Abstract
We state when inequalities are preserved (or reversed) after applying real functions to symbolic numerals (m, a). The lift f*(m,a) = (f(m), a) preserves magnitude order on domains where f is monotone, and reverses it when f is monotone decreasing. For quantitative bounds (e.g., Jensen), statements hold on the magnitude channel exactly as in classical analysis; alignment is carried or summarized separately and does not alter classical inequality signs. Under collapse phi(m,a)=m, all results reduce to the classical theorems.
Monotone maps and order through functions
Let <=_m denote magnitude order: (m1,a1) <=_m (m2,a2) iff m1 <= m2.
Rule (increasing). If f: D -> R is increasing on D and m1, m2 in D, then
(m1,a1) <=_m (m2,a2) ⇒ f*(m1,a1) <=_m f*(m2,a2).
Rule (decreasing). If f is decreasing on D, then
(m1,a1) <=_m (m2,a2) ⇒ f*(m1,a1) >=_m f*(m2,a2).
Branchwise monotonicity. If f changes monotonicity (e.g., even powers), decompose D into monotone components and apply the rule within each component (cross-ref 2.31, 2.34, 2.38).
Strength order note (S_beta). With S_beta(m,a) = m * (1 - beta * (1 - a)):
- If alignments are equal (
a1 = a2), the same preservation/reversal holds for<=_beta. - If
adiffers,S_beta-order can disagree with pure magnitude order. To make global claims, either (i) compare within a fixed alignment slice or (ii) setorder_metric="magnitude"for inequality steps.
Collapse check. phi( f*(m,a) ) = f(m); hence all classical order results are retrieved when a = +1.
Jensen-style statements (magnitude channel)
Let M be a real random variable (the magnitude). For any convex f on a convex domain D with M in D a.s.,
f( E[M] ) <= E[ f(M) ] # Jensen (magnitude)
For concave f, the inequality reverses. The symbolic lift records alignment as metadata:
E_s[ X ] := ( E[M] , a_bar ) # alignment summary (e.g., rapidity mean)
E_s[ f*(X) ] := ( E[ f(M) ] , a_bar,f )
Jensen applies to the magnitudes E[M] and E[f(M)] exactly; no inequality is asserted between a_bar and a_bar,f.
Common convex/concave examples (on magnitude):
- Convex on
R:m^2,exp(m),softplus(m). - Concave on
(0, inf):log(m). - Convex on
(-inf,0)and concave on(0,inf):sigmoid(m)andsoftsign(m)are S-shaped; apply Jensen piecewise if restricting to one curvature side.
Inequalities through composition
If f is increasing and g is convex on D, then for M in D:
g( E[M] ) <= E[ g(M) ] (Jensen)
f( g( E[M] ) ) <= f( E[ g(M) ] ) # apply increasing f
If f is decreasing, the inequality reverses in the second line. The same logic applies to deterministic inequalities m1 <= m2.
Worked examples
A) Increasing square on m >= 0 (order preserved).x1 = (1.2, +0.3), x2 = (1.8, -0.6). On [0,inf), f(m)=m^2 is increasing.
f*(x1) = ( 1.44 , +0.3 )
f*(x2) = ( 3.24 , -0.6 )
Since 1.2 <= 1.8, we have 1.44 <= 3.24 (order preserved on magnitudes).
B) Decreasing reciprocal on (0,inf) (order reversed).y1 = (2.0, +0.8), y2 = (5.0, 0.1), f(m)=1/m decreasing on (0,inf).
f*(y1) = ( 0.5 , +0.8 )
f*(y2) = ( 0.2 , 0.1 )
Since 2.0 <= 5.0, we get 0.5 >= 0.2 (reversal on magnitudes).
C) Jensen (convex square).M takes values {0, 2} with probabilities {0.5, 0.5}; alignments {a0, a2} arbitrary.
E[M] = 1, E[ M^2 ] = 2
f( E[M] ) = 1^2 = 1 <= E[f(M)] = 2
Symbolic: f*( E_s[X] ) = ( 1 , a_bar ) and E_s[ f*(X) ] = ( 2 , a_bar,f ).
The inequality is on magnitudes 1 <= 2; alignment tags are recorded, not compared.
Guards and policy (manifest)
order_metric = "magnitude" # for inequality steps; "S_beta" allowed with slice/assumptions
domain = list of intervals where f is monotone/convex
branch_policy = "declare_per_interval"
expectation.alignment = "rapidity_mean" # weights as in 2.44, if publishing a_bar
report_fields = ["phi(lhs)","phi(rhs)","alignment_summaries"]
zero_policy = "canonical_zero" # (0,+1) when a magnitude is exactly 0
Notes.
- For Jensen or convexity claims, state the domain (e.g.,
(0,inf)forlog). - If using
S_betaorder, restrict to equal-alignment slices or include explicit assumptions (e.g.,a1=a2).
Takeaway
Inequalities through functions in SSM track classical signs on the magnitude channel: increasing maps preserve order, decreasing maps reverse it, and Jensen’s inequality holds exactly on magnitudes. Alignment is carried or summarized independently, providing auditability without perturbing the classical inequality logic. Collapse (a = +1) yields the standard results verbatim.
Navigation
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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. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
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