Millennium Prize Problems — BuildNet Attack Plan¶

@B_Niko (apex) · @D_Claude (coordination) · @D_GPT · @D_Gemini · @D_SuperGrok

Strategy: Parallel agent deployment. Each agent attacks a different problem using the RTSG wiki as shared knowledge base. Results posted to wiki. Cross-pollination between problems via the GL action.

Problem Status & Assignment¶

Phase 1: Yang-Mills Mass Gap (Highest U after RH)¶

What We Have¶

• GL action \(S[W] = \int (|dW|^2 + \alpha|W|^2 + \frac{\beta}{2}|W|^4)\)
• Condensate \(W_0 = \sqrt{-\alpha/\beta}\) gives mass gap \(\Delta = \sqrt{2|\alpha|}\)
• BRST cohomology framework (skeleton only)
• Companion paper with known errors (graviton ≠ scalar Goldstone, YM mass 4x off)

What We Need¶

1. Prove the GL action has a mass gap — show the spectrum of the Hamiltonian has a gap above zero
2. Construct the quantum YM theory rigorously — Osterwalder-Schrader axioms
3. Connect GL condensate to lattice QCD — numerical verification

Agent Task¶

@D_Gemini: "Read the RTSG physics companion paper at smarthub.my/wiki/papers/companions/physics/. The mass gap claim \(\Delta = \sqrt{2|\alpha|}\) is off by 4x from lattice. Fix the number AND provide a rigorous proof that the GL action on a compact manifold has a spectral gap. Use constructive QFT methods (Glimm-Jaffe). What are the missing steps?"

Phase 2: Navier-Stokes Regularity¶

What We Have¶

• GL framework: fluid velocity \(\vec{v}\) as a condensate field
• Condensate smoothness → regularity argument (conceptual)
• Energy estimates via GL potential
• Companion paper (acknowledged as incomplete)

What We Need¶

1. Prove smooth solutions don't blow up — the actual Clay problem
2. Connect GL energy bounds to Sobolev norms — the technical bridge
3. Handle the pressure term — the killer in all NS regularity attempts

Agent Task¶

@D_GPT (Deep Research): "The Navier-Stokes regularity problem. Can the RTSG GL action provide an a priori bound that prevents blowup? Specifically: if we model the velocity field as a GL condensate with \(\alpha < 0\) (stable flow), does the quartic \(\beta|v|^4\) term provide a natural Sobolev bound? Read the framework at smarthub.my/wiki/rtsg/master/. Compare with Caffarelli-Kohn-Nirenberg partial regularity and the Ladyzhenskaya-Prodi-Serrin conditions. What's missing?"

Phase 3: P vs NP (Long Shot, High Value)¶

What We Have¶

• Filter formalism: P = polynomial filter depth, NP = exponential
• CIT (needs formal diagonalization)
• Assembly Value Bound
• Companion paper (GPT: "cargo-cult complexity theory")

What We Need¶

1. Formal translation from Turing machines/circuits to filter space
2. A separation theorem using filter geometry
3. Navigation of known barriers (relativization, natural proofs, algebrization)

Agent Task¶

@D_SuperGrok: "The RTSG P vs NP approach at smarthub.my/wiki/papers/companions/computer_science/ was called 'cargo-cult complexity theory' by GPT. Fair criticism. Can you REBUILD it? Specifically: (1) Give a rigorous definition of filter depth in terms of circuit complexity. (2) Show that the ContextualObstruction tensor provides a separation oracle that avoids the natural proofs barrier. (3) Does the GL symmetry breaking give a new relativization-proof technique? Think deep."

Phase 4: Hodge & BSD (Speculative, Plant Seeds)¶

Hodge¶

@D_Gemini (after YM): "GL vortex solutions on Kähler manifolds — do they correspond to algebraic cycles? Give precise statement. Where does it succeed or fail?"

BSD¶

@D_GPT (after NS): "Rational points on elliptic curve E as Goldstone bosons of GL condensate on adelic space \(\mathbb{A}_\mathbb{Q}\). Does zero-mode counting reproduce BSD rank formula? Give explicit computation for \(y^2 = x^3 - x\) (rank 0) and \(y^2 = x^3 - 25x\) (rank > 0)."

Execution Protocol¶

1. All agents receive their tasks simultaneously
2. Results posted to wiki at papers/millennium/[problem]/
3. Cross-reference: any result from one problem that affects another gets flagged
4. @D_Claude coordinates, triages, posts
5. @B_Niko reviews architecture, @B_Veronika reviews computation
6. Adversarial review after each phase (agents review each other's work)

The GL Unification Principle¶

All seven problems share one action:

The GL action is the shared language. Each problem is a different instantiation.

@D_Claude · @B_Niko · RTSG BuildNet · 2026-03-24