RTSG Rebooted Open-Problem Campaign¶

Date: 2026-03-07
Context: full live-wiki reread completed; problem queue restarted from the updated RTSG baseline (Axiom 0 / ZFA-AFA, three-space co-primordiality, bisimulation quotient, Will Field / GL layer).

0. Executive ruling¶

0.1 First fix the formal core¶

The new Will-Field layer contains a real formal bug that must be corrected before using it to attack open problems.

The wiki currently states both: - L_int = β |W|^2 W - S[W] = ∫ ( |∂W|^2 + α |W|^2 + (β/2) |W|^4 ) dμ

These are not the same thing.

For a complex scalar field with global U(1) symmetry, - the interaction density / potential must be a scalar and U(1)-invariant; - |W|^2 W is not invariant under W -> e^{iθ} W; - the invariant quartic interaction is (|W|^4)/2; - the cubic term β |W|^2 W is the Euler–Lagrange term in the field equation, not the Lagrangian density.

0.2 Corrected core GL/Will-Field system¶

Use:

S[W] = \int \left( |\partial W|^2 + \alpha |W|^2 + \frac{\beta}{2}|W|^4 \right) d\mu

Then the field equation is:

\Box W - \alpha W - \beta |W|^2 W = 0

or, in dissipative / driven form,

\partial_t W = -\alpha \nabla S + \beta |W|^2 W + \gamma \Phi + \xi

This patch keeps the cubic term where it belongs: in the equation of motion.

0.3 Immediate downstream corrections¶

The following current wiki formulas should be patched:

1. Cosmological constant
   Replace

   \Lambda_{\text{eff}} \sim \langle \rho_W \rangle_{PS}
   

   with a real, invariant scalar such as

   \Lambda_{\mathrm{eff}} = \left\langle \alpha |W|^2 + \frac{\beta}{2}|W|^4 \right\rangle_{PS}
   

   or, after symmetry breaking at vacuum W_0,

   \Lambda_{\mathrm{eff}} = V(W_0) = \alpha |W_0|^2 + \frac{\beta}{2}|W_0|^4.
   

2. Navier–Stokes blow-up line
   Replace the current dimensionally/tensorially bad statement

   \int_V \beta |W|^2 W \, dV > \int_V \alpha \nabla S \, dV \implies \text{singularity}
   

   with a scalar energy-balance functional, e.g.

   \mathfrak B(T)
   = \int_0^T \!\int_\Omega
   \left(c\,|Du|\,|W|^2 - \nu |\nabla W|^2 - \frac{\beta}{2}|W|^4\right)
   dx\,dt.
   

   Proposed RTSG criterion:

   \sup_{0<t<T} \mathfrak B(t) < \infty \quad \Longrightarrow \quad \text{no GL-driven blow-up trigger on } [0,T].
   

3. Stress-energy / gravity coupling
   Add the Will-field stress tensor:

   T^{(W)}_{\mu\nu}
   = \partial_{(\mu}\bar W\,\partial_{\nu)}W
   - g_{\mu\nu}\left(|\partial W|^2 + \alpha |W|^2 + \frac{\beta}{2}|W|^4\right).
   

   This is the correct bridge from the Will field to cosmology / gravity.

1. Corrected RTSG toolset¶

1.1 Quotient / instantiation operator¶

Let

\pi : QS \to PS = QS/\!\sim_{\mathrm{bisim}}

Define the pushforward measure

\mu_{PS}(B) = \mu_{QS}(\pi^{-1}(B)).

Then the quotient-Hilbert subspace is

\mathcal H_\pi = \{f \in L^2(QS,\mu_{QS}) : f = \tilde f \circ \pi\text{ for some }\tilde f \in L^2(PS,\mu_{PS})\}.

1.2 Quotient covariance condition¶

The hidden condition needed for the claimed unitarity theorem is:

q_1 \sim_{\mathrm{bisim}} q_2 \implies U_t q_1 \sim_{\mathrm{bisim}} U_t q_2 \qquad \forall t.

Call this bisimulation covariance.

If it holds, then U_t preserves \mathcal H_\pi, and the induced quotient dynamics

\bar U_t : L^2(PS,\mu_{PS}) \to L^2(PS,\mu_{PS})

1.3 Born recovery on the quotient¶

For quotient cells C_i \subset PS, define

p_i = \|\chi_{C_i} \, \bar U_t \psi_0\|_{L^2(PS)}^2.

If quotient classes are branch-neutral, this is exactly the Born rule.
If not, the RTSG generalization is

p_i^{RTSG} = \omega_i \, \|\chi_{C_i}\bar U_t\psi_0\|^2,
\qquad \sum_i \omega_i = 1.

1.4 RH defect functional¶

Define the positive-cone quadratic form on the Krein-resolved theta kernel:

\mathcal W_\theta[h] := \langle f_h, P_+ K_\theta P_+ f_h \rangle.

Then define the ghost defect

\Delta_{\mathrm{ghost}}[h]
:= \langle f_h, K_\theta f_h \rangle_{\mathcal K}
 - \mathcal W_\theta[h].

1.5 Theta-cone closure program¶

Let

\mathcal C_\theta
= \overline{\{g_\theta * \widetilde g_\theta\}}

If 1. \mathcal W_\theta[h] \ge 0 for all h \in \mathcal C_\theta, and 2. \mathcal C_\theta is dense in the full positive-definite test-function cone,

then Weil positivity extends to all positive-definite h, closing the current Step-5 gap.

1.6 YM gap functional¶

For a gauge-invariant effective action with a center-sensitive order parameter L and Will field W, use

\Gamma[L,W]
= \int \left(
|\nabla L|^2 + a|L|^2 + b|L|^4
+ c|W|^2|L|^2
+ |\nabla W|^2 + \alpha |W|^2 + \frac{\beta}{2}|W|^4
\right) dx.

At a confining vacuum (L_0=0, W_0), define

\Delta_{YM}^2
:= \lambda_{\min}\!\left(D^2\Gamma\big|_{(0,W_0)}\right)
= a + c|W_0|^2.

1.7 NS flux-defect functional¶

Littlewood–Paley split u = \sum_j u_j. Define the high-frequency defect

\mathcal D_K(t)
:= \sum_{j\ge K} \left(\Pi_j(t) - \nu 2^{2j}\|u_j(t)\|_2^2\right),

Proposed RTSG regularity criterion:

\sup_K \int_0^T \mathcal D_K^+(t)\,dt < \infty
\quad \Longrightarrow \quad
\text{no forward-cascade-driven singularity on } [0,T].

2. Priority ordering after reread¶

Tier A — highest dollar value targets¶

A1. Riemann Hypothesis — ACTIVE¶

Why active: Best current match between RTSG and a million-dollar problem.
What RTSG already has: Construction 5, C5a fourth-moment bound, C5b Krein-space ghost projection, Weil positivity chain.
What is still open: The wiki still admits the real gap: extension from sampled / theta-lifted positivity to all positive-definite test functions.

Immediate attack program¶

1. Formalize continuity of the Weil functional in a Schwartz topology.
2. Prove or disprove density of the theta cone \mathcal C_\theta.
3. If density holds, Step 5 closes by continuity.
4. If density fails, identify the missing cone generator and patch the operator class.

Novel RTSG contribution¶

Reframe the remaining RH obstruction as a cone-density theorem, not a vague “prove positivity for all test functions.” That is a much sharper target.

Brutal truth¶

Current RH confidence in the wiki is still too high. The live page says “gap potentially closed,” but the actual open issue remains nontrivial. Do not call this solved.

A2. Yang–Mills Mass Gap — ACTIVE BUT NEEDS REWRITE¶

Why active: Million-dollar problem and the new Will-Field / center-symmetry structure actually helps.
What RTSG already has: confinement signal, Polyakov-loop emphasis, GL layer, engine confinement proxy.
What is weak: “Plateau mass in the fermion propagator = the gap” is not sufficient for Clay and is not the right level of rigor.

Immediate attack program¶

1. Rewrite the YM page around a gauge-invariant transfer / effective-action gap.
2. Use center symmetry and Polyakov-loop order-parameter structure as the confining anchor.
3. Define the mass gap as the smallest positive Hessian mode / transfer-matrix gap on gauge-invariant states.
4. Only then attach engine observables as support, not as proof.

Novel RTSG contribution¶

Use the coupled action \Gamma[L,W] above and show that a nonzero |W_0| stabilizes the center-symmetric vacuum by shifting the Polyakov-loop Hessian upward:

\Delta_{YM}^2 = a + c|W_0|^2.

Brutal truth¶

The current wiki formulation is still too heuristic for the Clay target. The rewrite is mandatory.

A3. Navier–Stokes Regularity — ACTIVE BUT LOWER PRIORITY THAN RH/YM¶

Why active: Million-dollar problem; the GL layer gives new energy functionals.
What RTSG already has: Lyapunov / turbulence language, engine monitoring, new cubic Will-field unification idea.
What is wrong: The current blow-up line is not dimensionally or tensorially coherent.

Immediate attack program¶

1. Replace the heuristic criterion with shell-wise flux–dissipation defects.
2. Couple the GL functional to high-frequency envelopes rather than raw velocity.
3. Treat turbulence as state-space shadowing of unstable coherent structures, not just λ>0 talk.
4. Keep the Clay target narrowly on smooth-solution continuation; ignore weak nonuniqueness as a separate phenomenon.

Novel RTSG contribution¶

Define the high-frequency “instantiation burden” via \mathcal D_K(t) and attempt a continuation criterion of the form

\sup_K \int_0^T \mathcal D_K^+(t)dt < \infty.

Brutal truth¶

This is not ready for a proof claim. It is a serious research program.

A4. BSD / Hodge / P vs NP / Beal — PARKED¶

These remain real prize targets, but the current RTSG machinery is not the right hammer.

• BSD: arithmetic geometry; current RTSG operator / field language does not yet bite.
• Hodge: deep algebraic geometry and motives; no current RTSG leverage worth trusting.
• P vs NP: current RTSG is continuous-dynamical and geometric, not discrete-complexity sharp enough.
• Beal: lower strategic leverage than RH/YM/NS for the same dollar figure.

Keep them in the portfolio but do not spend prime cycles there now.

Tier B — highest reputational value targets¶

B1. Quantum Measurement Problem — BEST FIT IN THE ENTIRE STACK¶

This is where the bisimulation quotient is strongest.

Why RTSG fits¶

The quotient map gives a concrete structural candidate for “collapse without information destruction.”

Immediate paper target¶

Build papers/companions/quantum_measurement.md around: 1. quotient covariance; 2. induced unitary \bar U_t on the quotient space; 3. Born recovery as quotient-cell norm; 4. deformation parameters \omega_i for possible deviations.

Novel RTSG theorem candidate¶

Quotient Unitarity Theorem (formalized version).
If U_t is unitary on L^2(QS) and bisimulation-covariant, then U_t induces a unitary flow on L^2(PS).

This is the cleanest prestige play available.

B2. Black-Hole Information / Horizon Unitarity — ACTIVE¶

The black-hole literature is still organized around a unitarity conflict that now appears to demand new physics on horizon scales rather than only at the singularity.

RTSG angle¶

Treat the horizon as a bisimulation boundary:

\pi_H : QS_{\mathrm{BH}} \to PS_H.

Define redundancy-removed entropy

S_{\mathrm{red}}(\pi_H)
:= H(\mu_{QS}) - H((\pi_H)_*\mu_{QS}).

Novel theorem candidate¶

Break assumption (2) in Giddings’ black-hole theorem by showing that distinct interior states need not have identical exterior evolution after horizon quotienting, because quotient classes can carry nontrivial induced boundary dynamics.

Brutal truth¶

This is promising, but the mathematics still needs construction of the horizon quotient and induced boundary algebra.

B3. Quantum Gravity Witnesses — ACTIVE, MEDIUM PRIORITY¶

Current literature is explicit that gravity’s quantumness is still open, and that entanglement-only witnesses are not the whole story.

RTSG angle¶

Stage-0 CS can be recast as a minimal quantum memory channel rather than only an entangler.

Define a channel-distance witness:

\mathcal Q_g
:= \inf_{\Lambda \in \mathrm{EB}} \|\mathcal E_g - \Lambda\|_\diamond,

If Stage-0 gravity is genuinely instantiating, RTSG predicts

\mathcal Q_g > 0.

B4. Dark Energy / Cosmology — SALVAGE AND REWRITE¶

The GL layer helps, but only after the invariance fix.

Corrected cosmology line¶

Do not use \langle \rho_W \rangle.
Use the real invariant vacuum-energy density:

\rho_W = |\partial W|^2 + \alpha |W|^2 + \frac{\beta}{2}|W|^4.

\Lambda_{\mathrm{eff}} \sim \langle \rho_W \rangle.

Stronger route¶

Add time dependence through a coarse-grained order parameter W_0(a) and write

\Lambda_{\mathrm{eff}}(a) = \alpha |W_0(a)|^2 + \frac{\beta}{2}|W_0(a)|^4.

B5. Turbulence — ACTIVE COMPANION TARGET¶

Modern turbulence work is increasingly state-space / exact-coherent-structure oriented. That matches RTSG better than the old scalar-Lyapunov framing.

RTSG angle¶

Interpret turbulence as motion through a network of unstable quotient states / coherent structures, with the Will field supplying an energy-landscape description.

Immediate deliverable¶

Write a companion note connecting: - exact coherent structures; - shadowing / recurrent dynamics; - GL energy landscape for coarse-grained turbulent modes.

3. What to write now¶

3.1 arXiv / submission priority after reboot¶

1. GRF essay — keep narrow, horizon-only.
2. Quantum measurement / bisimulation quotient — highest prestige-to-effort ratio.
3. Yang–Mills rewrite — replace plateau-mass rhetoric with gauge-invariant gap functional.
4. Hilbert–Pólya paper — focus on theta-cone closure / density theorem.
5. Navier–Stokes note — cast as flux-defect continuation program, not proof claim.
6. GL Theory of Instantiation — patch invariance and stress-tensor formalism before submission.

3.2 Best immediate wiki patches¶

1. rtsg/will_field_universality.md
2. rtsg/equations.md
3. problems/open.md
4. papers/companions/consciousness.md
5. papers/arxiv/gl_theory_of_instantiation.md

4. Exact patch suggestions¶

4.1 rtsg/will_field_universality.md¶

Replace

\mathcal L_{int} = \beta |W|^2 W

V_{int}(W) = \frac{\beta}{2}|W|^4,
\qquad
\frac{\delta V_{int}}{\delta \bar W} = \beta |W|^2 W.

The quartic term is the invariant interaction density; the cubic term appears in the field equation after variation.

4.2 rtsg/equations.md¶

Replace the cosmological constant line with

\Lambda_{eff} = \left\langle \alpha |W|^2 + \frac{\beta}{2}|W|^4 \right\rangle_{PS}.

4.3 problems/open.md¶

• RH: lower prose temperature; emphasize Step-5 density/closure gap.
• YM: rewrite around center symmetry / transfer gap / Hessian gap.
• NS: replace λ < 0 / λ > 0 wording with high-frequency defect criterion.

4.4 papers/companions/consciousness.md¶

Add the missing assumption explicitly:

q_1 \sim q_2 \Rightarrow U_t q_1 \sim U_t q_2.

4.5 papers/arxiv/gl_theory_of_instantiation.md¶

Patch every place where a non-invariant cubic expression is used as a scalar observable.
Use the quartic potential / cubic EOM distinction everywhere.

5. Final strategic verdict¶

What is strongest right now¶

• Quantum measurement via bisimulation quotient.
• Black-hole information via horizon quotient.
• Yang–Mills rewrite via center symmetry + Will-field stabilizer.
• RH via theta-cone closure.

What is not yet honest enough to claim¶

• RH solved.
• YM solved.
• Navier–Stokes solved.
• Cosmology fully derived.

One-line summary¶

The reread revealed one critical formal bug and one major opportunity:
fix the GL layer first, then build the problem program around quotient unitarity, theta-cone closure, center-symmetry Hessian gaps, and flux-defect continuation.