Purpose. When a pure power law is borderline or inadequate, select among simple local families with a penalized fit score. Keep the magnitude decision conservative; never overstate.
Families (near x0).
1) Power: h(x) ~ c * |x - x0|^p
2) Power*log: h(x) ~ c * |x - x0|^p * L(x)^q, L(x) := log(1/|x - x0|)
3) Essential*: h(x) ~ c * exp( -alpha * |x - x0|^(-r) ), alpha>0, r>0 (*rare, guarded)
Use (3) only when pathology suggests essential-type flatness or blow-up that powers cannot capture.
Fitting lenses (ASCII). Use only samples with x != x0 and |h| > 0. Natural log implied.
# 1) Power (lens):
t := log|x - x0|; y := log|h|
fit: y = (log|c|) + p*t -> slope p, intercept log|c|
# 2) Power*log (two-step lens):
estimate p via Power lens
z := log L(x) = log( log(1/|x - x0|) )
y2 := log|h| - p*log|x - x0|
fit: y2 = (log|c|) + q*z -> slope q, intercept log|c|
# 3) Essential (scanned lens):
choose small grid r_guess in {0.5, 1.0, 1.5}
s := |x - x0|^(-r_guess); yE := -log|h|
fit: yE = alpha*s + const -> slope alpha >= 0
Unified scoring scale (AIC/BIC on y = log|h|). Regardless of lens used to estimate parameters, evaluate predictions on the same scale:
Power: y_hat = log|c| + p*log|x - x0|
Power*log: y_hat = log|c| + p*log|x - x0| + q*log log(1/|x - x0|)
Essential: y_hat = log|c| - alpha*|x - x0|^(-r)
e := y - y_hat
RSS := sum(e^2)
AIC = 2*k + n*log(RSS/n)
BIC = k*log(n) + n*log(RSS/n)
Parameter counts (typical).
Power: k = 2 # c, p
Power*log: k = 3 # c, p, q
Essential: k = 2 # per fixed r_guess (intercept encodes c, plus alpha)
Optional grid penalty: if you scan G distinct r_guess, add a small penalty (e.g., +log G to BIC or +2 to AIC) to remain conservative.
Minimal knobs (declare once; override only if you know better).
score_fn = BIC
delta_score = 2.0 # scores closer than this are a tie
R2_min = 0.85
r_grid = {0.5, 1.0, 1.5} # used only if pathology suggests essential
Lite mode (recommended default).
families considered: Power, Power*log
essential: disabled unless pathology = true
tie handling: use near-tie thresholds from 4A (T_p, T_q) after parameter estimation
defaults: T_p = 1e-3, T_q = 1e-3 # declare in manifest
Windowing rule (for f/g comparability). Fit f and g on identical neighborhoods/radii and sample points whenever possible; otherwise set REG = MULTI and print the conservative headline.
MULTI policy (model ambiguity).
If >=2 families tie within delta_score:
REG = MULTI
headline class: prefer ZERO over FINITE over INF # conservative total order
downstream can disambiguate with more data; do not overstate
Decision logic (magnitude first, then alignment).
1) Fit families 1–2 for both f and g. Consider 3) only if pathology warrants.
2) If a single family wins by > delta_score, decide magnitude using that family’s params:
- Power / Power*log: apply 0over0 rule with (p_f,q_f) vs (p_g,q_g)
(lexicographic on (Delta_p, Delta_q), using near-tie thresholds if declared)
- Essential (rare): if both f and g fit essential with same r (within tie), compare alphas:
if alpha_f > alpha_g -> < 0 , a_div >
if alpha_f = alpha_g -> < c_f / c_g , a_div > # track signs in c_f,c_g
if alpha_f < alpha_g -> < s * inf , a_div >, s := sign(c_f/c_g)
If only one side appears essential or r orders disagree -> REG = MULTI (headline conservative).
3) If ties remain or families disagree in class:
- set REG = MULTI
- headline: conservative class using ZERO < FINITE < INF
- optional audit: print per-family one-liners with (model, params, score deltas)
4) Alignment channel (metadata only):
a_div = tanh( atanh(a_f) - atanh(a_g) ) # clamp before atanh
do not let model disagreement change the headline magnitude
optional stability: report STAB[%] from bootstrap-over-families (separate from a_div)
Sign tracking. While lenses operate on |h|, always track signs separately from raw h. For INF, use DIR+ / DIR- per sign(c_f / c_g).
Print guidance.
# Single-winner:
< m_out , a_div >
SSMS: {CLASS}@{A-tag} DIV[a_div]@{A-tag} {DIR?}
# Disagreement/tie:
Headline: SSMS: {CLASS_conservative} DIV[a_div]@{A-tag} REG=MULTI
Audit (optional, separate lines):
POWER: {CLASS_p} score=...
PWR*LOG: {CLASS_pl} score=...
ESSENTIAL: {CLASS_e} r=... score=...
Safety and fallbacks.
Clamp before rapidity ops: a := clamp(a, -1 + eps_a, +1 - eps_a) # default eps_a = 1e-6
If no family passes R2_min and nested windows show non-shrinking variance:
REG = OSC (oscillatory) or REG = MULTI (NOFIT) rather than forcing a magnitude
Sided behavior:
run chooser on left (x<x0) and right (x>x0) separately
if classes differ -> REG = SIDED and print both one-sided results
Notes on signs and lexicographic consistency.
Magnitude and INF sign follow sign(c_f / c_g).
Lexicographic precedence is preserved: powers (p) dominate logs (q).
Chooser estimates (p,q); classification follows the lexicographic rule on (Delta_p, Delta_q).
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