Redundant frame algebra
The redundant-frame algebra describes finite analysis and synthesis systems, including their coefficient-space representations. It does not assume that analysis coordinates form an orthonormal basis.
Let a finite presentation be a pair
\[
A_X:X\longrightarrow K_X,
\qquad
S_X:K_X\longrightarrow X,
\qquad
S_XA_X=I_X.
\]
The coefficient-space identity is the generally oblique projection
\[
P_X=A_XS_X,
\qquad P_X^2=P_X.
\]
Every coefficient vector has the canonical gauge decomposition
\[
c=P_Xc+(I-P_X)c,
\qquad (I-P_X)c\in\ker S_X.
\]
Two coefficient vectors represent the same physical vector exactly when their difference lies in \(\ker S_X\). The positive semidefinite coefficient metric
\[
H_X=S_X^*S_X
\]
measures this quotient faithfully: \(\langle c,H_Xd\rangle= \langle S_Xc,S_Xd\rangle\), and gauge vectors have zero seminorm.
For another presentation \((A_Y,S_Y)\) of the same physical space, the canonical bridge \(R=A_YS_X\) is an exact quotient isometry:
\[
R^*H_YR=H_X.
\]
Moreover the reverse bridge composes with \(R\) to the corresponding oblique coefficient projection. Thus the bridges become genuine mutually inverse isometries after quotienting by the synthesis kernels, even though raw Euclidean coefficient norms can change drastically between presentations.
Maps and closure
For a physical operator \(T:X\to Y\), its canonical coefficient lift is
\[
\widehat T=A_YTS_X.
\]
It satisfies
\[
P_Y\widehat T P_X=\widehat T,
\qquad
S_Y\widehat T A_X=T.
\]
Consequently, physical maps are in bijection with the sandwiched coefficient matrices \(P_Y\operatorname{Mat}(K_Y,K_X)P_X\). The inverse map is \(L\mapsto S_YLA_X\).
If two maps use the same middle presentation, their lifts compose by ordinary matrix multiplication. If the output presentation of the first map and the input presentation of the second map are different presentations of the same physical space, the bridge
\[
R=A_{Y_\mathrm{in}}S_{Y_\mathrm{out}}
\]
is required:
\[
\widehat{U\circ T}=\widehat U\,R\,\widehat T.
\]
The lifted physical identity is \(P_X\), not the full identity on redundant coefficient space.
Coefficient blocks versus physical components
For a coefficient partition with selectors \(E_\alpha\), the small blocks
\[
E_\beta\widehat T E_\alpha
\]
assemble to \(\widehat T\) and obey ordinary block-matrix multiplication when the middle partition is aligned. These are the compositional blocks in coefficient space.
The older physical components
\[
C_\beta T C_\alpha,
\qquad C_\alpha=S E_\alpha A,
\]
still reconstruct \(T\), but in a redundant or oblique system the \(C_\alpha\) need not be orthogonal projectors. Their tempting one-index multiplication rule is therefore false in general. The verifier includes an explicit oblique counterexample so that the two block notions cannot be silently conflated.
Adjoints
The lift of the physical adjoint is
\[
\widehat{T^*}=A_XT^*S_Y.
\]
It is related to, but generally different from, the raw conjugate transpose of the coefficient lift:
\[
\widehat T^*H_Y=H_X\widehat{T^*}.
\]
Raw conjugate transpose becomes the correct lift after passing to the dual presentations \((S^*,A^*)\). In Parseval presentations the distinction disappears.
Exact and numerical semantics
The numerical layer carries tolerance-aware certificates and immutable NumPy
matrices. The exact layer uses immutable Fraction matrices and checks every
identity by equality, with Exactness.EXACT certificates. Floating-point
inputs are intentionally not treated as exact rational data.
This block is finite-dimensional. It does not claim infinite frame completion, measurable direct integrals, or analytic convergence of an infinite expansion.