how much time i have. i see coordinates as scalars because the elements have weight, and here there are 2n vectors for any n dimensions. Each pair of opposite vectors points towards the other but can never intersect. My thought is they probably spiral toward infinity so I’m calling this spiral sudoku: "At the still point of the turning world, neither flesh nor fleshless; neither from nor towards; at the still point, there the dance is.";)

Spiral Sudoku can be expressed procedurally with an additional constraint that element expression $p_x$ must start in the greatest $c < 0$ or least $c > 0$, with the least unexpressed element $p > 0$ for any region, and all subsequent $p_x$ must share a cell boundary with a previous $p_x$

The toroidal surface is attractive because it reinforces the idea of "no center"—every region and vertex is the same until the initial $p_x$, and the regions themselves have to expressible center.

Where n is number of dimensions: $p_x \cap n$

Wherever that first $p_x$ is expressed becomes the center vertex of the sudoku.

This becomes a stacking game around a single vertex, or, because spaces can be virtual, players may “jump” across “hyperspace” to the closest uncolonized vertices to express new $p_x$

If new p must only share a vertex with $p_x$ sparser aggregations may be formed with empty spaces between $p_x$. Element weights $v$ may be balanced where $p_x$ can also be the greatest unexpressed $p < 0$.

In M-games, the elements of Sudoku carry weight $v$, and elements sharing a common boundary or vertex with exterior regions "influence" these regions $v/2$.

Infinite spaces $ℤ$ may be “split” by arbitrarily assigning any cell coordinate 0; the $(c < 0 , 0 < c)$ disjoints will have the same cardinality:

$c < 0 > c$

In order to maintain the common boundaries, make the common boundaries 0.

$> 0$ is $[1,2,…]$ and $[-2, -1]$ is $< 0$

$c > 0 < c$ is the boundary

$[c > 0, .., c < 0]$

$[c_{-i} > 0, .., c_{-i} < 0] , [c_{i} > 0, .., c_{i} < 0]$

Now the sets expand infinitely inward—the center of regions can never be reached but connections are maintained.

(i don't know if this breaks anything. naively, i feel like this is "twisting" the paradox and stowing it in the center to solve the problem of the common boundaries. i’m more of an engineer so I can’t help thinking of infinity procedurally—how high i can count in how much time i have. i see coordinates as scalars because the elements have weight, and here there are 2n vectors for any n dimensions. Each pair of opposite vectors points towards the other but can never intersect. My thought is they probably spiral toward infinity so I’m calling this spiral sudoku: "At the still point of the turning world, neither flesh nor fleshless; neither from nor towards; at the still point, there the dance is.";)

Spiral Sudoku can be expressed procedurally with an additional constraint that element expression $p_x$ must start in the greatest $c < 0$ or least $c > 0$, with the least element $p > 0$ for any region, and all subsequent $p_x$ must share a boundary with a previous $p_x$

The toroidal surface is attractive because there is no center—every region and vertex is the same until the initial $p_x$

Where n is number of dimensions: $p_x \cap n$

Wherever that first $p_x$ is expressed becomes the center vertex of the sudoku.

This becomes a stacking game around a single vertex, or, because spaces can be virtual, players may “jump” across “hyperspace” to the closest uncolonized vertices to express new $p_x$

If new p must only share a vertex with $p_x$ sparser aggregations may be formed with empty spaces between $p_x$. Element weights v may be balanced where $p_x$ can be the greatest $p < 0$.

In M-games, the elements of Sudoku carry weight $v$, and elements sharing a common boundary or vertex with exterior regions "influence" these regions $v/2$.

Infinite spaces $ℤ$ may be “split” by arbitrarily assigning any cell coordinate 0; the $(c < 0 , 0 < c)$ disjoints will have the same cardinality:

$c < 0 > c$

In order to maintain the common boundaries, make the common boundaries 0.

$> 0$ is $[1,2,…]$ and $[…, -2, -1]$ is $< 0$

$c > 0 < c$ is the boundary

$[c > 0, .., c < 0]$

$[c_{-i} > 0, .., c_{-i} < 0] , [c_{i} > 0, .., c_{i} < 0]$

Now the sets expand infinitely inward—the center of regions can never be reached but connections are maintained.

(i don't know if this breaks anything. naively, i feel like this is "twisting" the paradox and stowing it in the center to solve the problem of the common boundaries. i’m more of an engineer so I can’t help thinking of infinity procedurally—how high i can count in how much time i have. i see coordinates as scalars because the elements have weight, and here there are 2n vectors for any n dimensions. Each pair of opposite vectors points towards the other but can never intersect. My thought is they probably spiral toward infinity so I’m calling this spiral sudoku: "At the still point of the turning world, neither flesh nor fleshless; neither from nor towards; at the still point, there the dance is.";)

Spiral Sudoku can be expressed procedurally with an additional constraint that element expression $p_x$ must start in the greatest $c < 0$ or least $c > 0$, with the least unexpressed element $p > 0$ for any region, and all subsequent $p_x$ must share a cell boundary with a previous $p_x$

The toroidal surface is attractive because there is no center—every region and vertex is the same until the initial $p_x$

Where n is number of dimensions: $p_x \cap n$

Wherever that first $p_x$ is expressed becomes the center vertex of the sudoku.

This becomes a stacking game around a single vertex, or, because spaces can be virtual, players may “jump” across “hyperspace” to the closest uncolonized vertices to express new $p_x$

If new p must only share a vertex with $p_x$ sparser aggregations may be formed with empty spaces between $p_x$. Element weights $v$ may be balanced where $p_x$ can also be the greatest unexpressed $p < 0$.