rs_ledger_is_unique
Any positive real satisfying the golden-ratio fixed-point equation, any dimension at least 2 with nonzero linking number, and any cycle length matching the Gray-code length on that dimension must coincide with the RS values phi, 3, and 8. Researchers closing uniqueness gaps in discrete conservative models cite this to rule out alternative ledgers. The term proof applies the three upstream uniqueness lemmas in sequence and substitutes the resulting dimension into the cycle equality.
claimLet $phi$ denote the positive root of $x^2 = x + 1$. For any real $alpha > 0$, natural numbers $d geq 2$ and $t$, if $alpha^2 = alpha + 1$, the linking number of $d$ is nonzero, and $t$ equals the Gray-code cycle length on $d$ dimensions, then $alpha = phi$, $d = 3$, and $t = 8$.
background
The theorem belongs to the Meta.LedgerUniqueness module, which treats Gap 9: the objection that other discrete conservative structures might exist besides the RS ledger. The module shows each component is forced by its own constraint: the cost fixed point yields only phi, irreducible linking occurs only in three dimensions, and Gray-code traversal of the cube requires exactly eight ticks. Key supporting definitions are linkingNumber (nonzero precisely when the dimension equals 3) and grayCodeCycleLength (equal to 2 to the power of the dimension). The upstream lemmas phi_unique_fixed_point and Q3_unique_linking_dimension supply the uniqueness statements for the first two components.
proof idea
The term-mode proof begins by introducing the three alternative values together with their three hypotheses. It constructs the conjunction by invoking phi_unique_fixed_point on the fixed-point pair, then Q3_unique_linking_dimension on the linking pair to obtain the equality d = 3. The resulting equality is substituted into the cycle-length hypothesis, after which the remaining goal is discharged directly by the rewritten hypothesis.
why it matters in Recognition Science
The declaration closes Gap 9 by proving that the RS ledger (phi, Q3, eight-tick) is the only structure satisfying the three constraints, thereby completing the module's main claim that every discrete conservative system is isomorphic to this ledger. It draws on the forcing-chain steps that single out phi as the unique self-similar fixed point, three dimensions for nontrivial linking, and the eight-tick period as the minimal Gray-code cycle on the cube. No scaffolding remains; the result is fully proved and feeds directly into the ledger-uniqueness statement.
scope and limits
- Does not establish uniqueness when the fixed-point or linking hypotheses are dropped.
- Does not apply to continuous or nondiscrete models.
- Does not derive the numerical values of phi, 3, or 8 from more primitive axioms.
- Does not consider alternative notions of linking beyond the supplied definition.
formal statement (Lean)
229theorem rs_ledger_is_unique :
230 ∀ (altPhi : ℝ) (altD : ℕ) (altT : ℕ),
231 -- If an alternative satisfies the same constraints...
232 (altPhi > 0 ∧ altPhi^2 = altPhi + 1) →
233 (altD ≥ 2 ∧ linkingNumber altD ≠ 0) →
234 (altT = grayCodeCycleLength altD) →
235 -- ...it must equal the RS values
236 altPhi = phi ∧ altD = 3 ∧ altT = 8 := by
proof body
Term-mode proof.
237 intro altPhi altD altT ⟨hPhiPos, hPhiEq⟩ ⟨hDPos, hDLink⟩ hT
238 constructor
239 · exact phi_unique_fixed_point altPhi hPhiPos hPhiEq
240 constructor
241 · exact (Q3_unique_linking_dimension altD hDPos).mp hDLink
242 · have hD3 : altD = 3 := (Q3_unique_linking_dimension altD hDPos).mp hDLink
243 rw [hD3] at hT
244 exact hT
245
246end LedgerUniqueness
247end Meta
248end IndisputableMonolith