The duality and stabilization conjecture for valid tiles

A valid tile for $n$ has the form $(c_{\mathrm{out}},c_{\mathrm{in}},(a_1,\ldots,a_n),\varepsilon)$, with the carry data and coefficient as in the tile model.

Duality and stabilization conjecture. (a) If a tile is valid for $n$, then appending a zero to its vector produces a valid tile for $n+1$. (b) If $c_{\mathrm{in}}\in\mathbb{Z}$, then validity is preserved by

$$(c_{\mathrm{out}},c_{\mathrm{in}},(a_1,\ldots,a_n),\varepsilon)\mapsto(n-2-c_{\mathrm{out}},n-2-c_{\mathrm{in}},(p-1-a_n,\ldots,p-1-a_1),\varepsilon).$$

For an END tile, validity is preserved by

$$(c_{\mathrm{out}},\mathrm{END},(a_1,\ldots,a_n),\varepsilon)\mapsto(n-2-c_{\mathrm{out}},\mathrm{END},(p-2-a_n,\ldots,p-2-a_1),\varepsilon).$$

These patterns are inferred from observed tile tables and are not proved in the source.