eulerChar3D_filledBox
plain-language theorem explainer
A filled 3-D lattice box has cubical Euler characteristic exactly 1 for every integer side triple. Foam-topology and cosmogenesis readers cite it as the contractible baseline: size never moves χ off the point value. The proof is a pure ring cancellation on the closed-form cell counts N0−N1+N2−N3.
Claim. For all integers $a,b,c$, $(a+1)(b+1)(c+1) - \bigl[a(b+1)(c+1)+(a+1)b(c+1)+(a+1)(b+1)c\bigr] + \bigl[ab(c+1)+a(b+1)c+(a+1)bc\bigr] - abc = 1$. That is, a filled axis-aligned box with those vertex, edge, square, and cube counts has Euler characteristic $\chi=N_0-N_1+N_2-N_3=1$, independent of the side lengths.
background
Phase 18 (FoamTopology) supplies the parameter-free topological readout of the recognition foam assembled in Phase 15 and frozen in Phase 17. The Euler characteristic of a finite cubical complex is the alternating cell sum $\chi=N_0-N_1+N_2-N_3$ (occupied vertices, unit edges, unit squares, unit cubes), matching the quantity computed by the cosmogenesis foam scripts. In-module, eulerChar is the weighted sum $\sum_c (-1)^{\dim c}$ over a finite cell set.
Section 1 of the module proves contractible normalization: a filled $d$-box reads $\chi=1$ independent of side lengths. A solid region, however large, is topologically a point, so any numeric deviation from 1 measures genuine topology (extra components, tunnels $b_1$, enclosed voids $b_2$), never mere size. The 1-D and 2-D siblings give the same identity in lower dimension; the 3-D case is the physically relevant one once spatial dimension is fixed at $D=3$.
Cell counts for an $a\times b\times c$ unit-cube box are standard: $N_0=(a+1)(b+1)(c+1)$; $N_1$ sums the three axis-parallel edge families; $N_2$ sums the three coordinate-plane square families; $N_3=abc$.
proof idea
One-line ring proof. The four multilinear polynomials in $a,b,c$ that express $N_0,N_1,N_2,N_3$ are expanded and cancelled in $\mathbb{Z}[a,b,c]$; every non-constant monomial coefficient vanishes and the constant term is 1. No combinatorial lemmas, induction, or cell-complex machinery are used: the identity is pure integer-ring arithmetic on the closed-form counts.
why it matters
This is the 3-D half of the contractible-normalization pair that makes the foam topology readout scale-free. With the 2-D filled-box identity and the Betti detectors (one strictly interior vertex removed drops $\chi$ from 1 to 0 in 2-D, or raises it from 1 to 2 in 3-D), it pins the three primitive topologies the numeric module reports: blob ($\chi=1$), hole ($\chi=0$), void ($\chi=2$).
Because the forcing chain fixes $D=3$ spatial dimensions (T8/T9), the solid 3-box is the physically relevant baseline for the assembled recognition foam and its cosmic-web genus analogue. The module is theorem-complete (0 sorry, 0 RS-internal axioms). No downstream Lean declaration currently depends on this theorem, but the classical foam scripts and the in-module one-void / freeze-out siblings treat $\chi=1$ as the solid reference against which tunnels and voids are measured.
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