rotational axes
First mathematical prototype
From Paths
to Invariants
The accepted low-single graph is now analyzed as a 72-variable constrained system. Model 0.4 computes matrix rank, nullity, canonical graph forms, left/right automorphisms, Pareto dominance, and legality certificates for rewrites.
under left/right reflection
exact bilateral symmetry
predicate coverage
Selected state
Loading…
SmithDefender
Linear mobility
Constraint rank
C(G) q̇ = 0
M(G) = 72 − rank C(G)Hard grips, posts, unsafe axes, and load couplings generate rows. Gaussian elimination removes redundant constraints.
Symmetry
Canonical form
canon(G) = min { σ(G) : σ ∈ Γ }
Γ = { identity, left↔right }States are equivalent only when typed edges, hard mobility attributes, operators, status, and outcome agree after reflection.
Partial order
Dominance
G₁ ⪰ₛ G₂ iff
Aₛ(G₁) ⊇ Aₛ(G₂)
Aᴅ(G₁) ⊆ Aᴅ(G₂)
and mobility/control/support
are no worse for Smith.This is Pareto dominance, not a weighted score. It identifies structurally unambiguous improvements.
Computed relations
Dominance and symmetry
Filter the computed partial order around the selected state. Exact equivalence classes remain distinct unless every typed attribute matches.