Abstract
This page fixes the default, copy-pasteable manifest for SSM and indexes all policy switches by family (lifts, selectors, numerics, transforms, aggregates, linear algebra, solvers). Settings ensure deterministic, collapse-safe behavior: magnitudes follow classical formulas; alignment is carried (unary) or aggregated in rapidity; guards and branches are explicit.
Manifest scope and precedence
- Precedence (highest first): call_site > function_level > page_level > global_defaults
- Determinism: all randomness off by default; no stochastic tie-breakers.
- Reporting: include active policy block with every published result.
Global defaults
version = "SSM-2.x"
phi(x) = m # collapse: phi(m,a)=m
mult_mode = "M2" # product: u_sum, a=tanh(sum u)
sum_alignment = "rapidity_weighted"
order_metric = "magnitude" # or "S_beta" with beta in [0,1]
beta = 1.0 # used only if order_metric="S_beta"
clamp_eps = 1e-6 # for atanh clamp
zero_policy = "canonical_zero" # (0,+1) anywhere exact zero arises
precision_targets.abs_tol = 1e-12
precision_targets.rel_tol = 1e-12
report_mode = "value" # switch per function if overflow risk
Unary lifts (f: R -> R)
alignment_policy = "carry" # f*(m,a) = (f(m), a)
domain_guard = "error" # "error" | "nan" | "clamp" | "warn"
branch = "principal" # set explicitly for multi-branch f
Stable numerics (selected):
sinh(m) = 0.5*(exp(m) - exp(-m))
cosh(m) = 0.5*(exp(m) + exp(-m))
asinh(m) = log( m + sqrt(m^2 + 1) )
acosh(m) = log( m + sqrt(m-1)*sqrt(m+1) ) # m>=1
acoth(m) = 0.5*log( (m+1)/(m-1) ) # |m|>1
sigmoid(m) = 0.5*(1 + tanh(m/2))
softplus(m) = m + log1p(exp(-m)) if m>0 else log1p(exp(m))
overflow_policy = "switch_to_log" # for Gamma/Beta etc.
Selectors and piecewise (sgn, H, ramp, clip, min/max/clamp)
selector_alignment = "carry_from_source"
heaviside_zero = "half" # 0 -> 1/2
selector_tie = "prefer_constant" # ties at boundaries
# Strength-based selectors (2.37):
order_metric = "S_beta" | "magnitude"
beta = 1.0
tie_rule = "prefer_higher_alignment" # else "prefer_left"|"prefer_right"
clamp_guard = "swap" # if L>U, swap bounds
Rounding operators (floor, ceil, trunc, round)
rounding_mode = "half_even" # "half_up" | "half_away_zero" | "half_down"
half_tolerance = 4*machine_epsilon
zero_canonicalize = true
Series & special functions
series_guard = "error" # at/beyond radius boundary
Gamma.domain = "m>0" # real principal by default
Beta.domain = "x>0 & y>0"
report_mode_gamma = "value" # auto-switch to "log_value" if needed
report_mode_beta = "value"
Transforms and generating functions (Z, Laplace, MGF/PGF/CGF)
alignment_summary.policy = "rapidity_mean"
alignment_summary.weights = "abs_m^gamma" # gamma = 1
alignment_summary.parameter_dependent = false # true to include kernel factors
diagnostic_alignment_channel = "off" # "Z_u" | "L_u" if enabled
Weighting rule (discrete): w_n = |m_n|^gamma
Weighting rule (continuous): w(t) = |m(t)|^gamma
Multivariate lifts & aggregators (f: R^k -> R)
aggregator = "sensitivity_scale_weighted"
gamma = 1.0
weights.w_i = |∂f/∂m_i| * |m_i|^gamma
weights.normalize = "divide_by_sum"
fallback_when_sum_w_zero = "mean_u" # else "+1"
align_internal_space = "rapidity"
recompute_weights = "final_m"
Vectors, matrices, spectral maps
matrix.add.weights = "|entry_magnitude|" # for sum alignment
matrix.mul.term_alignment = "M2" # u_sum per product term
matrix.mul.entry_weights = "|term_magnitude|"
trace.weights = "|m_ii|"
frobenius.weights = "|m_ij|^2"
det.weights = "|adj(phi(A))^T|" # cofactor sensitivities
spectral.domain.log = "SPD" # principal real log
spectral.domain.pow = "SPD_or_positive_spectrum"
eigen_alignment = "rayleigh_sensitivity" # or "diagonal_only"
near_singularity = "warn_and_report_condition"
Convolution & correlation (scalar kernels)
conv.type = "discrete" # or "continuous"
align.weights = "|k|*|m|^gamma"
align.gamma = 1.0
align.fallback = "mean_u"
conv.mode = "same" # "full" | "valid"
conv.pad = "zero" # "reflect" | "wrap"
causal = false # set true for real-time
stability.check = "L1" # sum |k| < inf or integral
Implicit & inverse functions; solvers
inverse.branch = "principal" # "negative"|"nearest_to_seed"|"alignment_sign"
domain_guard = "error"
newton.tol = 1e-12
newton.max_iter = 50
newton.line_search = "off"
newton.bracket = "none" # supply [L,U] if used
derivative.mode = "analytic" # or "secant"
derivative.safeguard.eps_fprime = 1e-12
implicit.alignment.policy = "sensitivity_mean"
implicit.sensitivity = "|(∂F/∂p_i)/(∂F/∂m)|"
fallback_alignment = "mean_u"
Inequalities and expectations
order_metric_for_inequalities = "magnitude" # restrict to domain of monotonicity
expectation.alignment_summary = "rapidity_mean"
Publishing and reproducibility
report_manifest = true
report_fields = ["phi(value)", "alignment", "domain", "guard", "branch", "weights", "order_metric"]
numeric_backend = "IEEE754-double"
machine_epsilon = 2.220446049250313e-16
seed = "none" # ensure determinism
Default guard table (index)
asinh, acosh, acoth -> domain_guard="error"
coth -> domain excludes m=0 (no fill)
erf/erfc -> domain_guard="error" (R)
Bessel Y_nu, K_nu -> zero_guard="error" at m<=0
polylog Li_s(z) -> domain_guard="error" outside principal real slice
Lambert W -> branch="W0" (m>=-1/e), "W-1" (-1/e<=m<0), domain_guard="error"
Heaviside -> heaviside_zero="half"
ramp/ReLU, clip -> selector_tie="prefer_constant"
round -> rounding_mode="half_even"
min/max/clamp (S_beta) -> beta=1, tie_rule="prefer_higher_alignment", clamp_guard="swap"
series -> series_guard="error"
Gamma/Beta -> use log-space if overflow_policy triggers
convolution -> align.weights="|k|*|m|^gamma", gamma=1
correlation -> per-term u_pair=u_x+u_y, weights=|m_x*m_y|
matrix det/log near-sing. -> "warn_and_report_condition"
Minimal example (call-site override, ASCII)
# Global defaults active
# Local policy for clip and sigmoid in this cell:
selector_tie = "prefer_input"
sigmoid.impl = "tanh_half"
softplus.split_threshold = 0
x = (m,a) = ( -0.0, 0.3 )
y = clip*( x ; L=-1, U=2 ) # uses prefer_input at ties
z = sigmoid*( y ) # tanh_half implementation
# Report includes active overrides and global context
Takeaway
The defaults above make SSM plug-and-play: classical magnitudes, principled alignment propagation, explicit domains/branches, and stable numerics. Override only where a task demands it; always publish the active manifest so results are reproducible and collapse-safe.
Navigation
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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, rapidity-weighted sums for additions/aggregations, clamp_eps = 1e-6, |a| < 1. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
Explore further
https://github.com/OMPSHUNYAYA/Symbolic-Mathematics
Disclaimer
Observation-only; not for safety-critical decisions.
SHUNYAYA 