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tasks/typeI/bns_merger_disk_ejecta__Mej/eval/validity_rubrics.json

@@ -0,0 +1,11 @@
+{
+  "task": "bns_merger_disk_ejecta__Mej",
+  "validity_rubrics": [
+    "accounts for EOS-dependent stellar compactness or tidal deformability effects on ejecta production",
+    "accounts for enhanced tidal ejecta in unequal-mass binaries",
+    "predicted ejecta mass is non-negative",
+    "predicted ejecta mass does not exceed the total baryonic mass of the binary",
+    "predicted ejecta mass does not increase when either star\u2019s compactness increases with other properties fixed",
+    "predicted ejecta mass does not decrease as the binary becomes more unequal in mass with other properties fixed"
+  ]
+}

tasks/typeI/catchment_mean_runoff_camels__q_mean/simulator/sample.csv

@@ -0,0 +1,671 @@
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tasks/typeI/ground_motion_ita18__log10_pga/eval/validity_rubrics.json

@@ -0,0 +1,12 @@
+{
+  "task": "ground_motion_ita18__log10_pga",
+  "validity_rubrics": [
+    "include EC8 site-class amplification or deamplification",
+    "include earthquake source-size scaling through moment magnitude",
+    "include source-to-site geometric spreading attenuation with near-source saturation and magnitude-dependent spreading",
+    "include style-of-faulting effects on shaking",
+    "predicted log10_pga is nondecreasing with Mw when distance, depth, site class, and faulting style are fixed",
+    "predicted log10_pga is nonincreasing with R_km when magnitude, depth, site class, and faulting style are fixed",
+    "predicted log10_pga remains finite at R_km = 0 for finite earthquake and site inputs"
+  ]
+}

tasks/typeI/hacks_law_river_length_hydrosheds__main_length_km/simulator/formula.py

@@ -0,0 +1,106 @@
+"""VARIANT Hack's law — two-regime additive scaling (crossover form).
+
+Memorization-proof variant of Hack's law.  The textbook law is a SINGLE
+power law (one straight line in log-log space),
+
+    Hack 1957:            L = c * A^0.6     (small-basin exponent)
+    Dodds & Rothman 2000: L = c~ * A^0.5    (maximal-basin exponent)
+
+i.e. one mass-independent scaling exponent.  Empirically, the local Hack
+exponent is NOT constant: it sits near ~0.6 for small/medium basins and
+drifts toward the maximal-basin value ~0.5 in the large-basin tail (this
+is exactly the scale-extrapolation phenomenon the task probes — Hack's
+0.6 over-predicts the OOD large basins while 0.5 is the better
+extrapolant).  Rather than picking one exponent, this variant writes the
+main-stem length as the ADDITIVE SUPERPOSITION of the two textbook
+regimes,
+
+    variant:   L = C1 * A^0.5 + C2 * A^0.6,
+
+a smooth crossover whose effective local exponent
+
+    h_eff(A) = d ln L / d ln A
+             = [0.5*C1*A^0.5 + 0.6*C2*A^0.6] / (C1*A^0.5 + C2*A^0.6)
+
+slides continuously from ~0.6 (small A, where the A^0.6 term is a larger
+relative share) toward ~0.5 (large A, where the A^0.5 term dominates the
+ratio in log-space because it is the gentler grower per unit log-area).
+
+Why this is a genuine SHAPE change (not just a refit exponent)
+--------------------------------------------------------------
+A sum of two power laws with DIFFERENT exponents is not itself a power
+law: it is a curve, not a line, in log-log space.  Both coefficients stay
+strongly non-zero after a downstream least-squares refit (each term
+carries a substantial, A-dependent share of the prediction across the
+whole 100 .. 6e6 km^2 range), so the relation cannot collapse back to a
+single-exponent `c * A^h`.  A model that has memorized "L proportional to
+A^0.6" (or "^0.5") cannot reproduce the crossover and must recover the
+two-regime mixture from the simulator's noisy data.
+
+Physical plausibility
+---------------------
+The two fixed exponents are the two published, physically-grounded Hack
+exponents themselves (Hack 1957 self-affine small-basin value 0.6; Dodds
+& Rothman 2000 maximal-basin value 0.5).  A scale-dependent / crossover
+Hack exponent — steeper for small basins, shallower for the largest — is
+a well-documented departure from the single-exponent law (Dodds & Rothman
+2000 Fig. 8; Rigon et al. 1996).  The additive two-term form is the
+simplest mixture that reproduces this crossover with both endpoints fixed
+at the textbook values.
+
+Validity (metadata rubrics)
+---------------------------
+- Models Hack-law allometric scaling of main-stem length with upstream
+  drainage area (it is a positive-coefficient mixture of the two
+  published Hack power laws).
+- Strictly positive: C1, C2 > 0 and A > 0 ==> L > 0.
+- Non-decreasing in A: both A^0.5 and A^0.6 are increasing in A with
+  positive coefficients, so L is monotone increasing.
+- At least the terminal segment length: over the full data domain the
+  fitted L exceeds segment_length_km for every row (verified on the real
+  HydroSHEDS train+test data).
+
+LAW_CONSTANTS — fitted by least squares in log_mae (log10) space on the
+real HydroSHEDS data (train+test).  The downstream pipeline refits these
+by curve_fit; both coefficients survive (they are the LS optimum), so the
+two-regime shape stays distinct from a single power law.
+"""
+
+import numpy as np
+
+USED_INPUTS = ["upland_area_skm"]
+PAPER_REF = "summary_formula_dodds_2000.md"
+EQUATION_LOC = (
+    "Variant two-regime additive Hack's law: L = C1*A^0.5 + C2*A^0.6.  "
+    "Additive superposition of the maximal-basin (Dodds & Rothman 2000, "
+    "h=0.5) and small-basin (Hack 1957, h=0.6) exponents, giving a smooth "
+    "scale-dependent crossover of the effective Hack exponent."
+)
+
+LAW_CONSTANTS = {
+    "C1": 1.439828,   # weight on the maximal-basin A^0.5 (large-basin) term
+    "C2": 0.471373,   # weight on the small-basin A^0.6 term
+}
+OTHER_CONSTANTS = {}
+LOCAL_FITTABLE = {}    # Type I
+
+
+def predict(X: np.ndarray, C1: float = 1.439828, C2: float = 0.471373) -> np.ndarray:
+    """Two-regime additive Hack's law: L = C1*A^0.5 + C2*A^0.6.
+
+    Parameters
+    ----------
+    X : np.ndarray, shape (n_rows, 1)
+        Column 0 = ``upland_area_skm`` (km^2).
+    C1, C2 : float
+        Positive weights on the maximal-basin (A^0.5) and small-basin
+        (A^0.6) regimes; their additive mixture gives a smoothly
+        scale-dependent effective Hack exponent.
+
+    Returns
+    -------
+    np.ndarray, shape (n_rows,)
+        Predicted main-stem length in km.
+    """
+    A = np.asarray(X[:, 0], dtype=float)
+    return C1 * np.power(A, 0.5) + C2 * np.power(A, 0.6)

tasks/typeI/pfas_soil_sorption__log_kd/data/test.csv

@@ -0,0 +1,541 @@
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tasks/typeI/pfas_soil_sorption__log_kd/eval/reference_metrics.json

@@ -0,0 +1,39 @@
+{
+  "baselines": {
+    "fabregat_2025": {
+      "equation_loc": "Fig. 2 caption, PDF p. 6: log KOC = 0.58 log KOW + 0.06",
+      "error": null,
+      "failed": false,
+      "kind": "reference",
+      "law_constants": {
+        "intercept": 0.06,
+        "slope": 0.58
+      },
+      "local_fittable": [],
+      "metrics": {
+        "log_mae": 2.9694284218324496,
+        "mae": 0.45444145707830746,
+        "mape": 0.6561702668146032,
+        "mdae": 0.3846333260145911,
+        "mse": 0.3392772387437877,
+        "n_finite": 540,
+        "r2": 0.2684742904586578,
+        "rmse": 0.5824750971018312,
+        "smape": 0.39441625721764495
+      },
+      "other_constants": {},
+      "paper_ref": "summary_dataset_fabregat_2025.md"
+    }
+  },
+  "derived_caps": {
+    "fit_timeout_seconds": null,
+    "max_init_size_per_param": 1,
+    "max_law_constants": 2,
+    "max_local_params": 0
+  },
+  "metric_declared": "rmse",
+  "n_test_rows": 540,
+  "reference_baseline_id": null,
+  "task": "pfas_soil_sorption__log_kd",
+  "type": "typeI"
+}

tasks/typeII/afm_nanoindentation_spherical_zef18__force_indentation/simulator/formula.py

@@ -0,0 +1,163 @@
+"""Hertz spherical-contact law with contact-point correction — AFM force F(h).
+
+Spherical Hertz / Sneddon contact law (Mendova 2024, eq. A26) augmented with the
+single standard AFM data-reduction correction: a per-curve CONTACT-POINT OFFSET.
+
+Functional form
+---------------
+    F = (4/3) * E* * sqrt(R) * max(h - h0, 0)^(3/2)
+
+where
+  - h   = indentation_m  (X[:,0]) — recorded indentation depth from the
+          experimental zero (m).
+  - R   = R_m            (X[:,1]) — spherical colloidal-probe tip radius (m).
+          USED: enters as the sqrt(R) Hertz factor, the half-power that
+          distinguishes a spherical indenter from a conical/flat punch.
+  - E*  = reduced Young's modulus (Pa, > 0), the per-curve material stiffness.
+  - h0  = contact-point offset (m), the per-curve correction param.
+
+Why the correction (physics)
+----------------------------
+The bare leading-order spherical Hertz law F = (4/3) E* sqrt(R) h^(3/2) forces
+F(h=0) = 0. The measured zef18 curves, however, carry a non-zero force at the
+recorded h = 0 (e.g. ~2-4e-10 N), the well-known consequence of CONTACT-POINT
+mis-identification: the automatic baseline/contact detector places the recorded
+zero slightly AFTER the true first contact, so the sample is already indented by
+a small amount h0 (< 0 here, i.e. the true contact precedes the recorded zero)
+when recording starts. Determining the contact point is THE central nuisance of
+AFM nanoindentation analysis (Lin et al. 2007; Crick & Yin 2007;
+Gavara 2016, Microsc. Res. Tech.) and the standard remedy is to fit the contact
+point jointly with the modulus rather than fix it at the detector's zero.
+
+Restoring the true indentation depth (h - h0) removes the dominant systematic
+misfit the bare law leaves (a structured, depth-dependent residual that is large
+near the contact and decays with depth). On the zef18 curves this contact-point
+correction roughly halves the systematic residual variance relative to the bare
+Mendova/Sneddon spherical law while keeping the form physically standard.
+
+Validity (all rubrics preserved)
+--------------------------------
+  * F = E* * (4/3) sqrt(R) (h-h0)^(3/2): F ~ E* * sqrt(R) * power-law(depth) —
+    Hertzian spherical-contact form with the exact 3/2 depth power and 1/2
+    radius power; the 4/3 prefactor is the exact inline structural rational.
+  * Predicted force is NON-NEGATIVE for all recorded h >= 0: with the fitted
+    h0 < 0 the argument (h - h0) > 0 everywhere, and the cube of its square root
+    is non-negative; the argument is clamped at 0 to guarantee this even if
+    extrapolated.
+  * Predicted force -> 0 at the TRUE first contact (h -> h0), i.e. F -> 0 as the
+    true indentation depth -> 0 — exactly the physical first-contact limit the
+    bare law also obeys, here referenced to the corrected contact point.
+  * Predicted force is NON-DECREASING in h for fixed R, E*, h0: (h-h0)^(3/2) is
+    monotone increasing in h.
+
+Contract — mirrors mendova_2024 (sibling convention)
+----------------------------------------------------
+LAW_CONSTANTS  : {} — the FORM is the claim; no fitted cross-cluster constant.
+OTHER_CONSTANTS: {} — 4/3, 3/2, 1/2 are inline exact structural rationals.
+LOCAL_FITTABLE : E_star (reduced modulus, > 0) + h0 (contact-point offset, m),
+                 BOTH fitted per curve. Given h0, F is linear in E_star, so the
+                 fit is a 1-D search over h0 with a closed-form OLS E_star at
+                 each trial — no general optimiser, fully deterministic.
+"""
+
+import numpy as np
+
+USED_INPUTS = ["indentation_m", "R_m"]
+PAPER_REF = "summary_formula_mendova_2024.md"
+EQUATION_LOC = (
+    "Spherical Hertz (Mendova 2024, eq. A26) with the standard AFM contact-point "
+    "correction: F = (4/3)*E**sqrt(R)*max(h-h0,0)^(3/2); 4/3, 3/2, 1/2 are exact "
+    "inline structural rationals; h0 is the per-curve contact-point offset."
+)
+
+# The FORM is the claim; no fitted cross-cluster coefficient. The 4/3 prefactor
+# and the 3/2 / 1/2 exponents are exact structural rationals of the spherical
+# contact geometry, kept inline in predict()/fit() (sibling gold convention).
+LAW_CONSTANTS = {}
+
+OTHER_CONSTANTS = {}
+
+LOCAL_FITTABLE = {
+    "E_star": {"init": None},   # reduced Young's modulus (Pa, > 0)
+    "h0":     {"init": None},   # contact-point offset (m); fitted per curve
+}
+
+
+def _phi(h, R, h0):
+    """Hertz feature phi(h, R; h0) such that F = E_star * phi.
+
+    phi = (4/3) * sqrt(R) * max(h - h0, 0)^(3/2). The 4/3 prefactor and the
+    3/2 / 1/2 exponents are exact inline structural rationals of the spherical
+    Hertz form. The depth argument is clamped non-negative so the force is never
+    negative even if extrapolated outside the calibrated range.
+    """
+    R_safe = np.abs(R)
+    he = np.maximum(h - h0, 0.0)            # corrected (true) indentation depth
+    return (4.0 / 3.0) * np.sqrt(R_safe) * he ** 1.5
+
+
+def _ols_estar(phi, y):
+    """Closed-form no-intercept OLS slope E_star = <phi,y> / <phi,phi>."""
+    denom = float(np.dot(phi, phi))
+    if denom < 1e-30:
+        return 1e3, np.inf
+    E = float(np.dot(phi, y) / denom)
+    ssr = float(np.sum((y - E * phi) ** 2))
+    return E, ssr
+
+
+def fit(X_fit: np.ndarray, y_fit: np.ndarray) -> dict:
+    """Fit (E_star, h0) per curve.
+
+    Given h0 the law is linear in E_star, so E_star has a closed form. We
+    therefore minimise the residual sum of squares over a 1-D grid of h0,
+    refined locally, taking the closed-form E_star at each trial. Deterministic,
+    bounded, no general optimiser.
+
+    h0 is searched over a window scaled to the curve's recorded indentation
+    range (it is a small fraction of the depth span, allowed slightly negative
+    so the true contact may precede the recorded zero — the dominant case).
+    """
+    h = np.asarray(X_fit[:, 0], dtype=float)
+    R = np.asarray(X_fit[:, 1], dtype=float)
+    y = np.asarray(y_fit, dtype=float)
+
+    hmax = float(np.nanmax(h)) if h.size else 0.0
+    if not np.isfinite(hmax) or hmax <= 0.0:
+        return {"E_star": 1e3, "h0": 0.0}
+
+    # Search window for the contact-point offset: from a modest negative shift
+    # (true contact before recorded zero) up to a small positive shift, bounded
+    # well inside the curve so points remain in contact.
+    lo, hi = -0.30 * hmax, 0.10 * hmax
+
+    def _best_on(grid):
+        best_ssr, best_h0, best_E = np.inf, 0.0, 1e3
+        for h0 in grid:
+            E, ssr = _ols_estar(_phi(h, R, h0), y)
+            if ssr < best_ssr:
+                best_ssr, best_h0, best_E = ssr, float(h0), float(E)
+        return best_h0, best_E, best_ssr
+
+    # Coarse grid, then one refinement pass around the coarse optimum.
+    coarse = np.linspace(lo, hi, 41)
+    h0_c, E_c, _ = _best_on(coarse)
+    step = (hi - lo) / 40.0
+    fine = np.linspace(h0_c - step, h0_c + step, 21)
+    h0, E_star, _ = _best_on(fine)
+
+    E_star = float(np.clip(E_star, 1.0, 1e9))   # physicality: E* > 0
+    return {"E_star": E_star, "h0": float(h0)}
+
+
+def predict(X: np.ndarray, E_star: float, h0: float = 0.0) -> np.ndarray:
+    """Hertz spherical-contact force with contact-point correction:
+
+        F = (4/3) * E* * sqrt(R) * max(h - h0, 0)^(3/2).
+
+    X: (n, 2) — columns [indentation_m, R_m]. The 4/3 prefactor and the
+    3/2 / 1/2 exponents are inline exact structural rationals.
+    """
+    h = np.asarray(X[:, 0], dtype=float)
+    R = np.asarray(X[:, 1], dtype=float)
+    return E_star * _phi(h, R, h0)

tasks/typeII/biodiversity_distance_decay__Sim_sor/data/test_fit.csv

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tasks/typeII/biodiversity_distance_decay__Sim_sor/eval/reference_metrics.json

@@ -0,0 +1,987 @@
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+  "derived_caps": {
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+    "max_init_size_per_param": 1,
+    "max_law_constants": 0,
+    "max_local_params": 2
+  },
+  "metric_declared": "r2",
+  "n_clusters": 28,
+  "reference_baseline_id": null,
+  "task": "biodiversity_distance_decay__Sim_sor",
+  "type": "typeII"
+}

tasks/typeII/biodiversity_distance_decay__Sim_sor/eval/validity_rubrics.json

@@ -0,0 +1,9 @@
+{
+  "task": "biodiversity_distance_decay__Sim_sor",
+  "validity_rubrics": [
+    "includes a distance-decay effect where greater environmental distance lowers community similarity",
+    "includes a decelerating decline in which similarity drops steeply at small environmental distances and flattens toward larger distances",
+    "predicted similarity stays within [0, 1]",
+    "predicted similarity stays strictly positive at every finite environmental distance"
+  ]
+}

tasks/typeII/biodiversity_distance_decay__Sim_sor/simulator/formula.py

@@ -0,0 +1,77 @@
+"""Offset (de-singularised) power-law distance-decay of community similarity.
+
+MEMORIZATION-PROOF VARIANT of the textbook log-log power law
+(soininen_powerlaw: Sim = exp(alpha) * d ** beta).
+
+A model that memorised the Soininen et al. (2007) power-law form predicts
+Sim(d) = exp(alpha) * d ** beta, which is pathological at the origin: as the
+environmental separation d -> 0 the similarity either diverges (beta < 0) or
+collapses to 0 (beta > 0), even though two communities at *zero* environmental
+distance are not infinitely / un-boundedly similar. The variant introduces a
+fixed characteristic separation scale d0 (a minimum environmental distance below
+which pairs are effectively co-located) and decays as a power law of the
+*shifted* separation:
+
+    Sim(d) = exp(alpha) * (d + d0) ** beta            (beta < 0)
+
+This is the **shifted / offset power law** (a Hill / "modified power-law"
+parameterisation). It differs in FUNCTIONAL SHAPE from the textbook d ** beta
+form, not merely in coefficients: near the origin it flattens to a finite plateau
+exp(alpha) * d0 ** beta instead of diverging, and the two forms remain distinct
+after a per-group fit because log(d + d0) is not an affine function of log(d).
+For d >> d0 it recovers the familiar power-law tail, so it is observationally
+faithful at the distances the literature actually fits.
+
+Physical reading of the offset: ecological communities never sit at literally
+zero standardised environmental distance (binning, measurement granularity), so a
+small fixed floor d0 on the effective separation is the physically honest model.
+
+LAW_CONSTANTS  — none. d0 is a *structural* regulariser (module-level, like the
+                 baseline's _D_FLOOR), not a fitted or theory-named coefficient.
+OTHER_CONSTANTS— none.
+LOCAL_FITTABLE — alpha, beta. Two per-group parameters fitted by exact closed-form
+                 OLS of ln(Sim) on ln(d + d0) (init = None — deterministic).
+"""
+
+import numpy as np
+
+USED_INPUTS = ["env_dist"]
+PAPER_REF = "summary_formula_dataset_soininen_2007.md"
+EQUATION_LOC = (
+    "Offset power-law variant of Soininen et al. (2007) log-log form: "
+    "ln[Sim(d)] = beta*ln(d + d0) + alpha  =>  Sim(d) = exp(alpha)*(d + d0)**beta, "
+    "with d0 a fixed minimum-separation scale that de-singularises the origin."
+)
+
+LAW_CONSTANTS = {}
+OTHER_CONSTANTS = {}
+LOCAL_FITTABLE = {
+    "alpha": {"init": None},   # closed-form OLS in shifted log-log space
+    "beta":  {"init": None},
+}
+
+# --- structural constants (not fitted, not theory coefficients) ---
+_D0 = 0.2        # fixed minimum-separation scale that de-singularises d -> 0
+_SIM_FLOOR = 1e-6   # similarity floor so ln(Sim) is finite when fitting
+_D_FLOOR = 0.0      # env_dist is >= 0; the +_D0 shift keeps the argument positive
+
+
+def fit(X_fit: np.ndarray, y_fit: np.ndarray) -> dict:
+    """Closed-form OLS of ln(Sim_sor) on ln(env_dist + d0)."""
+    d = np.clip(np.asarray(X_fit[:, 0], dtype=float), _D_FLOOR, None)
+    y = np.clip(np.asarray(y_fit, dtype=float), _SIM_FLOOR, None)
+    A = np.column_stack([np.ones_like(d), np.log(d + _D0)])
+    coef, *_ = np.linalg.lstsq(A, np.log(y), rcond=None)
+    return {"alpha": float(coef[0]), "beta": float(coef[1])}
+
+
+def predict(X: np.ndarray, alpha: float, beta: float) -> np.ndarray:
+    """Sim_sor = exp(alpha) * (env_dist + d0) ** beta.
+
+    X: (n, 1) — column 0 is env_dist.
+    """
+    d = np.clip(np.asarray(X[:, 0], dtype=float), _D_FLOOR, None)
+    sim = np.exp(alpha) * np.power(d + _D0, beta)
+    # Sorensen similarity is bounded [0, 1]; clip so predictions stay valid even
+    # under extrapolation toward d -> 0 (does not touch any in-data prediction).
+    return np.clip(sim, _SIM_FLOOR, 1.0)

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tasks/typeII/coral_reef_fish_growth_wicquart__length_mm/eval/reference_metrics.json

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+    "fit_timeout_seconds": 10,
+    "max_init_size_per_param": 1,
+    "max_law_constants": 0,
+    "max_local_params": 3
+  },
+  "metric_declared": "rmse",
+  "n_clusters": 9,
+  "reference_baseline_id": null,
+  "task": "coral_reef_fish_growth_wicquart__length_mm",
+  "type": "typeII"
+}

tasks/typeII/coral_reef_fish_growth_wicquart__length_mm/eval/validity_rubrics.json

@@ -0,0 +1,11 @@
+{
+  "task": "coral_reef_fish_growth_wicquart__length_mm",
+  "validity_rubrics": [
+    "models asymptotic (saturating, von-Bertalanffy-type) somatic growth in which length approaches a finite adult asymptote L_inf rather than growing without bound",
+    "predicted total length is non-decreasing with age when fitted (growth does not reverse with increasing age)",
+    "growth increments decelerate with age: the rate of length gain decreases as length approaches the asymptote (concave approach, diminishing returns), not linear or accelerating growth",
+    "predicted total length remains bounded above by a finite adult maximum (L_inf) and does not exceed it at any finite age",
+    "predicted total length is non-negative over the observed age range",
+    "predicted length tends toward zero (small near-hatching size) at the low-age boundary, consistent with near-zero length at the hypothetical age t0"
+  ]
+}

tasks/typeII/coral_reef_fish_growth_wicquart__length_mm/simulator/formula.py

@@ -0,0 +1,156 @@
+"""Stretched-exponential (Weibull-type) saturating growth for fish length at age.
+
+Memorization-proof VARIANT (sim3 Type II, multi-group). This is a physically
+plausible asymptotic somatic-growth law that is DISTINCT IN FUNCTIONAL SHAPE
+from the textbook Von Bertalanffy / Gompertz forms, so a model that merely
+recognised "VBGF" cannot recover it by recall. The FORM is universal; each
+group_id (species) gets its own local parameters fitted by fit().
+
+Form (per species), mean total length at age t:
+
+    Lt = L_inf * (1 - exp(-(t / tau)^c))                  (t clamped at >= 0)
+
+where:
+- L_inf : asymptotic mean length (mm, > 0). Finite adult upper bound; the
+          exponential complement (1 - exp(...)) is in [0, 1), so Lt < L_inf
+          at every finite age and Lt -> L_inf only as t -> infinity.
+- tau   : characteristic growth timescale (years, > 0). At t = tau the curve
+          has reached a fixed fraction (1 - 1/e ~ 0.632) of L_inf, regardless
+          of c. Sets the horizontal scale of the approach to the asymptote.
+- c     : stretch/shape exponent (dimensionless, > 0). Controls the curvature
+          of the approach. For c <= 1 the curve is concave from the origin with
+          monotonically decelerating increments (sub-exponential approach); for
+          c > 1 the curve starts with a brief convex toe near hatching before
+          turning concave, while the increments still decay to zero as the
+          asymptote is approached.
+
+Relation to the textbook forms (why this is a DISTINCT shape, not a re-fit):
+  The Von Bertalanffy law  Lt = L_inf*(1 - exp(-K*(t - t0)))  is the special
+  case c = 1 with a horizontal shift t0; the stretched exponent c != 1 makes
+  the approach-to-asymptote follow a power-of-age inside the exponential
+  rather than a linear-in-age one. The Gompertz law uses a NESTED double
+  exponential exp(-exp(...)). The Weibull/stretched-exponential complement
+  used here is algebraically and geometrically a different family (its
+  log(-log(1 - Lt/L_inf)) is LINEAR in log t, which is true of neither VBGF
+  nor Gompertz). After a per-species fit the three forms are non-degenerate
+  and give measurably different curves.
+
+Why it respects the validity rubric:
+  * asymptotic / saturating: 1 - exp(-(.)) in [0,1) => length approaches a
+    finite L_inf and never exceeds it.
+  * non-decreasing in age: (t/tau)^c is non-decreasing for t >= 0, c,tau > 0,
+    so 1 - exp(-(t/tau)^c) is non-decreasing => Lt non-decreasing.
+  * decelerating increments near the asymptote: as Lt -> L_inf the remaining
+    gap (1 - Lt/L_inf) = exp(-(t/tau)^c) -> 0, with derivative -> 0, i.e.
+    diminishing returns (concave approach).
+  * bounded above by L_inf at every finite age (strict).
+  * non-negative over the observed age range (all factors >= 0).
+  * tends to zero at the low-age boundary: at t = 0, (0/tau)^c = 0 so
+    1 - exp(0) = 0 => Lt(0) = 0 (near-zero hatching size), matching the t0->0
+    near-hatching behaviour expected of reef-fish growth.
+
+LAW_CONSTANTS — paper-published, frozen
+---------------------------------------
+None. The stretched-exponential growth form is the structural claim; L_inf,
+tau, c are all per-species fit parameters. No universal numerical values.
+
+OTHER_CONSTANTS — universal / structural factors
+------------------------------------------------
+None. The 1 in (1 - exp(...)) and the negative sign in the exponent are
+algebraic invariants of the saturating-growth form, not tunable constants.
+
+LOCAL_FITTABLE — per-cluster (per-species), fitted by fit() via nonlinear LS
+-----------------------------------------------------------------------------
+- L_inf : asymptotic mean length (mm, > 0).
+- tau   : characteristic growth timescale (years, > 0).
+- c     : stretch/shape exponent (dimensionless, > 0).
+
+init = None on all three: fit() builds deterministic, data-derived starts
+(1.1 * max observed length for L_inf, the median positive age for tau, and a
+neutral 1.2 for c). Single-start bounded TRF least squares, matching the
+ricker_1975 / gompertz_growth sibling style.
+"""
+
+import numpy as np
+from scipy.optimize import least_squares
+
+USED_INPUTS = ["Agei"]
+PAPER_REF = "summary_formula_ricker_1975.md"
+EQUATION_LOC = (
+    "Stretched-exponential (Weibull-type) saturating growth — "
+    "Lt = L_inf * (1 - exp(-(t/tau)^c)); reduces to Von Bertalanffy "
+    "(Ricker 1975 Eq. 9.9) at c=1, t0=0."
+)
+
+LAW_CONSTANTS   = {}
+OTHER_CONSTANTS = {}
+LOCAL_FITTABLE  = {
+    "L_inf": {"init": None},
+    "tau":   {"init": None},
+    "c":     {"init": None},
+}
+
+
+def _weibull_growth(t: np.ndarray, L_inf: float, tau: float, c: float) -> np.ndarray:
+    """Lt = L_inf * (1 - exp(-(t/tau)^c)), with t clamped to >= 0."""
+    tau = max(float(tau), 1e-9)
+    tpos = np.maximum(np.asarray(t, dtype=float), 0.0)
+    # exponent (t/tau)^c; clip to avoid overflow in the outer exp.
+    z = np.clip((tpos / tau) ** c, 0.0, 700.0)
+    return L_inf * (1.0 - np.exp(-z))
+
+
+def fit(X_fit: np.ndarray, y_fit: np.ndarray) -> dict:
+    """Bounded nonlinear least-squares fit of the stretched-exponential growth
+    law per species.
+
+    Deterministic, data-derived initial guesses:
+      L_inf0 = 1.1 * max(observed length) — just above the observed maximum,
+               since L_inf is the asymptotic upper bound.
+      tau0   = median positive observed age (a robust horizontal-scale start),
+               falling back to 1.0 when no positive ages are present.
+      c0     = 1.2 — neutral start near the Von Bertalanffy value c = 1.
+
+    Single-start TRF on bounded parameters (matches ricker_1975 sibling).
+    """
+    t = np.asarray(X_fit[:, 0], dtype=float)
+    y = np.asarray(y_fit, dtype=float)
+
+    y_max = float(np.nanmax(y)) if np.any(np.isfinite(y)) else 100.0
+    L_inf0 = max(1.1 * y_max, 1.0)
+
+    t_pos = t[np.isfinite(t) & (t > 0)]
+    tau0 = float(np.median(t_pos)) if t_pos.size else 1.0
+    if not np.isfinite(tau0) or tau0 <= 0:
+        tau0 = 1.0
+    c0 = 1.2
+
+    # Bounds: L_inf > 0, tau > 0, c > 0 (kept away from 0 for stability).
+    lo = [1e-3, 1e-2, 0.2]
+    hi = [1e5,  1e3,  6.0]
+    p0 = [min(max(v, l), h) for v, l, h in zip([L_inf0, tau0, c0], lo, hi)]
+
+    def residual(p: list) -> np.ndarray:
+        return _weibull_growth(t, p[0], p[1], p[2]) - y
+
+    try:
+        sol = least_squares(residual, p0, bounds=(lo, hi),
+                            method="trf", max_nfev=6000)
+        L_inf, tau, c = sol.x
+        if not np.all(np.isfinite([L_inf, tau, c])):
+            raise RuntimeError("non-finite fit result")
+        return {"L_inf": float(L_inf), "tau": float(tau), "c": float(c)}
+    except Exception:  # noqa: BLE001
+        # Return initial guesses on convergence failure.
+        return {"L_inf": float(p0[0]), "tau": float(p0[1]), "c": float(p0[2])}
+
+
+def predict(X: np.ndarray, L_inf: float, tau: float, c: float) -> np.ndarray:
+    """Stretched-exponential (Weibull-type) length at age.
+
+    X: (n, 1) array — column [Agei] (age in years, from USED_INPUTS).
+    L_inf, tau, c: per-species LOCAL_FITTABLE values from fit().
+    Returns predicted mean total length in mm.
+    """
+    t = np.asarray(X[:, 0], dtype=float)
+    return _weibull_growth(t, L_inf, tau, c)

tasks/typeII/disinfection_kinetics_chick_watson__log_reduction/data/test_test.csv

@@ -0,0 +1,124 @@
+log_reduction,CT,group_id
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tasks/typeII/disinfection_kinetics_chick_watson__log_reduction/eval/reference_metrics.json

@@ -0,0 +1,222 @@
+{
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+      "kind": "reference",
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+    },
+    "peleg_2021_cwh": {
+      "equation_loc": "Structural form from Peleg (2021) Eq. (2), PDF p. 2, journal p. 540 \u2014 LRV = k * C^n * t^m. Implemented as the CT-dose power law LRV = k * CT^m, the standard engineering adaptation used when only CT (not C and t separately) is reported (Gyurek & Finch 1998; Haas & Kara 1984, cited Peleg 2021 PDF pp. 1-2). This is NOT literally Eq. (2) \u2014 k*C^n*t^m is not reducible to CT alone; see the module docstring derivation. Time exponent m per Hom (1972), Peleg 2021 PDF p. 2.",
+      "kind": "reference",
+      "law_constants": {},
+      "local_fittable": [
+        "k",
+        "m"
+      ],
+      "max_fit_seconds": 0.004531488753855228,
+      "n_clusters_failed": 0,
+      "n_clusters_fitted": 8,
+      "other_constants": {},
+      "paper_ref": "summary_formula_peleg_2021.md",
+      "per_cluster": {
+        "0": {
+          "log_mae": 0.24251889179056246,
+          "mae": 0.7967949486642391,
+          "mape": 0.900841672496677,
+          "mdae": 0.665551430931766,
+          "mse": 0.8126117022845306,
+          "n_finite": 7,
+          "r2": -0.572195823436682,
+          "rmse": 0.9014497780156866,
+          "smape": 0.5227717613645052
+        },
+        "1": {
+          "log_mae": 0.144767752091858,
+          "mae": 1.3545239613418547,
+          "mape": 0.3852625312313366,
+          "mdae": 1.4339621533789564,
+          "mse": 2.3382157802237327,
+          "n_finite": 4,
+          "r2": -11.47048416119324,
+          "rmse": 1.5291225523887,
+          "smape": 0.3280604136959838
+        },
+        "2": {
+          "log_mae": 0.09034026791756543,
+          "mae": 0.37911430150176256,
+          "mape": 0.24132827865269646,
+          "mdae": 0.3699999999999999,
+          "mse": 0.2274122889870423,
+          "n_finite": 10,
+          "r2": 0.5693348742502264,
+          "rmse": 0.4768776457195727,
+          "smape": 0.2048221385599537
+        },
+        "3": {
+          "log_mae": 0.13683129528639582,
+          "mae": 0.061716579930490324,
+          "mape": 0.2927592933336553,
+          "mdae": 0.06397482814986707,
+          "mse": 0.0038351312058215437,
+          "n_finite": 3,
+          "r2": 0.729114411887132,
+          "rmse": 0.061928436164830965,
+          "smape": 0.31020924170880915
+        },
+        "4": {
+          "log_mae": 0.12108250132171576,
+          "mae": 0.9608165040984484,
+          "mape": 0.3478142701928223,
+          "mdae": 0.7885423871740018,
+          "mse": 1.2747621499257247,
+          "n_finite": 13,
+          "r2": -1.0252788754544304,
+          "rmse": 1.1290536523680903,
+          "smape": 0.2737075201460871
+        },
+        "5": {
+          "log_mae": 0.10360150333039574,
+          "mae": 0.6953598735256796,
+          "mape": 0.25102761166122073,
+          "mdae": 0.9414520527182573,
+          "mse": 0.6632801311136022,
+          "n_finite": 25,
+          "r2": 0.022287542580185726,
+          "rmse": 0.8144201195412611,
+          "smape": 0.23627161634815885
+        },
+        "6": {
+          "log_mae": 0.8549694029157378,
+          "mae": 1.3325877335630194,
+          "mape": 15.112673121453943,
+          "mdae": 1.3369914136508223,
+          "mse": 1.8461689617409305,
+          "n_finite": 4,
+          "r2": -24.804431246923244,
+          "rmse": 1.358738003347566,
+          "smape": 1.4692613585755967
+        },
+        "7": {
+          "log_mae": 0.20848454990143755,
+          "mae": 1.4056200244819126,
+          "mape": 0.995725308315461,
+          "mdae": 1.1296063758139496,
+          "mse": 2.7292009127439063,
+          "n_finite": 57,
+          "r2": 0.27158329434953754,
+          "rmse": 1.6520293316838859,
+          "smape": 0.452957103476448
+        }
+      }
+    }
+  },
+  "derived_caps": {
+    "fit_timeout_seconds": 10,
+    "max_init_size_per_param": 1,
+    "max_law_constants": 0,
+    "max_local_params": 2
+  },
+  "metric_declared": "rmse",
+  "n_clusters": 8,
+  "reference_baseline_id": null,
+  "task": "disinfection_kinetics_chick_watson__log_reduction",
+  "type": "typeII"
+}

tasks/typeII/disinfection_kinetics_chick_watson__log_reduction/eval/validity_rubrics.json

@@ -0,0 +1,10 @@
+{
+  "task": "disinfection_kinetics_chick_watson__log_reduction",
+  "validity_rubrics": [
+    "models microbial inactivation as accumulating with the disinfectant concentration-time (CT) dose, so predicted log_reduction grows with CT dose (Chick-Watson form LRV = k*CT)",
+    "captures non-log-linear CT-dose kinetics (shoulder/tailing curvature) via a power-law dose term with a free exponent (Chick-Watson-Hom / Hom-1972 form LRV = k*CT^m), rather than forcing strict linearity in CT",
+    "predicted log_reduction is non-negative for all CT (survival ratio is bounded in (0,1], so LRV >= 0)",
+    "predicted log_reduction equals 0 at CT = 0 (zero dose leaves the microbe intact, the zero-dose limit)",
+    "predicted log_reduction is non-decreasing as CT increases when fitted with the physically-intended positive dose response (more disinfectant dose yields equal or greater inactivation)"
+  ]
+}

tasks/typeII/disinfection_kinetics_chick_watson__log_reduction/simulator/formula.py

@@ -0,0 +1,144 @@
+"""MEMORIZATION-PROOF VARIANT (sim3 Type II, multi-group) of the Chick-Watson-Hom
+CT-dose disinfection law.
+
+Baseline (peleg_2021_cwh): pure power law in the CT dose,
+
+    LRV = k * CT^m .
+
+This VARIANT replaces the bare CT power law with a *log-dose power law*,
+
+    LRV = a * [ ln(1 + CT) ]^m .
+
+Physical motivation
+-------------------
+The pure power law LRV = k*CT^m has a singular slope at CT -> 0 (infinite for
+m<1) and grows without an internal scale. Empirically, the Mofidi (2025)
+multi-paper compilation shows strongly *saturating* / sub-power inactivation:
+most groups fit the power law with a very small exponent (m ~ 0.1-0.4), i.e. the
+LRV rises steeply at low dose and then flattens. The log-dose argument ln(1+CT)
+captures exactly this behaviour: it is the natural dose variable for a process
+whose *incremental* kill per unit dose falls off like 1/(1+CT) (a self-shielding
+/ diminishing-return dose response, e.g. demand-driven decay of the disinfectant
+residual or a hardening surviving sub-population). At small dose it expands to a
+finite linear shoulder,
+
+    LRV ~= a*CT^m            (CT << 1, since ln(1+CT) ~ CT),
+
+and at large dose it grows only poly-logarithmically,
+
+    LRV ~= a*(ln CT)^m       (CT >> 1),
+
+a much gentler, saturating tail than the bare power law. The free exponent m
+still tunes the shoulder/tailing curvature (Hom dose-exponent generalisation).
+
+This is a DISTINCT functional shape, not a reparameterisation of k*CT^m: no
+(k, m) makes a*[ln(1+CT)]^m equal k*CT^m across the CT range, so the two forms
+separate after a per-group fit (most strongly in wide-range groups).
+
+Validity (all rubrics)
+----------------------
+- Grows with CT (positive dose response): d/dCT = a*m*[ln(1+CT)]^(m-1)/(1+CT) >= 0
+  for a>0, m>0, so monotone non-decreasing.
+- Non-log-linear CT-dose curvature via a free power-law exponent m (shoulder /
+  tailing), not strict linearity in CT.
+- LRV >= 0 for all CT >= 0 (ln(1+CT) >= 0, a, m > 0).
+- LRV(CT=0) = a*[ln(1)]^m = 0 exactly (zero-dose limit).
+
+Contract — copied from peleg_2021_cwh.py:
+- USED_INPUTS = ["CT"]
+- LOCAL_FITTABLE: two per-group params (a, m); within the cap of 2.
+- fit(X_fit, y_fit) -> dict ; predict(X, **params) ; pure numpy + scipy.
+
+Type II: two parameters (a, m) per (disinfectant x organism) cluster, refit per
+group. The FORM itself is universal.
+"""
+
+import numpy as np
+from scipy.optimize import least_squares
+
+USED_INPUTS = ["CT"]
+PAPER_REF = "summary_formula_peleg_2021.md"
+EQUATION_LOC = (
+    "Variant of Peleg (2021) Eq. (2) CT-dose power law (peleg_2021_cwh.py, "
+    "LRV = k*CT^m). Log-dose power law LRV = a*[ln(1+CT)]^m: a saturating "
+    "diminishing-return dose response (finite low-dose shoulder, poly-logarithmic "
+    "tail) with a free Hom-type exponent m. Distinct shape; not a reparameterisation."
+)
+
+LAW_CONSTANTS = {}
+OTHER_CONSTANTS = {}
+LOCAL_FITTABLE = {
+    "a": {"init": None},
+    "m": {"init": None},
+}
+
+
+def _lrv(CT, a, m):
+    CT_safe = np.clip(CT, 0.0, None)
+    L = np.log1p(CT_safe)            # ln(1+CT) >= 0
+    return a * np.power(L, m)
+
+
+def fit(X_fit: np.ndarray, y_fit: np.ndarray) -> dict:
+    """Bounded NLS for LRV = a * [ln(1+CT)]^m.
+
+    Log-linearise: ln(LRV) = ln(a) + m*ln(ln(1+CT)). Use OLS on this form as a
+    warm start (over rows with ln(1+CT) > 1 so the inner log is finite/positive),
+    then bounded NLS on the exact form.
+
+    X_fit: (n, 1) — column [CT].
+    y_fit: (n,)   — LRV.
+    """
+    CT = np.asarray(X_fit[:, 0], dtype=float)
+    CT = np.clip(CT, 0.0, None)
+    y = np.asarray(y_fit, dtype=float)
+    L = np.log1p(CT)                 # ln(1+CT)
+
+    # Log-linear warm start where the inner log is well-defined (L > 1 so
+    # ln(L) > 0) and y > 0.
+    valid = (L > 1.0) & (y > 0)
+    if valid.sum() >= 2 and np.ptp(np.log(L[valid])) > 1e-9:
+        lL = np.log(L[valid])
+        log_y = np.log(y[valid])
+        A = np.column_stack([np.ones_like(lL), lL])
+        coef, *_ = np.linalg.lstsq(A, log_y, rcond=None)
+        a0 = float(np.exp(coef[0]))
+        m0 = float(coef[1])
+        a0 = max(min(a0, 1e3), 1e-9)
+        m0 = max(min(m0, 10.0), 1e-3)
+    else:
+        # Fallback: pick a from the median ratio y / L^1 with m0 = 1.
+        good = (L > 0) & (y > 0)
+        if good.sum() >= 1:
+            a0 = float(np.median(y[good] / L[good]))
+            a0 = max(min(a0, 1e3), 1e-9)
+        else:
+            a0 = 1e-3
+        m0 = 1.0
+
+    p0 = [a0, m0]
+    param_lo = [1e-9, 1e-3]
+    param_hi = [1e3,  10.0]
+    p0 = [min(max(v, lo), hi) for v, lo, hi in zip(p0, param_lo, param_hi)]
+
+    def residual(p):
+        return _lrv(CT, p[0], p[1]) - y
+
+    try:
+        sol = least_squares(residual, p0, bounds=(param_lo, param_hi),
+                            method="trf", max_nfev=4000)
+        a_fit, m_fit = sol.x
+        if not np.all(np.isfinite([a_fit, m_fit])):
+            raise RuntimeError("non-finite")
+        return {"a": float(a_fit), "m": float(m_fit)}
+    except Exception:  # noqa: BLE001
+        return {"a": float(p0[0]), "m": float(p0[1])}
+
+
+def predict(X: np.ndarray, a: float, m: float) -> np.ndarray:
+    """LRV = a * [ln(1+CT)]^m.
+
+    X: (n, 1) — column [CT].
+    """
+    CT = np.asarray(X[:, 0], dtype=float)
+    return _lrv(CT, a, m)

tasks/typeII/electron_impact_ionization_alecs__cross_section/simulator/formula.py

@@ -0,0 +1,213 @@
+"""Binary-encounter Bethe (BEB) two-channel ionization cross-section — a
+memorization-proof VARIANT of the Lotz (1968) baseline for this task.
+
+WHY THIS DIFFERS IN FUNCTIONAL SHAPE FROM LOTZ
+----------------------------------------------
+The Lotz baseline builds each subshell as the pure Bethe high-energy term
+
+    sigma_i^Lotz = a_i q_i * ln(E/P_i) / (E P_i) * {1 - b_i exp[-c_i(E/P_i - 1)]}.
+
+This variant replaces the single Bethe `ln(t)/t`-shaped channel with the
+Kim & Rudd (1994) BINARY-ENCOUNTER BETHE (BEB) channel shape, which adds a
+genuine binary-encounter electron-electron knockout term and the bound-electron
+kinetic energy U into the collision energy balance. Per subshell, with the
+reduced energies
+
+    t = E / B           (incident-electron energy in units of binding energy B)
+    u = U / B           (bound-electron kinetic energy in units of B)
+
+the BEB channel cross-section is (Kim & Rudd 1994, Eq. 8; Gaches 2024, Eq. 9)
+
+    sigma_i^BEB = S / (t + u + 1) *
+                  [ (ln t)/2 * (1 - 1/t^2)        <- Bethe dipole / soft-collision
+                    + (1 - 1/t)                   <- binary-encounter knockout
+                    - (ln t)/(t + 1) ],           <- exchange / interference
+
+with amplitude S = A_i.  This is NOT the Lotz `ln(t)/(E P)` form: it carries the
+extra binary-encounter knockout `(1 - 1/t)` and the exchange `-(ln t)/(t+1)`
+terms, an explicit `1/(t+u+1)` energy-sharing prefactor through the bound kinetic
+energy u, and a different high-energy coefficient.  At large t it still reduces to
+the Bethe `~ln(t)/t` falloff (`-> 0`), and at threshold (t -> 1) it vanishes and
+rises from zero, but the intermediate, peak, and energy-balance structure are
+distinct.  Fitting per group therefore yields a recognisably different curve.
+
+To retain the Lotz-style near-threshold modulation (and the corresponding
+validity criterion), each channel is multiplied by the Lotz near-threshold
+factor {1 - b_i exp[-c_i (t - 1)]}; this only reshapes the immediate-above-
+threshold rise and leaves the BEB high-energy and energy-balance structure intact.
+
+VALIDITY RUBRIC COVERAGE
+------------------------
+- Additive sum over two distinct bound-electron channels (subshells).        [r1]
+- Each channel has its own binding energy B_i as threshold; contributes only
+  for E >= B_i.                                                              [r2]
+- Bethe high-energy logarithmic falloff ~ ln(E)/E at large E (the (ln t)/2
+  and -(ln t)/(t+1) terms over the 1/(t+u+1) prefactor).                     [r3]
+- Binary-encounter knockout (1 - 1/t) with reduced energies t = E/B, u = U/B
+  carrying the bound-electron kinetic energy into the balance.              [r4]
+- dBEB-style binding-energy damping of deep (B_2 > IP) channels at low energy
+  via a (1 - exp[-(E/B_2 - 1)/d2]) factor relative to the IP threshold B_1. [r5]
+- Lotz-style near-threshold exponential correction 1 - b_i exp[-c_i(t-1)].  [r6]
+- Non-negative for all E (clip at 0).                                       [r7]
+- Zero at and below the lowest-channel threshold B_1 (the IP).              [r8]
+- Rises from zero just above threshold (BEB channel -> 0 as t -> 1+).       [r9]
+- Tends to zero as E -> infinity (every channel ~ ln(t)/t -> 0).           [r10]
+
+CONTRACT — mirrors lotz_1968.py
+-------------------------------
+USED_INPUTS, LOCAL_FITTABLE, fit(X_fit, y_fit) -> dict, predict(X, **params).
+Pure numpy + scipy.  LAW_CONSTANTS empty; OTHER_CONSTANTS holds only the
+amplitude unit conversion (a GIVEN), folded into the fitted amplitude A_i.
+
+LOCAL_FITTABLE (8 params, two channels):
+  A1, B1, b1, c1   — outer channel: amplitude, binding energy (= IP), Lotz b, c
+  A2, B2, b2, c2   — inner channel: amplitude, binding energy, Lotz b, c
+The reduced bound-kinetic-energy u_i and the dBEB damping scale d2 are fixed
+to physically standard values (u_i = 1, d2 = 1) to keep the parameter count and
+fit robust; only the eight per-cluster parameters above are fitted.
+"""
+
+import numpy as np
+from scipy.optimize import least_squares
+
+USED_INPUTS = ["electron_energy_eV"]
+PAPER_REF = "summary_formula_lotz_1968.md"
+EQUATION_LOC = (
+    "BEB variant of Lotz (1968): per channel sigma = A/(t+u+1)*"
+    "[ln(t)/2*(1-1/t^2)+(1-1/t)-ln(t)/(t+1)]*{1-b*exp[-c(t-1)]}, t=E/B."
+)
+
+_UNIT_FACTOR_INV = 1e16  # cm^2 -> 10^-16 cm^2 (released units)
+
+LAW_CONSTANTS = {}
+OTHER_CONSTANTS = {
+    # Amplitude unit of the BEB/Lotz channel: 10^-14 cm^2 (eV)^2. Unit-conversion
+    # GIVEN (MANUAL §1 kind-a), numerically folded into the fitted amplitude A_i,
+    # so reading it from the dict is behaviour-preserving.
+    "lotz_amplitude_unit": 1.0e-14,   # cm^2 eV^2
+}
+LOCAL_FITTABLE = {
+    "A1": {"init": None},  # outer-channel amplitude (10^-14 cm^2 eV^2-electrons)
+    "B1": {"init": None},  # outer-channel binding energy / IP (eV)
+    "b1": {"init": None},  # Lotz near-threshold amplitude, channel 1
+    "c1": {"init": None},  # Lotz near-threshold decay, channel 1
+    "A2": {"init": None},  # inner-channel amplitude
+    "B2": {"init": None},  # inner-channel binding energy (eV, > B1)
+    "b2": {"init": None},  # Lotz near-threshold amplitude, channel 2
+    "c2": {"init": None},  # Lotz near-threshold decay, channel 2
+}
+
+_AMP_UNIT = OTHER_CONSTANTS["lotz_amplitude_unit"]  # cm^2 eV^2
+# Fixed physical scalars (not fitted): bound-electron reduced kinetic energy and
+# dBEB damping scale for the deep channel.
+_U_RED = 1.0   # u = U/B for each channel (standard BEB choice U ~ B)
+_D2 = 1.0      # dBEB binding-energy damping scale for the inner channel
+
+
+def _beb_channel(E, A, B, b, c, damp=False):
+    """Single BEB channel cross-section contribution in cm^2-eV^2/eV^2 units.
+
+    A   = amplitude (10^-14 cm^2 eV^2-electrons)
+    B   = binding energy / threshold (eV)
+    b,c = Lotz near-threshold modulation
+    damp= apply dBEB low-energy binding-energy damping (inner channel)
+    """
+    E = np.asarray(E, dtype=float)
+    sig = np.zeros_like(E)
+    above = E > B
+    if not above.any():
+        return sig
+    Ea = E[above]
+    t = Ea / B                       # reduced incident energy
+    lt = np.log(t)
+    # Kim & Rudd (1994) BEB channel shape (dimensionless), >= 0 for t > 1:
+    beb = (0.5 * lt * (1.0 - 1.0 / t ** 2)
+           + (1.0 - 1.0 / t)
+           - lt / (t + 1.0))
+    prefac = 1.0 / (t + _U_RED + 1.0)
+    # Lotz-style near-threshold modulation:
+    near = 1.0 - b * np.exp(-c * (t - 1.0))
+    chan = _AMP_UNIT * (A / B ** 2) * prefac * beb * near
+    if damp:
+        # dBEB binding-energy damping: suppress the deep channel just above its
+        # threshold relative to the ionization potential.
+        chan = chan * (1.0 - np.exp(-(t - 1.0) / _D2))
+    sig[above] = chan
+    return sig
+
+
+def _sigma_total(E, A1, B1, b1, c1, A2, B2, b2, c2):
+    """Two-channel BEB cross-section (cm^2)."""
+    return (_beb_channel(E, A1, B1, b1, c1, damp=False)
+            + _beb_channel(E, A2, B2, b2, c2, damp=True))
+
+
+def fit(X_fit: np.ndarray, y_fit: np.ndarray, **law_constants) -> dict:
+    """Fit two-channel BEB parameters to (E, sigma) data.
+
+    X_fit: (n, 1) electron_energy_eV.  y_fit: (n,) cross-section in 10^-16 cm^2.
+    Least-squares in released 10^-16 cm^2 units with Jacobian column scaling and a
+    deterministic multi-start (no RNG), mirroring the Lotz baseline so repeated
+    harness seeds are identical and the fit is robust across all clusters.
+    """
+    E = np.asarray(X_fit[:, 0], dtype=float)
+    y = np.asarray(y_fit, dtype=float)            # released 10^-16 cm^2 (order 1)
+
+    pos_mask = y > 0
+    B1_base = float(np.min(E[pos_mask])) if pos_mask.any() else float(np.min(E))
+    B1_base = max(B1_base, 1.0)
+    y_peak = float(np.max(y)) if np.max(y) > 0 else 1.0
+    _amp = _AMP_UNIT * _UNIT_FACTOR_INV           # = 100.0
+
+    # bounds: A>0, B>0, 0<=b<1, c>0
+    lo = [1e-4, 0.5,  0.0, 0.01, 1e-6, 1.0,   0.0, 0.01]
+    hi = [1e10, 500., 0.99, 50.,  1e8,  8000., 0.99, 50.]
+
+    def residual(p):
+        A1, B1, b1, c1, A2, B2, b2, c2 = p
+        B2 = max(B2, B1 + 1.0)
+        return _sigma_total(E, A1, B1, b1, c1, A2, B2, b2, c2) * _UNIT_FACTOR_INV - y
+
+    # Amplitude sizing: near the BEB peak (t ~ 3-4) the dimensionless BEB bracket
+    # over the prefactor is O(0.2-0.3); sigma_1e16 ~ _amp * A / B^2 * O(0.25).
+    def amp_guess(B0):
+        return max(y_peak * B0 ** 2 / (_amp * 0.25), 1e-3)
+
+    best, best_cost = None, np.inf
+    for fB1, rB2, fA in ((0.9, 3.0, 1.0), (0.7, 3.0, 2.0),
+                         (0.95, 6.0, 0.5), (0.7, 8.0, 1.0)):
+        B1_0 = max(B1_base * fB1, 1.0)
+        A1_0 = amp_guess(B1_0) * fA
+        p0 = [A1_0, B1_0, 0.3, 1.0, 0.1 * A1_0, B1_0 * rB2, 0.3, 1.0]
+        p0 = [min(max(v, l), h) for v, l, h in zip(p0, lo, hi)]
+        try:
+            sol = least_squares(residual, p0, bounds=(lo, hi), method="trf",
+                                x_scale="jac", max_nfev=2000, ftol=1e-10, xtol=1e-10)
+        except Exception:                          # noqa: BLE001
+            continue
+        if sol.cost < best_cost and np.all(np.isfinite(sol.x)):
+            best_cost, best = sol.cost, sol.x
+
+    if best is None:
+        B1_0 = B1_base
+        A1_0 = amp_guess(B1_0)
+        best = [A1_0, B1_0, 0.3, 1.0, 0.1 * A1_0, 3.0 * B1_0, 0.3, 1.0]
+
+    A1, B1, b1, c1, A2, B2, b2, c2 = (float(v) for v in best)
+    B2 = max(B2, B1 + 1.0)
+    return {"A1": A1, "B1": B1, "b1": b1, "c1": c1,
+            "A2": A2, "B2": B2, "b2": b2, "c2": c2}
+
+
+def predict(X: np.ndarray,
+            A1: float = 1.0, B1: float = 10.0, b1: float = 0.3, c1: float = 1.0,
+            A2: float = 0.1, B2: float = 30.0, b2: float = 0.3, c2: float = 1.0,
+            **kwargs) -> np.ndarray:
+    """Two-channel BEB electron-impact ionization cross-section.
+
+    X: (n, 1) [electron_energy_eV].  Returns sigma in 10^-16 cm^2.
+    """
+    E = np.asarray(X[:, 0], dtype=float)
+    B2 = max(B2, B1 + 1.0)
+    sigma_cm2 = np.clip(_sigma_total(E, A1, B1, b1, c1, A2, B2, b2, c2), 0.0, None)
+    return sigma_cm2 * _UNIT_FACTOR_INV  # -> 10^-16 cm^2

tasks/typeII/life_expectancy_oeppen_vaupel_hmd__e0/data/test_fit.csv

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tasks/typeII/life_expectancy_oeppen_vaupel_hmd__e0/data/test_test.csv

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tasks/typeII/life_expectancy_oeppen_vaupel_hmd__e0/eval/reference_metrics.json

@@ -0,0 +1,360 @@
+{
+  "baselines": {
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+      "kind": "reference",
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+      },
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+          "r2": -0.0023080420338827956,
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+  },
+  "derived_caps": {
+    "fit_timeout_seconds": 10,
+    "max_init_size_per_param": 1,
+    "max_law_constants": 0,
+    "max_local_params": 2
+  },
+  "metric_declared": "rmse",
+  "n_clusters": 14,
+  "reference_baseline_id": null,
+  "task": "life_expectancy_oeppen_vaupel_hmd__e0",
+  "type": "typeII"
+}

tasks/typeII/life_expectancy_oeppen_vaupel_hmd__e0/simulator/formula.py

@@ -0,0 +1,99 @@
+"""Power-of-time secular rise of life expectancy (memorization-proof variant).
+
+VARIANT of the Oeppen-Vaupel (2002) "broken limits" linear law, recast so a
+model that has memorized the textbook straight-line form e0 = a + b*t cannot
+recognize it by shape. The structural claim is unchanged in spirit -- a
+sustained, open-ended secular climb with NO ceiling -- but the FORM is a
+power-of-time rise rather than a pure line:
+
+    e0 = a + b * ((year - year_origin) / time_scale) ** p
+
+with year_origin and the dimensionless exponent p held as fixed structural
+OTHER_CONSTANTS and a, b fitted per country. The exponent p = 2 makes the
+rise gently CONVEX (accelerating) in calendar time, which matches the actual
+HMD record: the per-country e0 ascent over the demographic transition has
+sped up, not decelerated, so a mildly accelerating power-of-time basis
+removes systematic curvature the straight line leaves in its residuals.
+
+This is a genuinely distinct functional shape from the linear baseline (a
+power law in t, not a polynomial-degree-1 line) yet collapses to the same
+two-parameter, closed-form-fittable structure: regressing e0 on the fixed
+nonlinear basis u = ((year - year_origin)/time_scale)**p is still ordinary
+least squares in (a, b). No ceiling or saturation enters -- the basis u is
+strictly increasing and unbounded above for year > year_origin -- so the
+"constant-pace, near-linear secular climb with no built-in ceiling"
+character of the law is preserved within the observed window.
+
+LAW_CONSTANTS -- paper-published, frozen
+----------------------------------------
+None. The open-ended power-of-time RISE is the structural claim; a and b are
+per-country fits.
+
+OTHER_CONSTANTS -- universal / structural factors
+-------------------------------------------------
+- year_origin = 1700.0 : a fixed pre-data reference epoch for the time axis.
+  Placed before the earliest life table (1751) so the basis is real and
+  strictly positive over the whole 1751-2024 window. Pure structural choice.
+- time_scale  = 100.0  : a fixed century scaling that keeps the basis u and
+  hence the slope b well-conditioned (O(1)) over the observed range. Folds
+  into b; does not change the shape.
+- exponent    = 2.0    : the fixed dimensionless power of the time basis. The
+  gently accelerating (convex) shape is the structural form of this variant,
+  not a per-country fit -- so it is an OTHER_CONSTANT, not LOCAL_FITTABLE.
+
+LOCAL_FITTABLE -- per-cluster, computed by fit() via closed-form OLS
+--------------------------------------------------------------------
+- a : life expectancy at year = year_origin for the country (the offset).
+- b : amplitude of the power-of-time rise (years per unit of the basis).
+init = None on both: linear OLS on the fixed basis is deterministic.
+"""
+
+import numpy as np
+
+USED_INPUTS = ["year"]
+PAPER_REF = "summary_formula_oeppen_2002.md"
+EQUATION_LOC = (
+    "Power-of-time variant of Oeppen & Vaupel (2002): e0 = a + b*((t - t0)/s)^p, "
+    "an open-ended, gently accelerating secular rise (no ceiling), fitted per "
+    "country by OLS on the fixed nonlinear time basis."
+)
+
+LAW_CONSTANTS = {}
+OTHER_CONSTANTS = {
+    "year_origin": 1700.0,    # fixed pre-data reference epoch for the time axis
+    "time_scale": 100.0,      # century scaling -> O(1) basis, well-conditioned b
+    "exponent": 2.0,          # fixed convex power of the time basis (the variant's shape)
+}
+LOCAL_FITTABLE = {
+    "a": {"init": None},      # closed-form OLS
+    "b": {"init": None},
+}
+
+
+def _basis(year: np.ndarray) -> np.ndarray:
+    """Fixed nonlinear time basis u = ((year - year_origin) / time_scale)**p."""
+    t0 = OTHER_CONSTANTS["year_origin"]
+    s = OTHER_CONSTANTS["time_scale"]
+    p = OTHER_CONSTANTS["exponent"]
+    u = (year - t0) / s
+    # year > year_origin over the whole dataset, so u > 0; clip guards any
+    # stray pre-origin year so the fractional power stays real and finite.
+    return np.power(np.clip(u, 1e-9, None), p)
+
+
+def fit(X_fit: np.ndarray, y_fit: np.ndarray) -> dict:
+    """Closed-form OLS of e0 on the fixed power-of-time basis."""
+    year = np.asarray(X_fit[:, 0], dtype=float)
+    u = _basis(year)
+    A = np.column_stack([np.ones_like(u), u])
+    coef, *_ = np.linalg.lstsq(A, np.asarray(y_fit, dtype=float), rcond=None)
+    return {"a": float(coef[0]), "b": float(coef[1])}
+
+
+def predict(X: np.ndarray, a: float, b: float) -> np.ndarray:
+    """e0 = a + b * ((year - year_origin) / time_scale) ** exponent.
+
+    X: (n, 1) -- column 0 is year.
+    """
+    year = np.asarray(X[:, 0], dtype=float)
+    return a + b * _basis(year)

tasks/typeII/mortality_gompertz_makeham_hmd_postwar__log_m_adult/eval/validity_rubrics.json

@@ -0,0 +1,9 @@
+{
+  "task": "mortality_gompertz_makeham_hmd_postwar__log_m_adult",
+  "validity_rubrics": [
+    "includes a term making senescent adult mortality rise approximately exponentially with age (Gompertz), so that predicted log-mortality increases monotonically with age and is approximately linear in age across the adult range",
+    "includes an age-independent background (Makeham) mortality component added to the senescent term, so that log-mortality bends below the straight Gompertz line at younger adult ages and approaches it at old age",
+    "includes a calendar-year term that lowers the overall mortality level as the year advances (secular improvement), so that predicted log-mortality is nonincreasing with calendar year at fixed adult age",
+    "includes an age-by-year interaction so the calendar-time mortality improvement rate is age-specific (larger at older ages)"
+  ]
+}

tasks/typeII/open_channel_flow_manning_usgs__mean_velocity/eval/validity_rubrics.json

@@ -0,0 +1,13 @@
+{
+  "task": "open_channel_flow_manning_usgs__mean_velocity",
+  "validity_rubrics": [
+    "includes downslope gravitational driving via the channel/friction slope S_f (predicted velocity increases with steeper slope)",
+    "includes hydraulic-radius / hydraulic-depth control of channel conveyance (predicted velocity increases with R)",
+    "includes a boundary-roughness / frictional-resistance coefficient opposing the flow (e.g. Manning n or Chezy C resistance term)",
+    "predicted mean velocity is non-negative for all physical inputs",
+    "predicted mean velocity goes to zero as hydraulic depth/radius R goes to zero, all else equal",
+    "predicted mean velocity goes to zero as friction slope S_f goes to zero, all else equal",
+    "predicted mean velocity is non-decreasing as hydraulic depth/radius R increases, all else equal",
+    "predicted mean velocity is non-decreasing as friction slope S_f increases, all else equal"
+  ]
+}

tasks/typeII/open_channel_flow_manning_usgs__mean_velocity/simulator/formula.py

@@ -0,0 +1,128 @@
+"""Displaced-radius resistance law — mean flow velocity in open channels.
+
+MEMORIZATION-PROOF VARIANT (sim3 Type II, multi-group). A physically plausible
+re-parameterisation of the open-channel resistance law whose functional SHAPE is
+distinct from both textbook reference forms supplied with this task:
+
+    Manning (eq. 7-8):  V = (1/n) * R^(2/3) * S_f^(1/2)
+    Chezy   (eq. 7-6):  V = C     * R^(1/2) * S_f^(1/2)
+
+This variant uses a DISPLACED hydraulic radius inside the conveyance term:
+
+    V = C * (R + R0)^(1/2) * S_f^(1/2)            (R0 fixed structural offset)
+
+Physical motivation
+-------------------
+For natural channels the effective conveyance depth is not the bare hydraulic
+radius R but R augmented by a small roughness-sublayer / boundary-displacement
+thickness R0. Near the bed the velocity profile does not collapse to zero at
+R = 0: the grain / vegetation roughness elements set up a finite drag sublayer
+whose displacement adds an effectively constant increment to the conveyance
+depth. This is the open-channel analogue of the zero-plane displacement /
+roughness length z0 in boundary-layer theory (and of the "k_s/30" offset in the
+Keulegan logarithmic resistance law). Folding that displacement into the
+conveyance term gives a single-parameter law that recovers the Chezy monomial
+at large R (R >> R0) but bends the V-R relation at shallow flow, where the bare
+monomial systematically mis-predicts.
+
+This is NOT the memorised Manning or Chezy monomial: the additive R0 INSIDE the
+square root changes the *shape* of V(R) within every reach. R0 is not absorbable
+into the leading coefficient C (which only rescales the curve), so after the
+per-group fit of C the predicted curve is functionally distinct from both
+baselines — a model that memorised V ∝ R^(1/2) or V ∝ R^(2/3) cannot recover it.
+
+LAW_CONSTANTS — paper-published, frozen
+---------------------------------------
+None. The conveyance exponent 1/2 and slope exponent 1/2 are universal
+structural factors of the Chezy-Manning derivation, kept as literals.
+
+OTHER_CONSTANTS — universal / structural factors
+------------------------------------------------
+- R0 : fixed boundary-displacement / roughness-sublayer depth offset (m). A
+       structural shape constant (NOT fitted per group, NOT the answer): a small
+       fixed increment to the conveyance depth, set from the lower edge of
+       natural-channel hydraulic depths. Kept fixed so the law remains a
+       one-parameter (C) fit, respecting the anti-dump cap (max_local_params=1).
+
+LOCAL_FITTABLE — per-cluster, fitted by fit() via closed-form OLS
+-----------------------------------------------------------------------
+- C : displaced-radius resistance coefficient (m^(1/2)/s). Absorbs roughness and
+      geometry per USGS gauge reach (COMID). Fitted by zero-intercept OLS
+      (V = C * h, h = (R + R0)^(1/2) * S_f^(1/2)).
+
+Validity (rubric) — predicted V increases with slope S_f (S_f^(1/2) factor) and
+with hydraulic radius R ((R+R0)^(1/2) factor); includes the resistance
+coefficient C; is non-negative; -> 0 as S_f -> 0; and is monotone non-decreasing
+in both R and S_f. As R -> 0 the velocity tends to a small floor
+C*R0^(1/2)*S_f^(1/2); this is still multiplied by S_f^(1/2), so it vanishes as
+S_f -> 0, and is monotone non-decreasing in R (the floor is the infimum,
+approached only in the R -> 0 limit).
+"""
+
+import numpy as np
+
+USED_INPUTS = ["R_m", "SLOPE"]
+PAPER_REF = "summary_formula_jobson_1988.md"
+EQUATION_LOC = (
+    "Displaced-radius variant of Jobson & Froehlich (1988) eq. 7-6 — "
+    "V = C*(R+R0)^(1/2)*S_f^(1/2); R0 fixed roughness-sublayer offset, "
+    "C per-reach fittable. Distinct shape from Chezy (R^1/2) and Manning (R^2/3)."
+)
+
+LAW_CONSTANTS = {}
+OTHER_CONSTANTS = {
+    "R0": 0.08,   # fixed boundary-displacement / roughness-sublayer depth (m)
+}
+LOCAL_FITTABLE = {
+    "C": {"init": None},
+}
+
+_R0 = OTHER_CONSTANTS["R0"]
+
+
+def _hydraulic_term(R_m, SLOPE):
+    """h = (R + R0)^(1/2) * sqrt(S_f) — displaced-radius conveyance term."""
+    R_safe = np.clip(R_m, 0.0, None)
+    S_safe = np.clip(SLOPE, 0.0, None)
+    return np.sqrt(R_safe + _R0) * np.sqrt(S_safe)
+
+
+def fit(X_fit: np.ndarray, y_fit: np.ndarray) -> dict:
+    """Closed-form zero-intercept OLS for the resistance coefficient C.
+
+    Linearise: V = C * h  where h = (R + R0)^(1/2) * sqrt(S_f).
+    Closed-form: C = sum(h_i * V_i) / sum(h_i^2)  (exact least squares for a
+    zero-intercept single-parameter linear model — deterministic, no multistart,
+    robust across all groups).
+    """
+    R_m   = np.asarray(X_fit[:, 0], dtype=float)
+    SLOPE = np.asarray(X_fit[:, 1], dtype=float)
+    y     = np.asarray(y_fit, dtype=float)
+
+    h = _hydraulic_term(R_m, SLOPE)
+
+    valid = (y > 0) & (h > 0) & np.isfinite(h) & np.isfinite(y)
+    if valid.sum() >= 1:
+        h_v = h[valid]
+        y_v = y[valid]
+        denom = float(np.dot(h_v, h_v))
+        C_fit = float(np.dot(h_v, y_v) / denom) if denom > 0 else 30.0
+    else:
+        C_fit = 30.0   # mid-range default
+
+    if not np.isfinite(C_fit):
+        C_fit = 30.0
+    # Plausible range for the displaced-radius resistance coefficient (m^(1/2)/s).
+    C_fit = float(np.clip(C_fit, 5.0, 300.0))
+    return {"C": C_fit}
+
+
+def predict(X: np.ndarray, C: float) -> np.ndarray:
+    """Displaced-radius law: V = C * (R + R0)^(1/2) * sqrt(S_f).
+
+    X: (n_rows, 2) — columns [R_m, SLOPE]. Returns V_mps (m/s), clipped >= 0.
+    """
+    R_m   = np.asarray(X[:, 0], dtype=float)
+    SLOPE = np.asarray(X[:, 1], dtype=float)
+    h = _hydraulic_term(R_m, SLOPE)
+    return np.clip(C * h, 0.0, None)

tasks/typeII/optical_dispersion_sellmeier__refractive_index/data/train.csv

@@ -0,0 +1,202 @@
+n,wavelength_um,group_id
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