posting_dalembert
The d'Alembert identity asserts that the posting potential satisfies Π(xy) + Π(x/y) = 2 Π(x) Π(y) for positive reals x and y. Workers deriving additive scale composition in Recognition Science hierarchies cite it to justify multiplicative structure on geometric sequences without assuming linearity. The proof is a direct algebraic reduction after unfolding PostingPotential as Jcost plus one, clearing denominators via field_simp, and verifying the identity by ring.
claimLet $Π(x) = ½(x + x^{-1})$ denote the posting potential. For positive real numbers $x$ and $y$, $Π(xy) + Π(x/y) = 2 Π(x) Π(y)$.
background
The posting potential is defined here as Π(x) := Jcost(x) + 1, where Jcost is the Recognition cost function. This shift converts the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y), with J(x) = ½(x + x^{-1}) - 1, into the simpler d'Alembert form. The module derives additive scale composition from the forced RCL, closing Proposition 4.3 of the phi paper without presupposing linearity of the recurrence ℓ_{k+2} = f(ℓ_{k+1}, ℓ_k).
proof idea
The proof is a term-mode reduction. It unfolds PostingPotential and Jcost, introduces the facts that x and y are nonzero from the positivity hypotheses, applies field_simp to clear fractions, and finishes with ring to confirm the algebraic identity.
why it matters in Recognition Science
This supplies the core identity used by posting_scales_compose to obtain Π(σ^{j+k}) + Π(σ^{j-k}) = 2 Π(σ^j) Π(σ^k) on geometric sequences. It thereby advances the derivation of additive scale composition from closure, as described in the module documentation for closing Proposition 4.3. In the framework it realizes the extensivity property forced by the RCL at T5 (J-uniqueness) and supports the eight-tick octave at T7.
scope and limits
- Does not hold for x = 0 or y = 0.
- Does not address non-positive arguments.
- Does not derive the underlying RCL, which is taken from prior modules.
- Does not specify any particular value for a scale ratio.
- Does not extend to non-geometric sequences.
Lean usage
posting_dalembert (σ ^ j) (σ ^ k) (pow_pos hσ j) (pow_pos hσ k)formal statement (Lean)
80theorem posting_dalembert (x y : ℝ) (hx : 0 < x) (hy : 0 < y) :
81 PostingPotential (x * y) + PostingPotential (x / y) =
82 2 * PostingPotential x * PostingPotential y := by
proof body
Term-mode proof.
83 unfold PostingPotential Jcost
84 have hx_ne : x ≠ 0 := ne_of_gt hx
85 have hy_ne : y ≠ 0 := ne_of_gt hy
86 field_simp [hx_ne, hy_ne]
87 ring
88
89/-! ## Extensivity in the Self-Similar Regime -/
90
91/-- For a geometric scale sequence with ratio σ, the posting potential
92at level k is Π(σ^k). The d'Alembert identity becomes:
93
94 Π(σ^{j+k}) + Π(σ^{j-k}) = 2 Π(σ^j) Π(σ^k)
95
96This is the RS-native form of "scale composition is governed by
97the posting potential's multiplicative structure." -/