T_min
T_min(d) equals the vertex count 2^d of the d-dimensional hypercube and therefore the period of any Hamiltonian cycle on that graph. Mass-ladder derivations and temperature-ratio calculations in chemistry cite this to anchor the vacuum rung at D=3. The definition is a direct one-line reference to the standard hypercube vertex function.
claimLet $T(d)$ denote the number of vertices in the $d$-dimensional hypercube $Q_d$. Then $T(d) = 2^d$.
background
The module derives baseline rung integers, octave offsets, and related quantities from the combinatorics of the 3-cube $Q_3$, upgrading prior boundary assumptions to derived status using only the input $D=3$. T_min(d) is the Hamiltonian cycle period on $Q_d$, which coincides with the vertex count $V=2^d$. Upstream definitions in AlphaDerivation, PlanckScaleMatching, and PMNSCorrections each state the identical formula cube_vertices(d) := 2^d.
proof idea
The definition is a one-line alias that directly invokes the imported cube_vertices function.
why it matters in Recognition Science
T_min supplies the period that yields the octave offset B-15 as -T_min = -8 at D=3, fixing the vacuum rung on the phi-ladder. It is used to derive nontriviality_from_cost and to define haberBoschTempCost in chemistry. This step closes the eight-tick octave (T7) from cube geometry and feeds the mass formula yardstick * phi^(rung - 8 + gap(Z)).
scope and limits
- Does not prove existence of a Hamiltonian cycle on the hypercube.
- Does not restrict the domain to D=3; the definition holds for any natural number d.
- Does not incorporate J-cost, phi scaling, or rung arithmetic.
- Does not compute edge, face, or wallpaper counts of the cube.
Lean usage
theorem T_min_at_D3 : T_min 3 = 8 := by unfold T_min; simp [cube_vertices]formal statement (Lean)
96def T_min (d : ℕ) : ℕ := cube_vertices d
proof body
Definition body.
97
98/-- At D = 3: T_min = 8. -/