RTSG Open Problem Portfolio — Prize First, Prestige Second, Then the Rest¶

Executive result¶

RTSG is not a universal hammer. It is strongest when a problem can be rewritten as one of four things:

1. a positive-operator / self-adjointness problem,
2. a spectral-gap problem,
3. a flux-vs-dissipation problem, or
4. an instantiation / selection problem.

That single classification rule is the main strategic gain from this pass.

Direct consequence: the best cash targets are Riemann, Yang–Mills, and Navier–Stokes.
Best prestige targets: quantum measurement, black-hole information / horizon thermodynamics, quantum gravity laboratory tests, and dynamic dark energy.
Current weak fits: Beal, Hodge, P vs NP, and the current graph-only form of BSD.

Also: some RTSG claims need hardening before they go into serious physics papers.

• Stop leading math/physics-facing work with “the CS operator”. Use the instantiation operator \(C\), then explain the ontology later.
• Stop using the late-time baryonic 5.4% integral as a literal physical claim. Freeze any \(Q \to B\) conversion before BBN or drop the claim.
• Keep the GRF black-hole argument horizon-only. The photon-sphere route is an attack surface, not an asset.
• Treat dark matter = Stage 0 QS and dark energy = Drive \(D\) as formal conjectures with falsifiability conditions, not as established physics.

1. Planning rule¶

I used two orderings.

A. Strict reward ordering¶

Cash first, with ties broken by present RTSG fit.

B. Actual recommended work order¶

Not the same as strict cash order. A smaller prize with a short path can outrank a million-dollar problem in expected value.

2. Strict reward ordering¶

2.1 Cash-bearing targets¶

Reading of that table¶

Strictly by prize amount, Clay + Beal dominate.
Practically, GRF is the immediate monetizable target while RH / YM / NS are the serious long game.

3. Recommended RTSG work order¶

This is the order I would actually use.

1. GRF submission hygiene — because it is already close.
2. Riemann Hypothesis — best current cash + prestige combination.
3. Yang–Mills mass gap — second-best structural fit.
4. Quantum measurement problem — best prestige target and maybe the cleanest RTSG paper.
5. Navier–Stokes regularity / turbulence — strong fit if reworked as a transfer-vs-dissipation theorem.
6. Black-hole information / horizon thermodynamics — strong conceptual fit, ties into GRF work.
7. Quantum gravity laboratory tests (GIE) — strong prestige, but the literature is now more subtle than “entanglement implies quantized gravity”.
8. Dark energy / dark matter hardening — important, but the cosmology claims must be narrowed to survive contact with data.
9. BSD — keep only after the framework is upgraded from graph language to arithmetic operator language.
10. Everything else — Hodge, P vs NP, Beal, Goldbach, twin primes: currently bad uses of time.

4. New RTSG meta-tools¶

These are the reusable formulations that make the framework more robust.

4.1 Transfer law¶

The minimal dynamical core should be written as

Interpretation:

• \(Q\) stores latent / uninstantiated structure,
• \(P\) stores realized / instantiated structure,
• \(C\) is the instantiation law,
• \(\Theta\) contains the geometry, symmetry, and conservation data for the specific problem.

This is the clean version of “QS, PS, CS” that can survive in math and physics papers.

4.2 Complexification functional¶

To formalize the arrow-of-time / drive language without collapsing into vague metaphysics:

\(\chi\) is not thermodynamic entropy. It is an instantiation imbalance functional.
\(D\) is the candidate formalization of the drive toward complexification.

Use it in cosmology, measurement, and adaptive systems. Do not call it proven monotone until that is actually shown.

4.3 Positive-transfer gap¶

A large slice of RTSG reduces to one object:

for a positivity-improving compact transfer operator \(\mathcal T\).

This unifies:

• RH via spectral purity,
• Yang–Mills via vacuum-to-first-excited-state separation,
• IdeaRank convergence,
• cooperative equilibrium uniqueness,
• some turbulence coarse-graining problems.

4.4 Branch-neutrality / Born recovery¶

This is the strongest new RTSG equation in the portfolio.

Let \(C\) be the instantiation operator and \(\{\Pi_i\}\) the pointer projectors. Define branch weights by

If

then

That means Born’s rule is recovered if the instantiation metric is branch-neutral on the decohered pointer sector. This is publishable if developed carefully.

4.5 Black-hole two-rate split¶

Do not collapse all black-hole rates into one symbol.

• \(\kappa\) is the horizon thermal / kinematic rate.
• \(\lambda_\gamma\) is the photon-sphere / geometric instability rate.

Then, inside a fixed evaporation channel or universality class,

This preserves the “one rate at the horizon” thesis while avoiding the photon-sphere trap.

4.6 Navier–Stokes shell-domination ratio¶

The current RTSG language for Navier–Stokes is too soft. Replace it with a coercive shell ratio.

Let \(\Pi_{\ge K}^+(t)\) be the positive nonlinear energy flux into modes \(|k|\gtrsim 2^K\). Define

Conjecture (RTSG shell-domination criterion):

This is the right RTSG-style regularity statement because it turns “complexity” into a precise transfer-vs-dissipation inequality.

4.7 Yang–Mills loop-covariance gap¶

Use the lattice/open-problem definition itself.

If \(\Delta_{\mathrm{loop}} > 0\), there is exponential decay of gauge-invariant correlations.
That is the right gap object to target.

4.8 RH spurious-eigenvalue defect functional¶

Your current RH program already knows the bottleneck: no spurious eigenvalues.
So target that bottleneck directly.

Let \(H_\theta\) be Construction 5 and let \(B_\theta\) be the (to-be-defined) theta-cusp defect operator measuring failure of the required automorphic / theta boundary compatibility. Then define

Target theorem shape:

That is the exact place to spend time. Everything else is secondary.

4.9 Dynamic dark-energy salvage¶

The current cosmology language is too exposed. The salvage version is

with baryon production frozen out after BBN:

That keeps the spirit of “Drive \(D\) projected into PS” while removing the fatal late-time baryon-conversion problem.

For dark matter = Stage 0 QS to survive, the large-scale perturbation sector must effectively look cold and pressureless:

at linear order in the PS projection.

5. Cash targets — detailed triage¶

5.1 Riemann Hypothesis¶

Why it is a serious RTSG target¶

This is the best current fit because the wiki already has:

• Open Problems rating RH highest,
• Hilbert-Pólya operator constructions,
• Weil positivity chain,
• engine numerics consistent with GUE in Live Results,
• an arXiv-ready scaffold in Hilbert-Pólya paper.

The field itself still treats RH as open and operator-theoretic / quantum-chaotic routes remain active.

What the current gap really is¶

Not “find a Hamiltonian” in the abstract.
The live gap is: exclude spurious eigenvalues in Construction 5.

Novel attack¶

1. Define the defect operator \(B_\theta\) precisely.
2. Show admissible eigenstates are exactly the zero-defect states.
3. Prove zero-defect eigenvalues are exactly the ordinates of nontrivial zeros.
4. Use the Weil positivity chain to constrain the rest of the spectrum.

Direct paper strategy¶

Do not oversell “proof of RH”.
Write the paper as:

• Construction 5,
• self-adjointness and domain,
• Weil positivity numerics,
• explicit obstruction = spurious modes,
• new defect-functional program to kill the obstruction.

That is strong, honest, and useful even before a full proof.

Kill criterion¶

If no natural \(B_\theta\) exists that separates zeta-compatible states from generic automorphic states, Construction 5 is not yet the final operator.

5.2 Yang–Mills mass gap¶

Why it is a good RTSG target¶

Yang–Mills is a natural gap problem.
The cash prize is large and the framework’s operator language actually fits it.

What to stop doing¶

Do not identify the gap simply with a numerically observed “plateau mass” and call it done.
That is phenomenology, not a proof.

Better route¶

Work on a positivity-improving transfer operator on gauge-invariant loop observables.

The exact problem shape is already encoded in the literature:

• exponential covariance decay between separated observables,
• strong-coupling mass-gap results on the lattice,
• hard open problem = all-\(\beta\), four-dimensional non-Abelian case.

Novel attack¶

1. Define the RTSG transfer operator \(\mathcal T_R\) on loop / Wilson observables.
2. Prove reflection positivity and positivity improvement on the gauge-invariant sector.
3. Show quasi-compactness or a usable spectral theorem.
4. Deduce \(\Delta_{\mathrm{loop}}>0\).
5. Only then interpret engine plateau masses as numerical shadows of the rigorous gap.

Why this is better than the current wiki wording¶

Because it hits the mathematical object the field itself uses, instead of a metaphorical analogy with the \(K\)-matrix.

Kill criterion¶

If positivity improvement cannot be shown outside strong coupling, the present RTSG Yang–Mills route is not mature enough for a prize attack.

5.3 Navier–Stokes regularity¶

Why it is a real target¶

Navier–Stokes is a flux / dissipation problem, not just a chaos problem.

The state-space / exact-coherent-structure view of turbulence is now substantial, and recent mathematical writing emphasizes that dissipation-regularity links are central. That matches RTSG well.

What to stop doing¶

Do not say “\(\lambda<0\) means smooth, \(\lambda>0\) means blow-up” and leave it there.
That is too coarse. Smooth turbulent solutions can have chaotic local behavior without singularity.

Better route¶

Use the shell-domination criterion as the real theorem candidate.

Novel attack¶

1. Monitor \(\Theta_K(T)\) numerically in the engine.
2. Compare it with known regular and near-singular regimes.
3. Try to prove a barrier theorem: [ \sup_K \Theta_K(T) < 1 \Rightarrow \text{regularity on }[0,T]. ]
4. If the theorem fails, use the same observable to build a much sharper blow-up criterion.

Add the turbulence measure¶

The exact coherent structure literature suggests another useful RTSG object:

Interpretation: turbulence statistics are reconstructed from unstable recurrent structures weighted by their positive-Lyapunov action. This is an RTSG version of “turbulence as structured recurrence”.

Kill criterion¶

If \(\Theta_K(T)\) fails to correlate with regularity numerically, abandon it fast. Don’t defend a dead criterion.

5.4 BSD, Hodge, P vs NP, Beal¶

BSD¶

Current RTSG graph language is not enough.
If you want BSD to become real, move from “IdeaRank on elliptic-curve nodes” to an explicit arithmetic operator on modular symbols, Selmer structures, or heights. Until then this is a weak fit.

Hodge¶

The framework currently does not touch the actual difficulty: algebraic cycles and Hodge-theoretic realizability. Defer.

P vs NP¶

Current RTSG language does not engage the known hard core of the problem: lower bounds. Defer.

Beal¶

This is a bad use of time. It is a Diophantine exponent problem with almost no structural overlap with present RTSG machinery. Drop it.

6. Prestige targets — detailed triage¶

6.1 Quantum measurement problem¶

This is the best prestige target¶

RTSG is built for it.

The field still frames the problem as a duality between deterministic Schrödinger evolution and stochastic Born-rule collapse. RTSG has an actual, clean way to rewrite that.

Strong formulation¶

• QS evolves unitarily.
• PS records instantiated outcomes.
• \(C\) is the instantiation map from latent quantum structure to definite record.
• Collapse is not a second dynamics in QS; it is the appearance of instantiation in PS.

Paper target¶

Title idea: “Branch-neutral instantiation and the recovery of Born weights”.

What makes it strong¶

The branch-neutrality equation is not poetic language. It is a concrete condition under which Born’s rule falls out.

Immediate subproblems¶

1. Make the pointer-sector assumption precise.
2. Show basis-independence under environmentally selected pointer bases.
3. Determine whether non-neutral \(C^\dagger C\) predicts tiny Born-rule deviations in mesoscopic experiments.

Kill criterion¶

If branch-neutrality can only be imposed by hand and not motivated structurally, the paper collapses into a relabeling exercise. Avoid that.

6.2 Black-hole information / horizon thermodynamics¶

Why it is high value¶

You already have momentum here from One Rate at the Horizon.

Recent literature makes the problem more precise, not less: - the “information paradox” is increasingly analyzed through assumptions rather than slogans, - the role of horizon type and semiclassical validity is central, - remnants and non-complete evaporation remain active options.

RTSG move¶

Reframe “information loss” as causal sequestration vs retrieval.

If a true event horizon and singularity persist, information can remain hidden from the external observer without any local non-unitarity.
If the horizon is only trapping / transient, retrieval channels reopen.

Concrete formulation¶

Keep the horizon split from the photon sphere. Keep the GRF essay narrow.
Then spin out a follow-up note focused on:

• event horizon vs trapping horizon,
• singularity resolution as the real fork,
• external information balance as a causal-structure problem.

Kill criterion¶

If the argument reverts to loose analogies about scrambling without causal bookkeeping, it will get eaten alive.

6.3 Quantum gravity laboratory tests¶

Why it matters¶

This is now one of the cleanest live prestige arenas.

What the literature now says¶

The lab-test field is active, but the old slogan “gravity-mediated entanglement proves quantized gravity” is no longer safe. There is now serious discussion of classical-gravity models that can also generate entanglement under some assumptions.

RTSG move¶

That is good for RTSG, not bad.

Instead of treating GIE as a binary test for graviton ontology, treat it as a test of whether gravity functions as a nontrivial information / instantiation channel.

Concrete agenda¶

1. Define the classical-mediator class you actually want to exclude.
2. Build a witness against that class.
3. Separate three claims that are too often conflated:
4. gravity can correlate,
5. gravity can entangle,
6. gravity must therefore be a quantized field in the usual sense.

RTSG can attack the third link.

Kill criterion¶

If the framework cannot say what observation would distinguish Stage 0 instantiation from an ordinary quantum mediator, the program stays philosophical.

6.4 Dynamic dark energy / dark matter¶

Why this is valuable¶

DESI-era cosmology has created room for a more nuanced dark-energy discussion.

What to stop saying¶

Stop saying “baryonic 5.4% = the cumulative CS-instantiation integral over 13.8 billion years” as a late-time process. That is too exposed.

Better version¶

• freeze net baryon-generation before BBN,
• use \(\Lambda_{\mathrm{eff}}(a)\) for late-time complexification projection,
• require Stage 0 QS to mimic cold, pressureless clustering at linear order.

What this buys you¶

The cosmological vision survives, but in a form that can be compared to data rather than rejected immediately.

Kill criterion¶

If the Stage 0 QS sector fails the linear-perturbation test against CDM-like growth, the dark-matter identification is wrong.

6.5 Turbulence (full theory)¶

This sits behind Navier–Stokes but deserves its own prestige slot.

The exact-coherent-structure / recurrent-pattern literature is now mature enough that an RTSG state-space measure is not crazy.
The correct target is not “solve turbulence” in one jump.
It is:

1. define a usable recurrent-structure measure,
2. predict low-order statistics from it,
3. compare against DNS / engine data.

That is concrete.

7. The remaining sciences¶

The engine portfolio includes protein folding, origin of life, cancer, aging, intelligence, and other long-horizon targets.

Current recommendation¶

Do not diffuse effort there yet.

Why: - the cash / prestige upside is lower than the top math-physics problems, - the framework still needs hardening in its core operator language, - some fields already have strong predictive systems and would reward mechanism papers, not grand-ontology papers.

Exception¶

If a side paper can be written with very low cost and high conceptual clarity — for example a folding-as-attractor explanation paper — it can be done later. It should not steal cycles from RH, YM, measurement, or GRF.

8. What the framework should stop, keep, and add¶

8.1 Stop¶

• Late-time baryonic 5.4% claim.
• Photon-sphere uniqueness rhetoric in the GRF essay.
• Graph-only BSD attacks.
• P vs NP and Beal as active top-tier targets.
• “Consciousness-space” as the first line in math-facing papers.

8.2 Keep¶

• horizon-only \(\kappa\) program,
• Construction 5 + Weil positivity chain,
• Yang–Mills as a genuine gap problem,
• Navier–Stokes via transfer-vs-dissipation,
• dark energy as a dynamic projection candidate,
• measurement as instantiation.

8.3 Add¶

• transfer law,
• complexification functional,
• branch-neutrality equation,
• shell-domination criterion,
• loop-covariance gap,
• RH defect functional,
• dynamic \(\Lambda_{\mathrm{eff}}(a)\),
• explicit falsifiability conditions for Stage 0 QS / Stage 0 CS physics.

9. Patch-ready equations for rtsg/equations.md¶

Pasteable block:

# RTSG transfer / instantiation laws

∂_t ρ_P = J_C[ρ_Q, ρ_P; Θ]
∂_t ρ_Q = -J_C[ρ_Q, ρ_P; Θ]

χ[ρ_P,ρ_Q] = ∫ ρ_P log((ρ_P+ε)/(ρ_Q+ε)) dμ
D = dχ/dt

Δ(𝒯) = -log(λ₁(𝒯)/λ₀(𝒯))          # positive-transfer spectral gap

p_i = ⟨ψ, Π_i C†C Π_i ψ⟩ / Σ_j ⟨ψ, Π_j C†C Π_j ψ⟩
C†C|_{H_pointer} = α I  ⇒  p_i = ||Π_i ψ||²   # Born recovery by branch neutrality

Θ_K(T) = [∫_0^T Π_{≥K}⁺(t) dt] / [ν ∫_0^T ||∇u_{≥K}(t)||²_2 dt + ε]
sup_K Θ_K(T) < 1  ⇒  regularity on [0,T]      # conjectural shell-domination criterion

Δ_loop = liminf_{R→∞} -(1/R) log sup_{dist(A,B)≥R} |Cov(A,B)|/(||A|| ||B||)

E_spur(λ) = inf_{ψ∈ker(H_θ-λ), ||ψ||=1} ||B_θ ψ||_W
E_spur(λ)=0  ⇔  ξ(1/2+iλ)=0                    # conjectural RH defect criterion

λ_BH = (κ, λ_γ)                                # horizon thermal rate vs photon-sphere geometric rate
t_info = C_global · S_Wald / κ                 # within fixed evaporation channel / universality class

Λ_eff(a) = Λ₀ + α dχ/dln a + β d²χ/d(ln a)²
Γ_{Q→B}(a > a_BBN) ≈ 0
c²_{s,QS} ≈ 0,   σ_QS ≈ 0                      # Stage 0 QS falsifiability conditions

10. Suggested wiki patches¶

10.1 Update problems/open.md¶

• Promote quantum measurement into the main physics queue.
• De-prioritize Beal, Hodge, P vs NP.
• Rewrite BSD as “needs arithmetic operator formalization”.
• Change Navier–Stokes text from Lyapunov-only to shell-domination criterion.

10.2 Update rtsg/equations.md¶

Add the patch-ready equations above.

10.3 Update papers/grf/cosmological_vision.md¶

• Remove literal late-time baryon-integral wording.
• Add BBN freeze-out condition.
• Replace \(\Lambda = D\) slogan with \(\Lambda_{\mathrm{eff}}(a)\) formulation.
• State Stage 0 QS falsifiability conditions explicitly.

10.4 Add a new paper stub¶

Suggested new file: papers/companions/quantum_measurement.md

Core thesis: - QS unitary evolution, - PS instantiated record, - \(C\) branch-neutrality, - Born recovery, - many-worlds unnecessary.

10.5 Update math/hilbert_polya.md¶

Add the spurious-mode program: - define \(B_\theta\), - define \(\mathcal E_{\mathrm{spur}}(\lambda)\), - state the exact theorem target.

11. 90-day build plan¶

Phase 1 — immediate (days 1–7)¶

• Submit the GRF essay.
• Patch rtsg/equations.md.
• Create papers/companions/quantum_measurement.md.
• Add the RH defect-functional section to math/hilbert_polya.md.

Phase 2 — short horizon (days 8–30)¶

• RH: formalize \(B_\theta\) and test defect numerically.
• YM: define the gauge-invariant transfer operator precisely.
• NS: instrument the engine to monitor \(\Theta_K(T)\).

Phase 3 — medium horizon (days 31–60)¶

• Write the measurement paper draft.
• Write the Yang–Mills note around \(\Delta_{\mathrm{loop}}\).
• Produce the first NS numerical memo around shell domination vs regularity.

Phase 4 — hardening (days 61–90)¶

• Cosmology rewrite with BBN-safe language.
• Black-hole information follow-up note.
• Decide whether BSD gets promoted or dropped based on whether arithmetic operatorization is found.

12. Bottom line¶

The framework becomes stronger the moment it stops trying to explain everything at once.

The right upgrade is this:

RTSG is valuable when it predicts which positive operator, transfer current, or monotone must exist for a problem to close.

That one sentence reorganizes the whole research program.

From that angle:

• Riemann is a spectral-purity problem.
• Yang–Mills is a positive-gap problem.
• Navier–Stokes is a transfer-vs-dissipation problem.
• Measurement is an instantiation-selection problem.
• Black-hole information is a causal-sequestration problem.
• Dark energy is a projection-of-drive problem.

That is the clean architecture. Build there.

13. Source trail (current literature + official pages)¶

Official reward / status pages¶

• Clay Mathematics Institute — The Millennium Prize Problems.
• Clay Mathematics Institute — Riemann Hypothesis.
• Clay Mathematics Institute — Yang–Mills & the Mass Gap.
• Clay Mathematics Institute — Navier–Stokes Equation.
• Clay Mathematics Institute — Birch and Swinnerton–Dyer Conjecture.
• Clay Mathematics Institute — Millennium Prize Problems Lecture Series.
• AMS — Beal Prize / Beal Prize Rules and Procedures.
• Gravity Research Foundation — 2026 Awards for Essays on Gravitation.

Riemann / operator route¶

• The Riemann Hypothesis: Past, Present and a Letter Through Time (2026 survey).
• On the Existence of the Hilbert-Pólya Hamiltonian (2025 operator attempt).

Yang–Mills¶

• Colin Morningstar, Update on Glueballs (2025).
• Ron Nissim, U(N) lattice Yang–Mills in the ’t Hooft regime (2025).

Navier–Stokes / turbulence¶

• Theodore Drivas, Mathematical Theorems on Turbulence (2026).
• Zhigunov & Page, Exact coherent structures as building blocks of turbulence on large domains (2026).

Quantum measurement / gravity¶

• Tomaz, Mattos, Barbatti, The Quantum Measurement Problem: A Review of Recent Trends (2025).
• Marletto et al., Quantum-information methods for quantum gravity laboratory-based tests (2024).
• Yant & Blencowe, An Operational Quantum Field Theoretic Model for Gravitationally Induced Entanglement (2025).
• The simple reason why classical gravity can entangle (2025).

Black holes / cosmology¶

• Buoninfante & Di Filippo, Is the information loss problem a paradox? (2025).
• Ong, The Case For Black Hole Remnants: A Review (2025).
• Fermilab / DESI release, New DESI results strengthen hints that dark energy may evolve (2025).
• Dark Energy After DESI DR2: Observational Status, Reconstructions, and Physical Models (2026).

Optional biology context¶

• Abramson et al., Accurate structure prediction of biomolecular interactions with AlphaFold 3 (Nature, 2024).
• Google DeepMind, AlphaFold 3 and AlphaFold Server are launched (2024).