thirteen_natural_interpretations
Thirteen admits three equivalent natural-invariant interpretations in the three-dimensional cube: as vertices plus faces minus body, as edges plus one, and as itself. Researchers tracing the Recognition Science mass ladder cite this when mapping generation steps to Q3 invariants. The proof reduces the conjunction to native arithmetic decisions via a refine tactic.
claimFor the three-dimensional cube, $V + F - C = 13$, $E + 1 = 13$, and $13 = 13$, where $V, E, F, C$ denote the numbers of vertices, edges, faces, and the three-dimensional body.
background
The module narrows the open forcing step between proved Q3 cube invariants and their assignment as generation steps. SectorDependentTorsion already establishes that the values {13, 11, 6, 8} are cell counts on the three-dimensional cube at D=3, with the Euler characteristic χ(S²)=2 supplying the relations V-E+F=2 that rearrange to the two nontrivial identities for 13. Upstream results fix the discrete setting: RSNativeUnits sets c=1 and the phi-ladder, while CellularAutomata.step supplies the local rule that generates successor states on the recognition tape.
proof idea
The proof is a one-line term-mode wrapper. It refines the goal into the three conjuncts and discharges each arithmetic equality with native_decide.
why it matters in Recognition Science
The theorem supplies the Euler identity for 13, which the module summary lists as newly proved and which supports the conditional uniqueness of the chain (13, 11, 6, 8) modulo the edge-symmetric filter. It advances the T8 step that forces D=3 and the identification of generation steps with cube invariants on the phi-ladder. The remaining openness, quoted from the module, is the physical reason the Up sector selects the edge-symmetric opening rather than another pair summing to 21.
scope and limits
- Does not derive the edge-symmetric opening from first principles.
- Does not prove uniqueness of the chain without the structural filter.
- Does not connect the identities to the explicit phi-ladder mass formula.
formal statement (Lean)
217theorem thirteen_natural_interpretations :
218 -- V + F - C
219 cube_vertices' 3 + cube_faces' 3 - cube_body = 13 ∧
220 -- E + 1
221 cube_edges' 3 + 1 = 13 ∧
222 -- The value equals itself
223 (13 : ℕ) = 13 := by
proof body
Term-mode proof.
224 refine ⟨?_, ?_, rfl⟩ <;> native_decide
225
226/-! ## Summary of What This Module Proves
227
2281. **Middle pair uniqueness** (re-proved): {11, 6} is the only nonzero
229 natural-invariant pair summing to 17. See `middle_pair_unique_nonzero`.
230
2312. **Endpoint non-uniqueness** (newly proved): multiple endpoint pairs
232 satisfy a + d = 21. See `endpoint_pairs_not_unique`.
233
2343. **Conditional uniqueness** (newly proved): given the "edge-symmetric"
235 structural filter (chain starts with E+1, E-1), the chain is uniquely
236 (13, 11, 6, 8). See `current_chain_unique_modulo_edge_pair_filter`.
237
2384. **Euler identity** (newly proved): 13 = V+F-C = E+1 at D=3, by the
239 Euler characteristic χ(S²) = 2. See `euler_identity_Q3`.
240
241## Residual Openness
242
243The only unresolved step is: *why* the edge-symmetric pair (E+1, E-1) appears
244as the "opening" of the cyclic chain (i.e., why Up-quark generation steps
245are {V+F-C, E-A} = {13, 11} rather than e.g. {V+F, V-C} = {14, 7}).
246
247This reflects a physical identification of the U(1) gauge channel with the
248edge cells of Q₃ (which is proved separately in `Foundation/GaugeFromCube`).
249Under that identification, the edge-symmetric opening {E+1, E-1} is natural,
250but the Lean does not yet have a first-principles forcing proof that the Up
251sector specifically uses this opening.
252
253The epistemic status is therefore:
254- Structural uniqueness modulo the edge-symmetric filter: **THEOREM**.
255- Physical identification of Up sector with edge-symmetric opening: **HYPOTHESIS**.
256
257-/
258
259end StepValueEnumeration
260end Masses
261end IndisputableMonolith