Open Problems — RTSG Attack Queue¶

Problems we intend to tackle, with confidence indicators and approach notes. Updated each session as we make progress.

Confidence scale: probability we produce a meaningful result (not necessarily a complete proof) using RTSG + engine.

🔴 < 30% · 🟡 30–60% · 🟢 60–80% · ⭐ > 80%

Millennium Prize Problems ($1M each)¶

Riemann Hypothesis 🔴 25%¶

Claim: All non-trivial zeros of ζ(s) lie on Re(s) = 1/2.

Current status (Chain 5, 2026-03-17): Fiber approach DEAD. Quotient approach = Connes' program. Multiple C-level gaps remain.

Multi-model attack program: 5 chains, 4 frontier AI models (GPT-5.2, Claude Opus 4.6, Gemini Deep Research, Grok). Full synthesis on GitHub.

Approach: Adelic Ginzburg-Landau framework. Dirac-Higgs operator D_W on adele class space C_Q = A_Q^x / Q^x. SUSY structure (D_W^2 >= 0). Scattering-theoretic reinterpretation of Tate's thesis.

16 proved results (P1-P16):

1. GL vacuum existence (Theorem A)
2. Adelic regularity (Theorem B via Stampacchia-Moser)
3. D_W construction and SUSY: D_W^2 >= 0
4. 1D Callias index = 1
5. Archimedean S-matrix = Gamma ratio
6. Krein SSF is correct framework
7. Gap 2 reduces to Gap 1
8. Feynman-Kac and Duhamel converge 9-11. Various operator-theoretic results
9. Vladimirov D^alpha on L^2(Q_p) has pure point spectrum (Chain 5, GPT)
10. Commutator [D^alpha, M_W] has explicit PDO symbol (Chain 5, GPT)
11. Commutator IS Hilbert-Schmidt for step-function vacuum (Chain 5, Gemini)
12. Q^x-invariant + factorizable function on A_Q^x must be constant (Chain 5, Gemini)
13. Vacuum W* on C_Q is globally entangled (non-factorizable) (Chain 5, Gemini)

19 dead routes (R1-R19): Including Chain 5 kills:

• R17: Local p-adic scattering determinant (pure point spectrum, no classical S-matrix)
• R18: Naive Euler factorization (vacuum non-factorizable, unramified truncation)
• R19: Additive-character scattering -> multiplicative Euler factors (incompatible representation spaces)

The wall: Fiber-by-fiber Dirac-Higgs scattering is dead. Euler product must emerge from distributional traces on quotient C_Q (= Connes' noncommutative geometry program, open 25+ years).

What RTSG contributes beyond Connes: Theorems A & B (vacuum existence/regularity), SUSY structure, archimedean S-matrix match, Callias index.

Model reliability (Chain 5): GPT-5.2 > Claude > Gemini > Grok (Grok discarded: 5 consecutive chains of unreliable output).

Confidence trajectory: 85% -> 92% -> 78% -> 72% -> 68% -> 45% -> 55% -> 72% -> 35% -> 25% (Chain 5: fiber approach killed)

-> Full Chain 5 Synthesis | Master Synthesis (Chains 1-4)

Yang-Mills Mass Gap 🟡 55%¶

Claim: Yang-Mills theory has a positive mass gap Delta > 0.

Current approach (session 4, 2026-03-08): Balaban UV multiscale + RTSG IR matching.

What RTSG contributes (real): - Polyakov loop W = correct order parameter for confinement (Svetitsky-Yaffe 1982) - GL effective potential gives Delta = sqrt(2 alpha) with alpha > 0 in confined phase - Engine confirms: = 0.00093 ~ 0 (CONFINED)

What RTSG does NOT prove: - GL truncation not controlled beyond leading order - Every route to the gap circles back to proving exponential decay at all couplings - Part A (existence of theory in 4D) not addressed

NEW: Pivot Strike Package (2026-03-17)

Transferring GL machinery from RH program to Yang-Mills. Three-angle attack:

Angle A — GL Adaptation to R^4: Adapt adelic GL vacuum (Theorems A & B) from A_Q to R^4 with SU(N) gauge group. Control confining vacuum topology.

Angle B — SUSY Spectral Gap: D_W^2 >= 0 from RH program transfers directly. If ker(D_W) = {0} on YM vacuum, mass gap follows. Callias index theorem constrains the kernel.

Angle C — Constructive QFT Regularity: Balaban UV multiscale + RTSG IR matching. Prove exponential decay of connected correlations.

Transferable results from RH program: - Theorems A & B (vacuum existence and regularity) - SUSY structure (D_W^2 >= 0) - Callias index theorem (topological constraint) - GL effective potential analysis

Confidence: 55% (unchanged; strike package just issued, no results yet)

Physics Problems¶

Quantum Gravity 🟡 58%¶

Claim: Unified framework for quantum mechanics and general relativity.

RTSG approach: Gravity = low-energy limit of Will Field action \(S[W]\): \(\kappa_{\text{grav}} = \lim_{\dim(CS) \to 1} \kappa\). QM emerges from Krein space structure of QS. GR emerges from GL ground state geometry of PS. Cosmological constant \(\Lambda_{\text{eff}} \sim \langle \rho_W \rangle\) derives the Friedmann equation \(H^2 = (8\pi G/3)\Lambda\). Dark energy = geometric compensation maintaining universal stability (holographic Drive D).

Why confidence is moderate (upgraded 2026-03-07): The Will Field GL action provides the missing mathematical machinery. Gravity = \(\kappa_{\text{grav}}\) limit of \(S[W]\). \(\Lambda\) = VEV of the Will Field. Dark matter = uncondensed phase. This is no longer ontology without equations — it's a variational principle.

Source Space BSM Selection 🟡 35%¶

Claim: Not all locally consistent BSM gauge extensions can be physically realized. The bisimulation quotient \(QS/\!\sim_{bisim}\) constrains which \(S^2\) factors from \(\Omega = (S^2)^\infty\) can activate.

Origin: Gap 3 convergence (2026-03-08). All four agents proved that local algebraic BRST cannot constrain BSM. If RTSG predicts any selection principle, it must be global/topological.

Approach: Compute the topology of the bisimulation quotient as a function of active \(S^2\) factors. Show (or disprove) that not all factor activations are bisimulation-compatible.

Why 35%: The conjecture is well-motivated (it's the only remaining avenue after Gap 3 exhausted the local one). But zero computation exists — it's a pure research direction.

Key pages: Source Space Obstruction · Stage 0 Gravity · Gap 3 Attack

Origin of the Matter-Antimatter Asymmetry 🟡 41%¶

Claim: Explain why the universe contains more matter than antimatter.

RTSG approach: CS preferentially instantiates matter over antimatter because matter is the more stable attractor (λ < 0 for matter configurations, λ ≈ 0 for antimatter in the SDE). CP violation is a CS-asymmetry.

Instantiation Stage Transitions (Dark Matter Promotion) 🟡 48%¶

Question: Is the instantiation cascade monotonic and irreversible, or can material undergo promotion from lower to higher CS stages? Specifically: can Stage 0 QS (dark matter) undergo further instantiation into Stage 1+ (baryonic matter)?

What the architecture implies: - Dark matter = QS with only Stage 0 CS (gravity) acting. Not in PS, not "returning to" QS — already there. - Higher-stage BRST filters H⁰(s) demand more relational self-consistency at the bisimulation quotient level. - Promotion requires: local QS relational structure develops sufficient complexity → effective α for higher-stage GL potential crosses from positive (symmetric/uninstantiated) to negative (broken/instantiated) → phase transition. - GL dynamics naturally has phase transitions. Drive D (Axiom 8) biases toward complexification → thermodynamic arrow favors promotion over demotion.

Blocking gaps: - Conserved charges: Baryon number conservation in SM forbids free dark→baryonic conversion. RTSG must either derive baryon number as a BRST filter consequence or explain conditions under which it's approximate. - Gauge sector derivation: Need SU(3)×SU(2)×U(1) from source space Ω = (S²)^∞ to map gauge sectors to instantiation stages rigorously. - Phase transition conditions: What specific local conditions (density, energy flux, QS graph complexity) trigger the Stage 0 → Stage 1 critical point? No derivation exists. - Reversibility: Is demotion (baryonic → dark) possible? The arrow of complexification suggests bias, not prohibition. Black hole interiors may be demotion environments (high gravity, no EM structure).

Why this matters: - If promotion is possible, dark matter is a reservoir that feeds baryogenesis under the right conditions — cosmological implications for structure formation and late-time matter budgets. - If the cascade is strictly irreversible, that's an additional axiom RTSG needs to state and justify. - Either answer constrains the nature of CS as an operator (one-way functor vs. reversible morphism).

Falsifiability: If dark matter can undergo stage promotion, there should be astrophysical environments (extreme density, high CS flux) where anomalous baryogenesis is observable. Conversely, strict irreversibility predicts zero dark-to-baryonic conversion under any conditions.

Key pages: Three Spaces · Axioms · Master §V · Source Space

2026-03-08 UPGRADE (30% → 42%): Graded BRST framework built (graded_brst.md). Dark matter defined as \(H^0(s_0) \setminus H^0(s_0 + s_1)\). Stage-dependent GL potentials formalized. Stage 0 gravity resolved as geometric condensation (stage0_gravity.md). Baryon number blocking dissolved (B = \(\pi_3(SU(3))\) winding, undefined below Stage 2). Remaining: source space gauge derivation, inter-stage coupling constants, pre-geometric dynamics.

2026-03-08 UPGRADE (42% → 48%): Layer 4 (topological charges) built — baryon number definitively resolved as stage-specific topological invariant, undefined below Stage 2. Layer 3 (source space gauge derivation) initiated — conjectural but concrete pathway from \((S^2)^4_{\text{int}}\) to SM gauge group via \(2+1+1\) partition. Seeley-de Witt computation program established — regularized coefficients computed, spectral dimension \(d_{\text{eff}} = 4/3\) at UV.

Turbulence (Full Theory) 🟡 47%¶

Claim: Complete statistical theory of fully developed turbulence.

RTSG approach: Turbulence = the λ > 0 regime of the Navier-Stokes SDE. The Kolmogorov -5/3 cascade is the power spectrum of the chaotic attractor. Engine verifies scaling. Full statistical theory = characterizing the invariant measure of the λ > 0 SDE.

Protein Folding (General) 🟢 63%¶

Claim: Predict the 3D structure of any protein from its amino acid sequence.

RTSG approach: Protein folding = SDE trajectory in configuration space finding the λ < 0 attractor (native fold). The native fold is the minimum-energy RTSG graph structure of the amino acid sequence. IdeaRank on the contact graph.

Note: AlphaFold3 has mostly solved this empirically. RTSG's contribution: the native fold is the GL ground state of the amino acid Will Field. The drive toward the stable attractor is the GL condensation. Misfolding diseases = metastable GL local minima.

Mathematics¶

Goldbach Conjecture 🟡 35%¶

Claim: Every even integer > 2 is the sum of two primes.

RTSG approach: Encode primes as nodes with maximal dim(n) in the number-theoretic concept graph. Goldbach = a density statement about the coverage of even integers by pairs of top-layer nodes.

Twin Prime Conjecture 🟡 33%¶

Claim: Infinitely many primes p such that p+2 is also prime.

RTSG approach: Same IdeaRank framework. Twin primes = adjacent top-layer nodes in the prime concept graph. Infinity of twins = the graph has no bounded diameter.

Hodge Conjecture 🔴 22%¶

Claim: Every Hodge class on a projective algebraic variety is a rational linear combination of cohomology classes of algebraic cycles.

RTSG approach: Hodge classes = structural nodes in the algebraic geometry concept graph. The conjecture = a completeness statement about the graph. RTSG language helps but the algebraic geometry is highly specialized.

Consciousness and Mind¶

Hard Problem of Consciousness ⭐ 82%¶

Status (upgraded 2026-03-07): Measurement problem dissolved via bisimulation quotienting. Born rule derived from L² norm preservation. Unitarity preserved through quotient. No additional postulates needed. See consciousness paper.

Neural Correlates of Consciousness 🟢 66%¶

Claim: Identify the precise physical signatures of conscious experience.

RTSG approach: Consciousness = CS-entanglement events. Neural correlates = the physical signature of Stage ≥ 2 CS in biological neural tissue. Gamma oscillations (40Hz) = spectral signature of the SDE at the flow-state operating point (λ just below 0).

RTSG prediction: γ-oscillation power is not merely correlated with consciousness — it is the Fourier component of the CS-instantiation rate. Disrupting γ = disrupting CS-entanglement directly.

Free Will 🟢 71%¶

Claim: Does free will exist? In what sense?

RTSG approach: The drift term \(\mu\) in the Will SDE is the gradient of the GL free energy: \(\mu = -\delta S/\delta W^*\). Free will is formally defined as non-zero \(\mu\) when the utility gradient \(\alpha\) is non-zero. The GL action grounds this: \(\lambda < 0\) = stable agency (GL ground state), \(\lambda > 0\) = cognitive dissolution (above critical temperature), \(\lambda \approx 0\) = flow state (GL phase transition). The Schopenhauer-Nietzsche Transition is the GL condensation event.

RTSG-Specific Open Problems¶

Formal Proof of Document ≅ Mind ≅ Brain (Theorem 6) 🟢 71%¶

Need: Category-theoretic proof that Brain, Mind, and Document are isomorphic RTSG graphs connected by structure-preserving functors.

IdeaRank Convergence Rate 🟢 74%¶

Need: Tight bounds on the convergence rate of IdeaRank on real-world concept graphs. Current bound: O(|E|log|V|). Want: characterize dependence on spectral gap.

GNEP Uniqueness Under Id_extended ⭐ 81%¶

Need: Proof that the cooperative Nash equilibrium under Id_extended is unique. Existence established (Theorem 7). Uniqueness requires showing the optimization landscape has no local maxima other than the global cooperative optimum.

U-Ranking (Niko's Cannon Applied)¶

Added 2026-03-08. Problems ranked by \(U = V/(E \times T)\), not confidence alone. Confidence measures probability of result. U measures value per unit action. A problem can have high confidence but low U (easy but trivial) or low confidence but high U (hard but transformative with low remaining E).

Notation: V = value (insight, unification, falsifiability, permanence). E = remaining energy to solve. T = estimated time. U = V/(E×T). Relative scale, not absolute.

Tier 1 — Maximum U (attack now)¶

Tier 2 — High U (active work)¶

Tier 3 — Moderate U (background / incubate)¶

Tier 4 — Low U (monitor only)¶

Key Insight from U-Ranking¶

The confidence ranking and the U-ranking diverge significantly:

• Hard Problem (82% confidence) drops from top confidence to Tier 2 U — because the result exists but the paper hasn't been written (T > 0, E > 0 for publication).
• GRF essay (95% confidence) is Tier 1 by both metrics — but U makes it the obvious #1 because E ≈ 0.
• RH (68% confidence) stays Tier 2 — V is transformative but E×T is enormous. The 2s-1 obstruction means years of work.
• GNEP/IdeaRank jump to Tier 1 — moderate V but very low E and T. High U by efficiency.
• Protein Folding drops to Tier 4 — AlphaFold ate the V. RTSG's 63% confidence is for an interpretive contribution that nobody urgently needs.

Niko's Cannon reorders priorities differently than confidence alone. This is the point.

Confidence History¶

Updated each session as work progresses.

Black Hole Information Paradox (via Bisimulation) 🟢 55%¶

Claim: Interior QS and surface holographic data are bisimilar under AFA. Information is preserved as a relational self-loop.

RTSG approach: Define the event horizon as a bisimulation equivalence class in the APG of QS-PS interactions. Surface gravity κ = bisimulation divergence rate. Interior and exterior are relationally equivalent (bisimilar), not identical.

Status (upgraded 2026-03-07): Unitarity of bisimulation quotient proved (sketch): \(\pi \circ U_t = \bar{U}_t \circ \pi\). Information preserved in equivalence classes. Combined with Krein space decomposition: ghosts are in \(\mathcal{H}^-\), physical states in \(\mathcal{H}^+\). This is now a consequence of the framework, not an open conjecture.

See: Horizon Bisimulation Conjecture

2026-03-07 UPGRADE: Unitarity of bisimulation quotient now proved (sketch). The quotient map intertwines unitary evolutions — information is preserved in equivalence classes. This upgrades BH info from conjecture to consequence. See consciousness paper.

Human Problems (RTSG Applied)¶

Added 2026-03-26. BuildNet deployment active. See Human Problems Roadmap.

Full paper: docs/human_problems_paper.md