stringModeCount
The theorem establishes that the Recognition Science string modes form a finite set of cardinality five. Physicists modeling tachyon-free spectra would cite this to confirm the mode count equals configDim D = 5. The proof is a one-line decision procedure that enumerates the five constructors of the StringMode inductive type.
claimThe finite set of string modes consisting of tachyon-suppressed, massless, massive, winding, and momentum has cardinality five: $|$tachyon-suppressed, massless, massive, winding, momentum$| = 5$.
background
In the Tachyon-Free Spectrum from RS module the tachyon-free condition is the absence of modes with negative J-cost. The StringMode inductive type enumerates five canonical modes: tachyonSuppressed, massless, massive, winding, and momentum, each deriving Fintype. This count is set equal to configDim D = 5, matching the module statement that five modes arise once J(x) ≥ 0 excludes tachyons at the Lagrangian level.
proof idea
The proof is a one-line wrapper that applies the decide tactic to Fintype.card on the StringMode type. The inductive definition supplies exactly five constructors together with an automatic Fintype instance, so the decision procedure evaluates the cardinality directly to five.
why it matters in Recognition Science
This theorem supplies the five_modes field inside the tachyonFreeCert definition that certifies the full tachyon-free spectrum. It fills the module requirement that the RS recognition field naturally excludes tachyons once J ≥ 0 holds, linking the mode count to the forcing chain step that sets configDim D = 5. The result closes one concrete check in the S7 string-theory depth of the framework.
scope and limits
- Does not derive the physical properties or masses of each mode from the J-cost function.
- Does not prove J(x) ≥ 0; that is supplied by the separate jcost_nonneg lemma.
- Does not address embeddings into higher-dimensional string spectra or full superstring consistency.
- Does not provide dynamics for transitions between the five modes.
formal statement (Lean)
28theorem stringModeCount : Fintype.card StringMode = 5 := by decide
proof body
Term-mode proof.
29
30/-- J ≥ 0 always: tachyon-free condition. -/