Make the prime gap entry points runnable again - #204
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The (hypotheses, exp_pairs) signature replaced a hardcoded list of exponent pairs with "Must supply either hypotheses or exp_pairs", but the three callers - compute_prime_excep, prove_prime_gap2 and compute_best_zero_density - all call it with no arguments, so each dies there. Keep the pairs as a named constant and use them when the caller gives neither argument. They are the pairs the numerical optimisation in approx_bourgain_ep_to_zd found, and passing a hypothesis set instead is not equivalent: the hull computed from the literature set alone yields a weaker bound at 223 of 315 sampled sigma in [3/4, 1), and covers only sigma >= 53/66 rather than sigma >= 3/4.
prime_excep(hypotheses, theta, DISCRETIZATION=100) never reads theta: the loop body opens with "theta = j / DISCRETIZATION", so the parameter is shadowed on the first iteration. Its only caller, compute_prime_excep, passes just the hypothesis set and so raised a TypeError. Remove the parameter and say in the docstring that the function sweeps theta and plots the bound.
compute_gap2 takes each stationary point of alpha and beta and calls float() on the value there. sympy returns a real root of a cubic in the casus irreducibilis form - a sum of complex radicals - so float() raises "Cannot convert complex to float" and prove_prime_gap2 dies on the last few sigma intervals. Evaluate numerically and drop a residual imaginary part below 1e-9, and select the stationary points the same way, so a root whose realness sympy cannot decide symbolically is still used when it evaluates real.
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compute_prime_excep,prove_prime_gap2andcompute_best_zero_densityeach die on the first substantial line they reach. Three separate causes, one per commit:bourgain_ep_to_zdlost its default pairs. The(hypotheses, exp_pairs)signature replaced the hardcoded list withValueError: Must supply either hypotheses or exp_pairs, and all three callers call it with no arguments. The pairs are now a named constant used when the caller supplies neither argument.Passing a hypothesis set instead is not equivalent, which is why I did not simply do that at the call sites: on the literature-only exponent-pair set the computed hull gives a weaker bound at 223 of 315 sampled σ in [3/4, 1), and covers only σ ≥ 53/66 rather than σ ≥ 3/4. The pairs in the constant are the ones the numerical optimisation in
approx_bourgain_ep_to_zdfound.prime_excephas a deadthetaparameter. The loop body opens withtheta = j / DISCRETIZATION, shadowing it, and the only caller does not pass it, so the call was aTypeError. Dropped, with the docstring saying the function sweeps θ.compute_gap2callsfloat()on a real root written with complex radicals. sympy returns the real root of a cubic in casus irreducibilis form, sofloat(beta.subs(x, p))raisesCannot convert complex to floaton the last few σ intervals. Values and stationary points now go through a small helper that evaluates numerically and discards an imaginary part below 1e-9.Verified locally on python 3.13 with
pycddlib<3:derived.compute_prime_excep()runs to completion (~151 s) and prints the zero-density and energy tables plus the μ_PNT(θ) plot data;derived.prove_prime_gap2()runs to completion (~123 s), ending at[1011/1012, 1) 1.4348 …;zd.bourgain_ep_to_zd()with no arguments again returns the 10 estimates startingA(x) ≤ 11/(12(4x-3))on [3/4, 14/15).