Abstract
We lift SSM into infinite-dimensional settings by embedding symbolic numerals through a centre-aware strength map and working in rapidity for convergence. This yields inner products, Hilbert spaces, bounded operators, adjoints, and Parseval-type expansions that collapse exactly to classical functional analysis when a = +1.
Symbolic inner product (real-valued, default)
Fix beta in [0, 1]. Define the strength map
S_beta(m, a) = m * (1 - beta * (1 - a)) # S_0(m,a)=m; S_1(m,a)=m*a
For u = ((m_i, a_i)) and v = ((n_i, b_i)) over a common index set I, set
<u, v>_beta = sum_{i in I} [ S_beta(m_i, a_i) * S_beta(n_i, b_i) ] # real
Properties (inherited from classical l2 on strengths)
- Symmetry:
<u, v>_beta = <v, u>_beta - Linearity in each argument over
R - Positive-definiteness:
<u, u>_beta >= 0, with equality iffS_beta(m_i, a_i)=0for alli
Interpretation
The inner product is real. When a symbolic “scalar” is desired, lift c in R to (c, +1) explicitly (metadata), while computations remain real.
Hilbert space H_beta (completion)
Let
V_beta = { u : sum_i S_beta(m_i, a_i)^2 < infinity }
||u||_beta = sqrt( <u, u>_beta )
The completion H_beta of V_beta is a Hilbert space. Concretely, the map
P_beta : ((m_i, a_i)) -> ( S_beta(m_i, a_i) )
is an isometric isomorphism from H_beta onto the classical l2(I). Hence all Hilbert-space theorems apply.
Worked exampleu = ((2, +0.5), (1, -0.2)), v = ((3, +1.0), (4, +0.5)), beta = 1S_1(u) = { 1.0, -0.2 }, S_1(v) = { 3.0, 2.0 }<u, v>_1 = 1.0*3.0 + (-0.2)*2.0 = 2.6||u||_1 = sqrt(1.0^2 + (-0.2)^2) ≈ 1.020, ||v||_1 ≈ 3.606 (Cauchy–Schwarz holds).
Convergence (rapidity-aware)
A scalar sequence x_k = (m_k, a_k) converges to x = (m, a) iff
|m_k - m| -> 0 and |atanh(a_k) - atanh(a)| -> 0
Equivalently, for vectors u^(n) in H_beta: u^(n) -> u iff P_beta(u^(n)) -> P_beta(u) in l2.
Practical note
If a jitters near 0 in data, regularize in u = atanh(a) (TV smoothing) before forming strengths (see Section 2.17).
Operators and adjoints (beta-linear viewpoint)
We call T : H_beta -> H_beta beta-linear and bounded if there exists a bounded linear operator B on l2(I) such that
P_beta( T x ) = B( P_beta(x) ) for all x in H_beta
Then:
- Operator norm:
||T|| = sup_{||x||_beta=1} ||T x||_beta = ||B||_{l2->l2} - Adjoint:
T*corresponds toB*viaP_beta( T* y ) = B*( P_beta(y) ) - Self-adjointness:
T = T*iffBis self-adjoint onl2(real spectrum classically).
For symbolic reporting, lift eigenvalueslambdato(lambda, +1)as metadata.
Remark (symbolic “scalars”)
If you require multiplication by a symbolic c = (m_c, a_c), act componentwise with the scalar M2 rule on entries in the base space, then pass through P_beta. For functional-analytic results, the default field is R via strengths.
Fourier-type expansions and Parseval
Let (phi_k) be an orthonormal basis of H_beta (<phi_j, phi_k>_beta = delta_{jk}). For any f in H_beta:
f = sum_k c_k * phi_k, where c_k = < f, phi_k >_beta in R
||f||_beta^2 = sum_k c_k^2 # Parseval
If a symbolic coefficient is desired for presentation, report (c_k, +1) alongside the real expansion.
Bounded linear functionals (Riesz)
Every bounded linear functional L on H_beta has the form
L(f) = < f, g >_beta for a unique g in H_beta
Equivalently, L = P_beta^* L_class P_beta with L_class on l2. This supports standard variational methods (e.g., Galerkin) in symbolic settings.
Collapse consistency
If all alignments equal +1, then S_beta(m, +1) = m for any beta. Consequently:
<u, v>_betareduces to the classical Euclidean inner productsum_i m_i * n_iH_betacollapses tol2(I)(orL2in continuous settings)- Operators, adjoints, bases, Parseval all reduce to their classical forms
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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: gamma = 1, mult_mode = M2, clamp_eps = 1e-6, |a| < 1.
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