polyNorm_one
The polynomial norm of the constant 1 over any finite field F equals 1. Workers on the cost spectrum for function fields cite this base case when verifying that costs vanish on units before establishing additivity. The proof is a one-line wrapper that unfolds the norm definition and simplifies the resulting zero-exponent power.
claimLet $q = |F|$. For the constant polynomial $1$ in $F[X]$, the norm satisfies $q^{0} = 1$.
background
This module develops the cost spectrum on the polynomial ring $F[X]$ as the function-field analog of the integer prime cost spectrum. Every nonzero polynomial factors uniquely into monic irreducibles; the norm of a polynomial $f$ is defined as $q$ raised to its natural degree, with $q$ the cardinality of $F$. The cost $c(f)$ is then obtained by summing $J$ of the norms of the irreducible factors, each weighted by its multiplicity in the normalized factorization.
proof idea
The proof is a one-line wrapper. It unfolds the definition of polyNorm, which produces $q$ raised to the natural degree of the constant 1 (equal to zero), then applies simplification to obtain $q^0 = 1$.
why it matters in Recognition Science
This normalization anchors the base case for the multiplicative cost function on $F[X]$. It directly supports the module's main theorem that the cost of 1 vanishes, which is required for additivity $c(fg) = c(f) + c(g)$ to hold with $c(1) = 0$. In the Recognition Science setting it parallels the zero-cost unit in the integer spectrum, reinforcing non-negativity and the overall cost ladder before passage to physical constants.
scope and limits
- Does not compute the norm for non-constant or non-monic polynomials.
- Does not connect the norm directly to the cost summation.
- Does not treat the zero polynomial.
- Does not specify any particular finite field beyond its cardinality.
formal statement (Lean)
69@[simp] theorem polyNorm_one : polyNorm (1 : Polynomial F) = 1 := by
proof body
One-line wrapper that applies unfold.
70 unfold polyNorm; simp
71