"""IsoFLOPs Ladder, Law Fitting & the Forecast — YOUR implementation.

Fill in every `# TODO`. Signatures, docstrings and validation contracts are given; the
numerics are yours.

    pytest test_lab.py -v                       # red until you implement
    LAB_MODULE=solution pytest test_lab.py -v   # the reference (must be green)
    python solution.py                          # the worked example

ORDER OF WORK
  1. fit_line, fit_power_law, fit_quadratic, parabola_vertex   (the numerical spine)
  2. parametric_loss, loss_decomposition, schedule_bias, synthetic_ladder
  3. isoflop_optimum, isoflops_ladder, fit_scaling_exponents   (the IsoFLOPs method)
  4. residuals, squared_loss, huber_loss, fit_parametric       (the surface fit)
  5. analytic_optimum, optimum_exponent
  6. bootstrap_forecast, compute_multiplier_for_loss_delta     (uncertainty)
  7. extrapolation_variance, design_ladder, extrapolation_risk (design)
  8. loss_at_budget, compare_recipes, decision_is_supported    (the decision)

THE MONEY TEST is `test_nonuniform_bias_tilts_the_fitted_exponent`: a NON-UNIFORM
measurement bias must move the fitted exponent, while a UNIFORM one must not (it gets
absorbed into E). That contrast is the Kaplan -> Chinchilla lesson, reproduced.

DETERMINISM: every random draw goes through a seeded `random.Random(seed)`. The fitter
uses deterministic multi-start coordinate descent — no library optimizer, because a
library upgrade that moves your exponents moves a nine-figure recommendation.
"""

import math
import random

# Chinchilla-style parameters, used as the "ground truth" the synthetic ladder samples.
CHINCHILLA = (1.69, 406.4, 0.34, 410.7, 0.28)      # (E, A, alpha, B, beta)


def fit_line(xs, ys):
    """Ordinary least-squares fit of y = slope*x + intercept."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def fit_power_law(xs, ys):
    """Fit y = c * x^k by regressing log y on log x. Returns (c, k)."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def fit_quadratic(xs, ys):
    """Least-squares fit of y = a*x^2 + b*x + c via Cramer's rule on the normal
    equations. No libraries, fully deterministic."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def parabola_vertex(a, b):
    """x at the minimum of a*x^2 + b*x + c. Requires a > 0."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def parametric_loss(n_params, n_tokens, E, A, alpha, B, beta):
    """L = E + A/N^alpha + B/D^beta.

    E    irreducible loss (entropy of the data itself)
    A/N^a capacity term  — shrinks with more parameters
    B/D^b data term      — shrinks with more tokens
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def loss_decomposition(n_params, n_tokens, params=CHINCHILLA):
    """Split a loss into its three sources, with fractions. Reveals which lever to pull."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def schedule_bias(n_tokens, schedule_tokens, magnitude):
    """The LR-decay measurement bias, as a function of how far into the schedule you are.

    A learning-rate schedule decays to ~0 at `schedule_tokens`. Much of the final loss
    improvement happens in that decay. If you READ the loss at `n_tokens` from a run
    whose schedule targets `schedule_tokens > n_tokens`, the LR is still high and the
    measured loss is too HIGH by roughly

        magnitude * (1 - n_tokens / schedule_tokens)

    The critical property is that this is NOT a constant: a constant would simply be
    absorbed into the fitted E and change nothing. It varies across the ladder, and a
    non-uniform bias TILTS the fitted exponents. That is the whole mechanism behind the
    Kaplan -> Chinchilla correction.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def synthetic_ladder(budgets, sizes_per_budget, params=CHINCHILLA,
                     noise_std=0.0, lr_decay_bias=0.0,
                     bias_mode="fixed_schedule", fixed_schedule_tokens=None,
                     seed=0):
    """Generate a ladder of (C, N, D, L) measurements.

    `lr_decay_bias` > 0 injects the schedule-mismatch measurement bias described in
    `schedule_bias`. `bias_mode` selects WHICH points the bias hits hardest, because
    that is what determines how the fit is distorted:

      "fixed_schedule"   every measurement is read from a run whose LR schedule targets
                         `fixed_schedule_tokens` (default: the largest D in the ladder).
                         This is closest to "one run per model size, read intermediate
                         losses" — points with small D are penalized most.

      "per_budget"       the schedule targets the largest D at that budget. The bias is
                         then driven by position WITHIN each IsoFLOPs curve, so large-N
                         (small-D) configs are penalized most.

      "none"             no bias.

    Returns [(C, N, D, L), ...]. All randomness goes through a SEEDED generator, so the
    same seed produces byte-identical output.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def isoflop_optimum(points):
    """Given [(N, D, L), ...] at ONE fixed budget, fit a parabola in log10(N) and
    return the vertex. Returns (N_opt, quadratic_coeffs)."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def isoflops_ladder(ladder):
    """Group a ladder by budget and locate each budget's optimum.

    Input:  [(C, N, D, L), ...]
    Output: [(C, N_opt, D_opt), ...] sorted by C.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def fit_scaling_exponents(isoflops_results):
    """Steps 5-6: fit N_opt ∝ C^a and D_opt ∝ C^b, and check a + b ≈ 1.

    That check is free and catches FLOP-accounting bugs — it follows directly from
    C = 6ND, so any fit violating it is wrong somewhere.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def residuals(points, params, space="log"):
    """points: [(N, D, L_observed), ...]; params: (E, A, alpha, B, beta)."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def squared_loss(rs):
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def huber_loss(rs, delta=1e-3):
    """Quadratic near zero, LINEAR in the tails, so one diverged run cannot dominate
    the fit. This is what Chinchilla used, and it is not incidental."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def fit_parametric(points, init=None, objective="huber", space="log", iters=300,
                   seed_step=0.5, shrink=0.9):
    """Fit L = E + A/N^alpha + B/D^beta by deterministic multi-start coordinate descent.

    No randomness and no library optimizer, so the same input always produces
    byte-identical output — a property the tests assert. (This is not fussiness: if a
    library upgrade can move your fitted exponents, it can move a nine-figure flagship
    recommendation, and your process is not reproducible.)

    `init=None` uses data-driven starting points; pass an explicit tuple to force one.
    Returns (params, final_objective_value).
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def analytic_optimum(total_flops, E, A, alpha, B, beta):
    """Closed-form compute-optimal (N, D) under the parametric law.

    Minimizing L(N, C/6N) over N gives
        N_opt = [ (alpha*A)/(beta*B) * (C/6)^beta ] ^ (1/(alpha+beta))
    which is a power law in C with exponent beta/(alpha+beta).
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def optimum_exponent(alpha, beta):
    """The exponent in N_opt ∝ C^k. Chinchilla's ~0.5 corresponds to alpha ≈ beta."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def bootstrap_forecast(points, target_flops, n_boot=120, seed=0, ci=0.95):
    """Percentile bootstrap CI for the loss forecast at `target_flops`.

    Resample the ladder with replacement, refit, predict, repeat. Analytic intervals
    are unavailable because the model is nonlinear and the noise model is unspecified —
    which is itself one of Feinberg's stated open problems.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def compute_multiplier_for_loss_delta(delta_nats, loss=1.75, irreducible=1.69,
                                      alpha=0.5):
    """Roughly how much extra compute buys `delta_nats` of improvement.

    Near the operating point, L - E ~ C^-alpha, so a fractional compute increase of
    delta / (alpha * (L - E)) buys delta nats. The point of this function is to make
    'is my confidence interval narrow enough?' answerable in dollars.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def extrapolation_variance(log_c_points, target_log_c):
    """Variance of a linear extrapolation to target_log_c, up to a noise constant.

    OLS: Var(y_hat) ∝ 1/n + (x* - xbar)^2 / sum((x - xbar)^2)

    The second term is why SPREAD beats DENSITY: adding a point far from the mean
    grows the denominator quadratically.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def design_ladder(flagship_flops, ladder_budget_fraction=0.03, n_budgets=5,
                  min_flops=1e18):
    """Geometrically spaced ladder budgets consuming a fixed fraction of the flagship.

    Geometric spacing maximizes spread in log space for a given range, which is
    exactly what minimizes extrapolation variance.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def extrapolation_risk(ladder_max_flops, target_flops, ci_width):
    """An honest risk label for a forecast: decades of extrapolation x interval width."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def loss_at_budget(params, total_flops):
    """Compute-optimal loss this recipe achieves at a given budget."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def compare_recipes(baseline_params, candidate_params, target_flops,
                    scan_lo=16.0, scan_hi=28.0, scan_steps=120):
    """Compare two fitted laws AT THE TARGET, and locate any crossover.

    The crossover is the deliverable. 'Candidate wins above 1e23' is actionable;
    'candidate is better' silently assumes a scale, and curves cross.
    """
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError

def decision_is_supported(delta_nats, ci_width_nats):
    """A delta smaller than your uncertainty is not a result. Say so out loud."""
    # TODO: implement. See WARMUP.md and the docstring above.
    raise NotImplementedError


if __name__ == "__main__":
    print("Fill in the TODOs, then compare against: python solution.py")