Abstract
The symbolic framework established here provides a rigorous algebraic, analytic, and geometric foundation. We have extended classical mathematics across arithmetic, calculus, geometry, probability, information theory, Fourier analysis, PDEs, quantum mechanics, relativity, and thermodynamics—each expressing the same principle: a symbolic numeral (m, a) carries both classical size and alignment drift, enriching the system while collapsing back to classical mathematics when a = +1. In running text we write a for the standalone alignment symbol; inside tuples (m, a) it remains plain a.
Universal extension recipe (one-page summary)
1) Object lift
Replace every classical scalar q by a pair (q, a_q) with a_q in [-1, +1]. Vectors/matrices become arrays of such pairs.
2) Operation lift
Use the defaults from this section: rapidity addition for oplus (via the (U, W) accumulator) and M2 (rapidity-additive) for otimes, odiv. Inside algorithms, update u = atanh(a) and report back a = tanh(u).
3) Metric/ordering
Compare with S_beta(m, a) = m * (1 - beta * (1 - a)) and distances d_beta as defined earlier. Choose beta in [0,1] per application.
4) Dynamics
For time/space evolution, advance the magnitude channel with the classical rule and evolve the alignment rapidity u with a consistent diffusion/relaxation/coupling law. Clamp only at presentation (a = tanh(u)).
5) Collapse guarantee
Verify that setting all a = +1 recovers the classical specification to machine precision.
6) Manifest
Declare a_mapping, gamma (weights), clamp_eps, multiplication (M2 default), and any coupling terms between channels. See Appendix F (EV) for empirical templates and Appendix G (CF) for comparisons.
Domain sketches (ready-to-port templates)
Symbolic statistical mechanics
• Partition function (strength-weighted): Z_s = sum_i exp( - beta * E_i ) * w_i, with w_i = (1 - beta_a * (1 - a_i)).
• Observables: <O>_s = (1/Z_s) * sum_i O_i * exp( - beta * E_i ) * w_i.
• Collapse: beta_a = 0 (or a_i = +1) recovers classical ensembles.
Symbolic control theory
• State x = ((m_k, a_k))_k, control u = ((m_j, a_j))_j.
• Cost: J_s = sum_t [ x_t^T Q_s x_t + u_t^T R_s u_t ], with Q_s, R_s built from S_beta on entries.
• Riccati/LQR extensions: magnitude channel uses classical recursions; alignment rapidities propagate with linear–quadratic penalties on u_a to suppress drift.
Symbolic machine learning
• Loss: L_s = L_classical( m_pred, m_true ) + lambda * R_align( a_pred ), e.g., R_align = mean( 1 - a_pred ).
• Optimizer: update m by classical gradients; update u = atanh(a) with its own step (keeps |a| < 1 without clips).
• Reporting: train/val metrics as pairs (score, mean a).
Symbolic finance
• Return r_t = (m_t, a_t), portfolio weight w_i = (m_i, a_i).
• Risk-adjusted objective: maximize E[ S_beta(r_p) ] − kappa * Var[ S_beta(r_p) ], where r_p is the symbolic sum of asset returns.
• Stress tests: shock the u-channel (alignment rapidities) separately from magnitudes to expose fragility.
Symbolic biological dynamics
• Population/state y = (m, a).
• Growth: dm/dt = f(m), du/dt = g(u, m) (e.g., logistic growth with alignment relaxation or homeostasis).
• Early-warning: rising Var[a] or spatial gradients in u can precede classical tipping points.
Interoperability with classical tools
• Drop-in: set all a to +1 to use any classical solver unchanged.
• Lift-in: wrap classical steps with u-updates; no change to the classical code path is required for the magnitude channel.
• Export: always provide the collapsed result phi(m, a) = m alongside alignment summaries for audit.
Defaults and invariants (normative)
• Multiplication/division: M2 (rapidity-additive) is the default; M1 is a documented alternative.
• Addition: n-ary rapidity accumulator (U, W).
• Weights: w(m) = |m|^gamma with gamma = 1 unless declared otherwise.
• Zero-class: canonical representative (0, +1).
• Symbols: ⊕, ⊗, ⊘ with ASCII aliases oplus, otimes, odiv.
Reproducibility manifest (required fields)
• a_mapping = <method>; params = {...}; bounds = [-1, +1]; clamp_eps = 1e-6
• gamma = 1 (unless declared)
• multiplication = M2 (default) or M1 (explicitly declared)
• zero_class_policy = canonical (default) or averaged (declared)
• beta (ordering/metrics) = chosen in [0, 1] with justification
• Coupling terms for u-dynamics (if any), and numerical scheme (splitting, CFL)
For empirical validation and head-to-head comparisons (interval, fuzzy), see Appendix F (EV) and Appendix G (CF).
Takeaway
Shunyaya Symbolic Mathematics is not a one-off construct but a general calculus of stability—a language of centre and edge—capable of illuminating every branch of science and mathematics. Two guarantees anchor every extension:
(1) conservative collapse to classical results;
(2) predictive enrichment via the alignment channel that makes hidden fragilities measurable and auditable.
Navigation
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Next → Elementary algebraic functions — powers & roots (2.31)
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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