Develop a no-sorry proof of the five-dimensional Geode value

[DomTheDeveloper]

Goal

Turn the computational certificate in geode5_bounty_solution.zip into a genuine Lean proof of the exact value of G(1000,1000,1000,1000,1000).

Formal proof layers now developed

• symbolic construction of the kernel P_y(t) = ∏_{w=0}^4 (y^w - t);
• Newton power-sum quotient rows for t^k P_y'(t);
• the exact R0–R4 sparse remainder tables copied from the certificate, with computable metadata checks and symbolic equality goals;
• lower-triangular recurrence diagonal and determinant;
• exact coefficient-ring-linear polynomial integration;
• the integration-by-parts moment recurrence;
• rational factorial reduction of each alternating-sum term to the binomial–beta-integral term;
• all 480 CRT residue pairs, exact answer, upper bound, CRT combination, and conditional uniqueness theorem.

Verified status

The foundational symbolic moment-algebra module has passed the pinned Lean 4.27 checker with warnings as errors and no sorry. The remaining modules are being checked incrementally on the temporary DTD audit PR.

Required remaining gates

1. finish kernel-checking Integral, Recurrence, sparse remainder tables, reduction, and CRT;
2. prove the generalized-Catalan divisibility/integrality bridge for hyperCatalan5;
3. prove the full alternating-sum-to-moment coefficient identity;
4. formalize and execute the modular recurrence for all certified primes;
5. prove the required signed/positive size bound;
6. compose the final exact theorem and replace by sorry in the benchmark.

This PR must not be merged as a completed proof until the final theorem is sorry-free and its axiom audit is clean.