IndisputableMonolith.Cosmology.LatticeBallVolume
Closed-form cell counts for discrete L1 balls: the 2D diamond of radius t has 2t^2+2t+1 lattice points, and the 3D octahedron is the centered-octahedral number. Cosmology modules that compare carried state to world size cite these formulas. Proofs sum explicit 1D slice cardinalities and evaluate the resulting arithmetic series over the integers.
claimFor $t\in\mathbb{N}$, the discrete $\ell^1$ ball $\{x\in\mathbb{Z}^2:\|x\|_1\le t\}$ has cardinality $2t^2+2t+1$. The 3D ball $\{x\in\mathbb{Z}^3:\|x\|_1\le t\}$ has the centered-octahedral cardinality obtained by summing those 2D slice sizes. Each fixed-$x$ slice $\{y:\|x\|+|y|\le t\}$ is the interval $[-(t-|x|),t-|x|]$ of length $2(t-|x|)+1$.
background
In the Recognition cosmogenesis pipeline the coarsening "world" at radius $t$ is the integer $\ell^1$ ball: a diamond in 2D and an octahedron in 3D. The upstream module InterfaceComponentBound already proves that locked-domain count is at most interface size plus one on these shapes. Exact world volume is still needed so that carried-state cost can be shown sub-extensive relative to $\Theta(t^d)$ cells.
The combinatorial engine is slice decomposition. For $|x|\le t$ the $y$-slice of the diamond is exactly the symmetric interval $[-(t-|x|),t-|x|]$, hence contains $2(t-|x|)+1$ lattice points. Summing over $x$ recovers the closed forms. Named pieces inside the module include slice cardinality, diamond-as-sum, the 2D outer arithmetic sum, the octahedral sum, and the 3D triple-outer identity.
proof idea
Volume is reduced to finite arithmetic sums. A slice lemma identifies each 1D cross-section as an interval of known length $2(t-|x|)+1$. The diamond cardinality is rewritten as the sum of those lengths over $x\in[-t,t]$. An outer-sum lemma evaluates the resulting quadratic series to $2t^2+2t+1$. The 3D case stacks the same 2D diamonds (or an equivalent triple sum) and simplifies to the centered-octahedral closed form. All steps are elementary equalities on $\mathbb{N}$; no analysis or extra axioms.
why it matters in Recognition Science
Four downstream cosmology modules import these volumes as the extensive world size. PolarizedBirthDomains (Phase 50) cites the closed forms $2t^2+2t+1$ (2D) and the centered-octahedral count (3D) to prove the forced conjugate-birth field is held in $O(1)$ locked domains while the world is $\Theta(t^d)$. LatticeBallEdges uses the same geometry for the total adjacency ledger and the monochromatic-versus-interface split. PolarizedBirthInterface and PolarizedBirthInterfaceCount then pin the complementary interface: spine confinement and exact bichromatic edge counts $8t-4$ (2D) and $8t^2-8t+4$ (3D). The volume formulas are the missing extensive baseline that turns those $O(1)$ and spine bounds into constant recognition activity per cycle.
scope and limits
- Does not bound locked-domain or component counts (InterfaceComponentBound).
- Does not count edges or adjacencies (LatticeBallEdges).
- Does not specialize to the polarized birth charge field.
- Does not treat continuous volumes or ell^2 balls.
- Restricted to integer-lattice L1 balls in dimensions 2 and 3.