Graded BRST Decomposition and Stage-Dependent GL Potentials¶

Jean-Paul Niko · Sole Author

1. Motivation¶

The RTSG instantiation operator \(C\) is formalized as BRST cohomological reduction: physical space \(PS = H^0(s)\). But this treats instantiation as a single step — either a state passes the BRST filter or it doesn't. The physical universe tells us otherwise: dark matter gravitates but doesn't interact electromagnetically. Quarks are confined. The early universe went through sequential symmetry-breaking phase transitions as it cooled.

Instantiation is not a binary gate. It is a graded cascade.

2. The Gauge Structure¶

The Standard Model gauge group (after coupling to gravity) is a direct product:

After electroweak symmetry breaking (EWSB) at \(\sim 246\,\text{GeV}\):

with \(W^\pm, Z^0\) acquiring mass via the Higgs mechanism.

RTSG identification: Each factor of \(G\) corresponds to an instantiation stage — a distinct level of relational complexity at which QS structure passes into PS observability. The direct product structure is not accidental: it reflects the independence of the BRST operators at each stage.

Source space grounding: For gauge group \(G\) with maximal torus \(T\), the flag manifold \(G/T\) embeds in \((S^2)^{\text{rank}(G)}\) (see Source Space). The rank of the SM group determines how many \(S^2\) factors participate. The graded BRST decomposition is the algebraic shadow of this geometric embedding.

3. Layer 1 — Graded BRST Complex¶

3.1 Stage Assignment¶

We assign instantiation stages by complexity and energy scale, consistent with the cosmological phase transition sequence:

The total BRST operator decomposes as:

3.2 Algebraic Properties¶

Proposition 1 (Graded nilpotency — CORRECTED 2026-03-08 after @D_Gemini adversarial review).

For the internal gauge group \(G_{\text{int}} = SU(3) \times SU(2) \times U(1)\) (direct product), the internal BRST operators satisfy:

However, gravity couples to gauge via the semi-direct product:

Diffeomorphisms drag gauge bundles. The gravity BRST operator \(s_0\) does NOT anticommute with the gauge operators:

where \(\mathcal{L}_{c_0}\) is the Lie derivative along the diffeomorphism ghost. The total BRST operator \(s = s_0 + s_1 + s_2\) remains nilpotent (\(s^2 = 0\)) because the non-anticommutation terms cancel in the total square — this is the standard consistency of gauge + gravity BRST. But the Künneth factorization \(H^*(s) \cong H^*(s_0) \otimes H^*(s_1) \otimes H^*(s_2)\) FAILS.

⚠ Correction history: The original Prop 1 (this session, @D_Claude) assumed a strict direct product with \(\{s_j, s_k\} = 0\) for all \(j \neq k\). @D_Gemini correctly identified this as a fatal error: the SM is a semi-direct product, not a direct product. The internal sectors (\(s_1, s_2\)) do anticommute with each other, but neither anticommutes with gravity (\(s_0\)). This invalidates the Künneth decomposition and makes the spectral sequence non-trivially convergent.

Corollary (Hochschild-Serre replaces Künneth). The correct computational tool is the Hochschild-Serre spectral sequence for the semi-direct product \(\text{Diff}(M) \ltimes G_{\text{int}}\):

The \(d_2\) differential is now potentially nontrivial — it maps internal gauge deformations (\(E_2^{0,1}\)) to gravitational obstructions (\(E_2^{2,0}\)).

RTSG Hochschild-Serre Rigidity Conjecture (@D_Gemini, 2026-03-08): A consistent BSM gauge deformation (anomaly-free, internal-BRST-closed) is mapped by \(d_2\) to a nontrivial element of \(H^2(s_0)\) — a gravitational obstruction. The cross-bracket \([A_0, A_1] = \mathcal{L}_{A_0} A_1\) fails to be BRST-exact. This kills uncorrelated multi-sector BSM modifications and establishes SM rigidity as a consequence of the semi-direct coupling between gravity and gauge.

⚠ Status: Conjecture. The mechanism is mathematically well-motivated (the semi-direct product is established physics). Computing \(d_2\) explicitly for the SM field content is the next step. Assigned to the network.

3.3 The Instantiation Filtration¶

Define a decreasing filtration on the extended state space \(\Gamma\):

where:

• \(F^0\Gamma = \Gamma\) — all QS relational states (raw potentiality)
• \(F^1\Gamma = \ker(s_0) \cap \Gamma\) — gravitationally closed states
• \(F^2\Gamma = \ker(s_0) \cap \ker(s_1) \cap \Gamma\) — gravitationally and electromagnetically closed states
• \(F^3\Gamma = \ker(s_0) \cap \ker(s_1) \cap \ker(s_2) \cap \Gamma = \ker(s) \cap \Gamma\) — fully closed (physical) states

Proposition 2 (Hochschild-Serre spectral sequence — CORRECTED 2026-03-08 after @D_Gemini review).

The semi-direct product \(G = \text{Diff}(M) \ltimes G_{\text{int}}\) defines a Hochschild-Serre spectral sequence:

where \(\mathfrak{diff}\) acts on the internal cohomology via the Lie derivative. The \(d_2\) differential is potentially nontrivial — it maps internal gauge deformations to gravitational obstructions. The sequence need not degenerate at \(E_2\).

Physical content: \(E_2^{0,q}\) = diff-invariant internal gauge cohomology. \(d_2\) checks whether an internal deformation is compatible with the semi-direct structure (gravity dragging gauge). Deformations that break diff-covariance are killed by \(d_2\). Covariant deformations survive.

@D_Claude computation (2026-03-08): \(d_2\) kills only non-covariant deformations. A dark photon (\(U(1)'\)) survives. \(SU(5)\) GUT embedding survives. The SM is NOT rigid against covariant gauge extensions — only against non-covariant ones. This means BSM physics is permitted but must respect the equivalence principle (Stage 0 filter).

For the internal sector: \(H^q(\mathfrak{g}_{\text{int}}) = H^q(\mathfrak{su}(3)) \otimes H^q(\mathfrak{su}(2)) \otimes H^q(\mathfrak{u}(1))\) by Künneth (the internal sectors ARE a direct product — Gemini confirmed this). The grading within the internal sector is sterile. The non-trivial structure comes only from the gravity-gauge coupling.

⚠ Status of Gemini's Hochschild-Serre Rigidity Conjecture: Partially killed by @D_Claude's \(d_2\) computation. \(d_2\) does not obstruct generic BSM gauge deformations — only non-covariant ones. See ai_notes for full computation and debate. Awaiting Gemini's response.

3.4 Dark Matter as Cohomological Obstruction¶

Definition (Dark matter). In the graded BRST framework:

Dark matter consists of states that are BRST-closed under \(s_0\) (gravitationally instantiated — they have well-defined energy-momentum, they curve spacetime, they gravitate) but are not closed under \(s_1\) (electromagnetically invisible — they carry no conserved electromagnetic charge, they neither emit nor absorb photons).

This is not a postulate. It is a derived characterization from the graded structure.

Corollary (Classification of matter by stage):

Quarks live in \(H^0(s_0 + s_1) \setminus H^0(s)\) — electromagnetically instantiated but not color-confined as individual states. Only their color-singlet composites (hadrons) reach \(H^0(s)\).

4. Layer 2 — Stage-Dependent GL Potentials¶

4.1 One GL Action Per Stage¶

Each instantiation stage \(k\) has a complex scalar order parameter \(W_k\) and its own GL functional:

This is the same U(1)-invariant structure as the master Will Field action (see Will Field Universality), applied independently at each stage. The universality argument (unique leading-order nonlinear self-interaction under U(1) symmetry) holds for each \(W_k\) separately.

Physical identification of order parameters:

⚠ Stage 2 note: Confinement is the symmetric phase (\(\langle W_2 \rangle = 0\)), not the broken phase. This is opposite to Stages 0 and 1, where instantiation corresponds to symmetry breaking. For color, the confined (physical, hadron-forming) phase is the one where the \(\mathbb{Z}_3\) center symmetry is preserved. The deconfined (quark-gluon plasma) phase has \(\langle W_2 \rangle \neq 0\) — center symmetry broken. The engine confirms: \(\langle W \rangle = 0.00093 \approx 0\) → CONFINED ✓.

4.2 The Critical Parameter \(\alpha_k\)¶

Each \(\alpha_k\) is the control parameter governing the phase transition at stage \(k\). Near the critical point, Landau theory gives:

where:

• \(T\) = local temperature / energy density
• \(\rho\) = matter-energy density
• \(\mathcal{C}(QS)\) = QS relational graph complexity (a measure of the local informational richness of the QS substrate)
• \(T_c^{(k)}\) = critical temperature for stage \(k\)

Phase diagram for each stage:

Physical critical temperatures (from SM):

4.3 The Cascade Coupling¶

The stage potentials are not fully independent. They are coupled through their vacuum expectation values:

Proposition 3 (Cascade ordering). The effective \(\alpha_{k+1}\) depends on the state of stage \(k\):

where \(\gamma_{k,k+1}\) is an inter-stage coupling constant and \(f\) is a monotone function with \(f(0) > 0\).

Physical content: If stage \(k\) is uninstantiated (\(\langle W_k \rangle = 0\) for Stages 0,1), the inter-stage coupling adds a positive contribution to \(\alpha_{k+1}^{\text{eff}}\), pushing it further into the symmetric (uninstantiated) regime. You need prior stages to be instantiated before subsequent stages become accessible.

RTSG reading: The instantiation cascade is a sequential funnel. Gravity must condense before electromagnetism is meaningful. Electroweak structure must exist before confinement can organize hadrons. This is not imposed — it follows from the coupling structure of the GL potentials.

⚠ Status: Conjecture. The existence of inter-stage coupling is physically motivated (the SM's running couplings depend on the Higgs VEV, confinement scale depends on electroweak parameters), but the specific form of \(\gamma_{k,k+1} \cdot f(\langle W_k \rangle)\) has not been derived from the source space Lagrangian.

4.4 The Higgs Mechanism as Stage 1 GL Phase Transition¶

This is not an analogy. The Higgs mechanism is a Ginzburg-Landau phase transition:

The Higgs field \(\phi\) is the Stage 1 order parameter \(W_1\). EWSB is the moment \(\alpha_1\) crosses from positive (symmetric, no masses) to negative (broken, \(W^\pm/Z^0\) acquire mass). The Higgs VEV \(v = \sqrt{-\alpha_1/\beta_1} \approx 246\,\text{GeV}\) is the Stage 1 condensate amplitude.

Consequence for dark matter: Dark matter lives in \(H^0(s_0) \setminus H^0(s_0 + s_1)\). In GL language, the dark sector has \(\alpha_1^{\text{local}} > 0\) — the Stage 1 GL potential is in its symmetric phase locally around dark matter. The Higgs mechanism has not occurred for dark matter. This is why it doesn't interact electromagnetically — electroweak structure is literally uninstantiated in its vicinity.

4.5 Confinement as Stage 2 GL Phase Transition¶

The Polyakov loop / Svetitsky-Yaffe order parameter (already on wiki — see Master v4 §V):

Mass gap: \(\Delta = \sqrt{2\alpha_2} = 1/\xi_{W_2}\) in the confined phase (\(\alpha_2 > 0\)).

Engine status: \(\langle W \rangle = 0.00093 \approx 0\) ✓. Confinement verified.

5. Stage Transitions — Promotion and Demotion¶

5.1 Promotion (Dark \(\to\) Visible)¶

Promotion from Stage 0 to Stage 1 requires driving \(\alpha_1^{\text{local}}\) from positive (symmetric) to negative (broken) in the vicinity of dark matter. In physical terms: triggering a local electroweak phase transition within a dark matter-dominated region.

Required conditions (from Landau theory):

This requires either: - Sufficient energy density (\(\rho > \rho_c^{(1)}\)) in the dark sector, OR - Sufficient QS graph complexity (\(\mathcal{C} > \mathcal{C}_c^{(1)}\)) — the relational structure has become rich enough to support electromagnetic interactions, OR - Cross-stage catalysis — interaction with existing Stage 1 matter lowers the effective barrier

⚠ Blocking constraint: Conserved charges. In the Standard Model, baryon number is conserved (approximately — violated only by electroweak sphalerons at \(T > T_{\text{EW}}\)). If dark matter carries no baryon number, promotion to baryonic matter requires either:

1. Baryon number violation (possible above EW scale or via sphaleron processes), or
2. Dark matter already carries a hidden baryon-like quantum number that becomes visible upon Stage 1 instantiation, or
3. Baryon number is a topological invariant of the Stage 2 BRST complex (see Layer 4 future work) and is not defined until confinement occurs — in which case Stage 0 \(\to\) Stage 1 doesn't violate it because it hasn't been created yet.

Option 3 is the most natural within RTSG. If baryon number is \(\pi_3(SU(3))\) winding (as in the Standard Model's skyrmion/sphaleron picture), it only exists within the Stage 2 cohomology. Below Stage 2, there is no baryon number to violate.

5.2 Demotion (Visible \(\to\) Dark)¶

Demotion reverses the cascade: drive \(\alpha_k\) back through its critical point into the uninstantiated regime.

Stage 1 demotion = restore electroweak symmetry locally. This requires \(T > T_c^{(1)} \sim 10^{15}\,\text{K}\).

Stage 2 demotion = deconfinement. Requires \(T > T_c^{(2)} \sim 2 \times 10^{12}\,\text{K}\).

Both conditions are achieved in: - Early universe (before \(\sim 10^{-10}\,\text{s}\)): all stages were uninstantiated at sufficiently high temperature - Heavy-ion collisions (RHIC, LHC): brief deconfinement in quark-gluon plasma (\(T \sim 3 \times 10^{12}\,\text{K}\), Stage 2 demotion only) - Neutron star cores (conjectured): possible deconfined quark matter \(\to\) partial Stage 2 demotion - Black hole interiors (conjectured): extreme curvature may drive all \(\alpha_k\) back through critical points \(\to\) full demotion to Stage 0 or raw QS

Proposition 4 (Black holes as demotion environments — Conjecture). In the interior of a black hole, as \(r \to 0\), the effective temperatures diverge, driving \(\alpha_k \to +\infty\) for \(k = 1, 2\) (and \(\alpha_2 \to -\infty\), i.e., deconfinement). All matter is demoted to Stage 0 or below. The singularity is the point where even Stage 0 fails — the bisimulation quotient destabilizes.

⚠ This connects to the Horizon Bisimulation Conjecture: the horizon is a bisimulation equivalence class boundary. Interior and exterior are relationally equivalent but instantiation-grade differs.

5.3 The Arrow of Instantiation¶

Proposition 5 (Thermodynamic arrow — Conjecture). The Drive principle (Axiom 8) provides a thermodynamic bias toward higher instantiation stages. In an expanding, cooling universe:

where \(\Theta\) is the Heaviside function and the sum counts instantiated stages. The number of active instantiation stages increases monotonically (on average) as the universe cools and complexifies.

Physical content: The universe began fully uninstantiated (all \(\alpha_k > 0\) at Planck temperature). As it cooled, stages activated sequentially: gravity \(\to\) electroweak \(\to\) confinement. The Drive principle says this complexification is thermodynamically favored. Demotion is possible but requires extreme local conditions (black holes, heavy-ion collisions) that are rare and transient compared to the cosmic cooling trend.

6. Falsifiable Predictions¶

6.1 From the Graded Structure¶

6.2 From the Promotion Mechanism¶

7. Open Gaps (Honest)¶

1. Source space derivation of gauge sectors. ⚠ IN PROGRESS — see Source Space Gauges. Partition \(2+1+1\) of internal \((S^2)^4\) conjectured to produce \(SU(3) \times SU(2) \times U(1)\) via three-space projections. Key gap: \(SU(3)\) derivation from \(SL(2) \times SL(2)\) via \(\pi_Q\) breaking \(\mathbb{Z}_2\).

2. Inter-stage coupling constants. The cascade coupling \(\gamma_{k,k+1}\) in Proposition 3 is postulated, not derived. The SM's running couplings provide indirect evidence, but a first-principles derivation from the source space is missing.

3. Stage 0 order parameter. ✅ RESOLVED — see Stage 0 Gravity. \(W_0\) = bisimulation stability field on \((S^2)^4\). \(S_0[W_0]\) = Chamseddine-Connes spectral action (EH = kinetic, Λ = mass, Weyl² = quartic). Big Bang = geometric phase transition at \(\alpha_0 = 0\). Remaining sub-gaps: Seeley-de Witt on \((S^2)^\infty\), measure theory on QS, pre-geometric dynamics.

4. Stage 2 sign reversal. ✅ RESOLVED (2026-03-08). The sign reversal is not an artifact — it reflects a genuine structural distinction between two modes of instantiation:

   • Stages 0, 1: Instantiation by symmetry breaking. The condensate \(\langle W_k \rangle \neq 0\) creates new structure (spacetime geometry, electroweak masses). Instantiation = making structure visible by breaking the symmetric vacuum.
   • Stage 2: Instantiation by symmetry preservation. Confinement (\(\langle W_2 \rangle = 0\), center symmetry preserved) hides substructure (quarks) and presents only composites (hadrons). Instantiation = making composites visible by hiding their constituents.

   In BRST language: \(s_0\) and \(s_1\) filter by gauge invariance (states must be diffeomorphism/electroweak-invariant → condensate forms in the invariant sector). \(s_2\) filters by color singlet condition (states must be trivially colored → only composites survive). The sign reversal in \(\alpha_2\) is the algebraic signature of this distinction: creating structure (\(\alpha < 0\), broken) vs hiding substructure (\(\alpha > 0\), confined).

   RTSG reading: The universe uses both modes of instantiation. Low-stage CS creates. High-stage CS confines. Together they produce the layered observability structure of physical reality: geometry is visible (broken), electromagnetic charges are visible (broken), but color is hidden (confined). The two modes are complementary — paralleling cognitive complementarity (synthetic creates, analytical confines/verifies).

5. Topological charges. ✅ RESOLVED — see Topological Charges. Charges are stage-specific invariants: \(B\) undefined (not zero) for dark matter, created by Kibble-Zurek topology during promotion. \(B-L\) = inter-stage invariant.

6. QS graph complexity measure \(\mathcal{C}\). ✅ RESOLVED (2026-03-08). \(\mathcal{C} = -\sum_i p_i \log p_i\) where \(p_i = \lambda_i / \sum_j \lambda_j\) are the normalized eigenvalues of the local QS Laplacian. Spectral entropy: 0 for trivial structure, \(\log N_{\text{eff}}\) for maximal complexity. Computable, intrinsic (no partition choice), connects to source space \(\Delta=2\). See agents/ai_notes.md, 2026-03-08 self-assignment.

8. Relation to Existing Wiki Pages¶

• Master Reference §V: BRST as single operator \(\to\) this page extends to graded decomposition
• Will Field Universality: one GL action for the full Will Field \(\to\) here one GL per stage, same U(1) universality argument
• Source Space: \(G/T \hookrightarrow (S^2)^{\text{rank}(G)}\) provides the geometric origin of the grading
• Horizon Bisimulation: black holes as demotion environments connects here
• YM Honest Assessment: Stage 2 GL / Polyakov loop is this page's Stage 2
• Open Problems: "Instantiation Stage Transitions" problem at 🟡 30% \(\to\) this page is the attack

9. Summary Equations¶