Editing Openai/68ec50da-cf00-8005-b5f6-b683506e5853 (section)

===== Your pages make the same partition of AAA and then treat three situations for A⋆A_\starA⋆ (nonempty with some 3∤a3\nmid a3∤a; nonempty with all 3∣a3\mid a3∣a, split further depending on where 333 divides outside A5±A_5^\pmA5±). The LaTeX formalizes each into an inequality with named constants and separates the roles of: =====
* the prime 555 (giving δ5\delta_5δ5 and its 333-variants in Lemma 1’s corollaries);
* the prime 333 (either excluded or handled via the 3∣b3\mid b3∣b / 3∤b3\nmid b3∤b versions); and
* the large primes p≡1(mod4), p≥29p\equiv1\pmod 4,\ p\ge29p≡1(mod4), p≥29 (handled by Θ29(1)\Theta_{29}^{(1)}Θ29(1) or Θ29(2)\Theta_{29}^{(2)}Θ29(2)).

Where your handwritten constants (.008, .02301, .00374, .00218) appear, the LaTeX shows the exact symbolic expression they correspond to and plugs in the Python‑checked values (matching to the digits you wrote).

If you’d like me to compile this to PDF or tweak the macros (e.g., switch everywhere to the sharp Θ29(1)\Theta_{29}^{(1)}Θ29(1) or keep the looser Θ29(2)\Theta_{29}^{(2)}Θ29(2) to mirror the handwritten sketch), just say the word and I’ll update the source accordingly.