Plain English (1): The declaration states that any function F satisfying the Composition axiom (the multiplicative d'Alembert relation F(xy) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y) for positive x, y) and the Normalization axiom (F(1) = 0) must obey F(x) = F(1/x) for every positive x. In other words, the two axioms together force reciprocal symmetry.
Role in Recognition Science (2): Symmetry is required for the cost functional to be consistent with multiplicative structure on positive reals. It is an intermediate step showing that the canonical J(x) = (x + x^{-1})/2 - 1 satisfies the axioms and that any solution must share this reciprocity, which later supports the functional equation and uniqueness of J as the unique cost.
Reading the formal statement (3): The theorem is quantified over F : ℝ → ℝ equipped with type-class instances [Composition F] and [Normalization F]. It asserts ∀ {x : ℝ}, 0 < x → F x = F x⁻¹. The proof instantiates Composition.dAlembert at (1, x), substitutes Normalization.unit_zero to obtain F(x) + F(x^{-1}) = 2F(x), then rearranges by subtraction.
Dependencies visible in source (4): Relies directly on Composition.dAlembert and Normalization.unit_zero (both defined in the same module). It is invoked inside uniqueness_specification to supply the symmetry hypothesis for T5_uniqueness_complete. Related declarations in the module include J_symmetric (proved independently via field_simp) and the instance proofs that J itself satisfies both axioms.
What it does not prove (5): It does not establish that J is the only solution (that requires the additional regularity hypotheses in uniqueness_specification). It does not derive the axioms from more primitive principles, nor does it connect to downstream physics constants or the full forcing chain. The complete T5 proof lives in the imported CostUniqueness module, which is not expanded in this slice.