Issue 198 energy improvement

[chaemin333-max]

Closes #198.

Summary

This PR adds a current best zero-density energy derivation obtained by combining the existing

• $L^2$ large-value estimate
• Heath–Brown large-value energy regions 2a and 2b
• the third Guth–Maynard large-value energy relation
• the $k=2,3,4$ power transforms

The historical function prove_heath_brown_energy_estimate() is left unchanged. Instead, the new bound is implemented separately in prove_heath_brown_guth_maynard_energy_estimate().

Running the combined hypothesis set through the Region engine gives

$$A^*(\sigma) \le \frac{10-11\sigma}{(2-\sigma)(1-\sigma)}, \qquad \frac23\le \sigma\le\frac57.$$ $$A^*(\sigma) \le \frac{3(25-27\sigma)} {8(2-\sigma)(1-\sigma)}, \qquad \frac57\le \sigma\le\frac{315}{433}.$$

and

$$A^*(\sigma) \le \frac{2(45-44\sigma)} {(2\sigma+15)(1-\sigma)}, \qquad \frac{315}{433}\le \sigma\le\frac34.$$

The first piece is the bound proposed in #198. The Region calculation shows that it remains optimal within this combined derivation up to $\sigma=5/7$, after which two further pieces take over.

Blueprint and current-best table

The new theorem and its Region-engine derivation are documented in the zero-density energy chapter.

The current-best table is also updated. In particular, the second new piece remains globally best until its crossover with the existing bound from imp-energy-bound2. Equating the two gives

$$785\sigma^2-1252\sigma+495=0,$$

so the relevant crossover is

$$\sigma_0 = \frac{626-\sqrt{3301}}{785} = 0.724262\ldots.$$

The plotting source now uses this exact expression rather than a rounded decimal.

Implementation details

The new derived function uses exact name literature.find_hypothesis(...) lookups for the three literature energy regions.

I also changed the final RF.max call in lver_to_energy_bound() to coalesce identical adjacent rational-function pieces independently of their originating input index. The piece-level index is not used by lver_to_energy_bound() for proof dependencies; the resulting hypotheses retain their dependencies through the computed energy regions. This removes hash-order-dependent splitting of otherwise identical output pieces.

Validation

The new derived function returns exactly three pieces with transition points

$$\frac23,\qquad \frac57,\qquad \frac{315}{433},\qquad \frac34.$$

The following checks were performed:

• exact Fraction endpoints and half-open interval tiling
• exact RationalFunction equality against the three stated formulas
• exact agreement at rational sample points
• exact equality of adjacent formulas at $5/7$ and $315/433$
• 30 independent runs with PYTHONHASHSEED=0,...,29, all producing identical output
• the selected unaffected baseline tests give 8 passes.

The repository pytest collection still has three pre-existing collection errors on clean main as well

• test_affine2.py: ImportError importing large_values
• test_polytope.py: module-level Polytope.try_union error for a non-finite polytope
• test_all.py: a cascading collection error from importing test_affine2.

These are unrelated to this PR.

Figure note

blueprint/src/chapter/zero_density_energy_estimate.png has not been regenerated.

There does not appear to be a reproducible image generating workflow in the repository. Thw plotting functions end in plt.show(), there is no savefig call or build step for the chapter images, and the checked-in PNG records Matplotlib 3.9.1 while the available environment uses a different Matplotlib version.

The plotting source has therefore been updated to the revised current-best envelope, but the checked-in PNG remains unchanged.