oneTwentyFive_is_Dcubed
125 equals the cube of the configuration dimension in the Recognition Science cardinality spectrum. Cross-domain researchers cite this to confirm 125 arises as 5 cubed among the structured values generated from cube-generators, config dimension 5, and gap45. The proof is a direct decision procedure that verifies the arithmetic identity.
claim$125 = D^3$ where the configuration dimension $D$ equals 5.
background
The module collects witnesses showing that cardinalities of canonical RS domain types form a structured spectrum: {2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 16, 45, 70, 125, 216, 256, 3125, ...}. Each entry decomposes via sums, products, or powers of the cube-generators {2, 3}, the configuration dimension 5, and gap45. The local setting is the claim that RS produces this non-random numerical spectrum rather than arbitrary integers.
proof idea
The proof is a one-line wrapper that applies the decide tactic to the numerical equality 125 = 5^3.
why it matters in Recognition Science
This witness feeds cardinalitySpectrumCert, which assembles the full spectrum certificate via Dspatial_is_3, Dconfig_is_5, gap_as_D, eightTick_as_D, and cubeFaces_as_D. It supplies the explicit 125 = 5^3 slot required by the module's structural claim. The result aligns with the forcing chain's use of small integers and powers while keeping the spectrum closed under the listed RS primitives.
scope and limits
- Does not derive the configuration dimension 5 from the forcing chain or J-uniqueness.
- Does not interpret 125 in terms of mass formulas or phi-ladder rungs.
- Does not address other spectrum members such as 216 or 3125.
- Does not connect the equality to Berry creation threshold or alpha band.
formal statement (Lean)
82theorem oneTwentyFive_is_Dcubed : (125 : ℕ) = Dconfig^3 := by decide
proof body
Term-mode proof.
83
84/-- 216 = 6³. -/