hilbertNormBridge
plain-language theorem explainer
Re-exports the finite Hilbert-display bridge that maps native F_RS[i] amplitude vectors of length N+1 to ambient complex vectors, with squared norm as the shared real observable. Anyone citing valid comparison of Born-weight totals against Hilbert norms uses this carrier. The body is a one-line alias of the upstream norm bridge.
Claim. For each $N \in \mathbb{N}$, there is a valid-comparison bridge from finite native amplitudes $\psi : \mathrm{Fin}(N+1) \to F_{RS}[i]$ to finite Hilbert display vectors $\mathrm{Fin}(N+1) \to \mathbb{C}$, with real observable given by the native Born-weight sum on one side and the squared Euclidean norm on the other, such that observation after display equals the native protocol.
background
A valid-comparison bridge is a triple of maps (display, native observe, display observe) from a native type $N$ through a display type $D$ to an observable type $O$, required to commute: observing the displayed object recovers the native observable on every input.
Here the native side is a finite amplitude over the Recognition-Science complex field $F_{RS}[i]$, i.e. a vector indexed by $\mathrm{Fin}(N+1)$. The display side is the ordinary finite complex Hilbert carrier $\mathrm{Fin}(N+1) \to \mathbb{C}$. The observable is $\mathbb{R}$. Upstream, the norm bridge packages display of $F_{RS}[i]$ amplitudes, native observation as the sum of Born weights, and display observation as squared norm, with the commuting identity proved there.
This module collects concrete bridges (real, probability, Hilbert) so the valid-comparison doctrine has worked examples rather than only the abstract structure.
proof idea
One-line definitional wrapper: the term is exactly the upstream normBridge at the same $N$. No new proof obligations are discharged at this layer; the commuting law, display map, and both observers are inherited verbatim from Hilbert display completion.
why it matters
Feeds the local theorem that two native amplitudes are validly comparable under this bridge if and only if their Born-weight sums agree, and appears in the module headline that the doctrine has concrete bridges for real display, finite probability display, and finite Hilbert display. In the Primitive Recognition Calculus, this is the Hilbert-side witness that Born-type totals on native $F_{RS}[i]$ amplitudes match squared-norm observation after display, tying the finite distinction calculus to ordinary complex Hilbert language without smuggling extra structure.
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