Abstract
Quantum theory is built on Hilbert spaces with complex amplitudes. Shunyaya Symbolic Mathematics (SSM) generalizes this by attaching an alignment channel a to every amplitude, so both probability and stability are explicit. Under collapse phi(m,a) = m, all constructions reduce to standard quantum theory.
Symbolic state space (beta-Hilbert view)
Let H be a separable complex Hilbert space with orthonormal basis { |i> }.
A symbolic state is
psi = sum_i c_i otimes |i>
with symbolic scalars c_i = (m_i, a_i) (scalar ops use M2 by default).
Strength map (real scale control).
S_beta(m, a) = m * (1 - beta * (1 - a)), beta in [0,1]
Define the effective complex amplitude
A_i = S_beta(m_i, a_i) * exp(i * theta_i)
where theta_i is the classical phase. Alignment modulates only the real scale via S_beta; phases remain classical.
Normalization (beta-family).
sum_i |A_i|^2 = 1
Collapse: if all a_i = +1, then S_beta(m_i, +1) = m_i and we recover sum_i |m_i|^2 = 1.
Notes
• beta = 0 ignores alignment (classical norm).
• beta = 1 fully weights alignment (A_i = m_i * a_i * e^{i theta_i}).
• For dynamics and numerics, use u_i = atanh( clamp(a_i, -1+eps, +1-eps) ), then report in a via a = tanh(u).
Superposition and interference
Given symbolic states psi_1, psi_2 and symbolic scalars alpha, beta, the superposition
alpha otimes psi_1 oplus beta otimes psi_2
induces (in the effective-amplitude picture)
A = alpha_eff * A^(1) + beta_eff * A^(2)
where alpha_eff = S_beta(m_alpha, a_alpha) * e^{i theta_alpha} and similarly for beta_eff.
Misalignment attenuates or flips contributions even when magnitudes are large, producing drift-sensitive interference.
Measurement (symbolic Born rule)
Outcome probability for basis element |i>:
P_s(i) = |A_i|^2 / sum_j |A_j|^2
= [ S_beta(m_i, a_i) ]^2 / sum_j [ S_beta(m_j, a_j) ]^2
Collapse: if a ≡ +1, this is the classical Born rule.
Illustration (two-level, beta = 1).m = (1, 1), a = (+1, -0.5)S_1(1,+1)=1, S_1(1,-0.5)=-0.5 ⇒ ratio (1^2 : 0.5^2) = (1 : 0.25) ⇒ (0.8, 0.2).
Classically it would be (0.5, 0.5); symbolic weighting reveals instability in the second outcome.
Observables and expectation
For a classical self-adjoint observable O with matrix elements O_ij, define
E_s[O] = sum_{i,j} A_i^* * O_ij * A_j
Collapse: if a ≡ +1, this reduces to <psi|O|psi>.
Audit: also report E[a] or E[u] = E[atanh(a)] alongside E_s[O] for reproducibility.
Dynamics (symbolic Schrödinger form)
Classical:
i * dpsi/dt = H * psi
Symbolic (coefficient-wise):
i * dpsi/dt = H otimes psi
Let { |k> } be an eigenbasis of H. Write psi(0) = sum_k c_k(0) otimes |k>, and give each eigenvalue a symbolic tag lambda_k = (E_k, a_lambda,k).
Mode-wise ODEs (separable case).
d/dt c_k(t) = (-i) otimes lambda_k otimes c_k(t)
In rapidity for alignment:
u_k(t) = atanh(a_k(t))
du_k/dt = u_lambda,k (constant if a_lambda,k is time-invariant)
⇒ a_k(t) = tanh( u_k(0) + t * u_lambda,k )
Simultaneously, the classical phase accumulates as exp(-i * E_k * t).
Norm and beta-unitarity.
If the induced operator on effective amplitudes A is unitary, sum_i |A_i|^2 is preserved. When alignment sources/sinks are modeled (nonzero u_lambda,k), renormalize before applying the Born rule (standard practice with effective channels).
Collapse: if a_lambda,k = +1 and a_k(0) = +1 for all k, we recover standard Schrödinger dynamics.
Density operators, channels, and entanglement (sketch)
Symbolic density (pure state).
rho_s = |A><A| where A := (A_1, A_2, ...)
Mixed states: convex sums rho_s = sum_r p_r |A^{(r)}><A^{(r)}|.
Partial trace and reduced states proceed on rho_s as classical, since A lives in the usual complex space.
Symbolic noise channels.
Alignment dynamics can be modeled as classical CPTP maps on rho_s combined with a side ODE in u (e.g., exponential relaxation) whose parameters are declared in the manifest. This yields amplitude damping–like behavior driven by alignment decay, without altering classical CPTP structure.
Entanglement.
All classical entanglement measures apply to rho_s. Alignment summaries (e.g., E[u]) can be reported per subsystem to audit stability of entanglement without redefining the classical measures.
Uncertainty (conservative statements)
Let X, P be position and momentum. On the magnitude channel (or under collapse) the Heisenberg inequality holds:
Var[m_X] * Var[m_P] >= (hbar/2)^2
Alignment introduces a separate, nonnegative drift-variance budget (e.g., Var[u_X], Var[u_P]). Report both:
Report: ( Var[m_X] * Var[m_P] , Var[u_X] , Var[u_P] )
This preserves the rigorous classical bound while auditing stability dispersion; no new constant is asserted.
Worked mini-example (alignment-damped Rabi oscillation)
Two-level Hamiltonian H = (Omega/2) * sigma_x (classical).
Initial state: psi(0) = c_1(0)|1> + c_2(0)|2>, with c_1(0) = (1, +1), c_2(0) = (0, +1).
Suppose eigenmodes carry symbolic alignments with u_lambda,1 = 0, u_lambda,2 = -gamma (gamma > 0). Then
A_1(t) ≈ cos(Omega * t / 2)
A_2(t) ≈ sin(Omega * t / 2) * tanh( -gamma * t ) # effective scale; phase omitted
P_s(2; t) ∝ [ sin(Omega * t / 2) * tanh( -gamma * t ) ]^2
Collapse (gamma = 0) recovers standard Rabi flops; negative u_lambda,2 damps oscillations through alignment decay.
Takeaway
Symbolic quantum foundations retain all classical results under collapse while adding a centre-aware axis:
• States: amplitudes scaled by S_beta(m,a); phases classical.
• Measurement: probabilities depend on alignment-weighted strengths.
• Dynamics: alignment evolves additively in u, alongside classical phase.
• Auditability: report classical expectations plus alignment summaries.
This yields entropy- and stability-aware quantum models without breaking standard formalism when a = +1.
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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1 with rapidity u = atanh(a). All formulas are presented in plain text. Collapse uses phi(m,a) = m.
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