Gap 3 Attack — \(H^1\) and \(H^2\) at the Standard Model BRST Point¶

@B_Niko · Sole Author · Deployed by @D_Claude

1. The Question¶

The Standard Model defines a specific BRST operator \(s_{SM} = s_0 + s_1 + s_2\) on the extended state space \(\Gamma\) (fields + ghosts + antighosts + Nakanishi-Lautrup auxiliaries) for the gauge group:

The CS mechanics framework (cs_mechanics.md) identifies:

• \(H^1(s_{SM})\) = tangent space to \(\mathcal{M}_{CS}\) at the SM point = directions the instantiation rule can evolve = possible BSM physics
• \(H^2(s_{SM})\) = obstruction space = forbidden directions = BSM physics that is self-contradictory at second order

If \(H^1 = 0\): The SM is a rigid point in \(\mathcal{M}_{CS}\). No continuous deformations possible. No new gauge forces. The SM is the unique consistent instantiation rule (up to discrete choices).

If \(H^1 \neq 0\): There exist directions in which CS can evolve. These correspond to possible extensions of the SM — new gauge sectors, new stages in the instantiation cascade, BSM physics that is consistent with BRST.

If \(H^2 \neq 0\): Some infinitesimal deformations hit obstructions at second order. These are BSM directions that look consistent at first order but fail. This constrains which extensions of the SM are actually realizable.

2. What Is Known (Literature)¶

2.1 BRST Cohomology of Yang-Mills¶

For a Yang-Mills theory with gauge group \(G\) on a manifold \(M\), the BRST cohomology is related to the Lie algebra cohomology of \(\mathfrak{g}\):

where \(\mathcal{F}\) is the space of local functionals (field-dependent). This is the content of the descent equations (Stora-Zumino).

2.2 Anomalies as \(H^1\)¶

The chiral anomaly is an element of \(H^1(s)\): it is a BRST-closed functional (satisfies the Wess-Zumino consistency condition \(s \mathcal{A} = 0\)) that is not BRST-exact. The anomaly cancellation condition in the SM (\(\sum Y^3 = 0\) per generation) is the statement that this element of \(H^1\) vanishes for the SM field content.

Key result (Barnich-Brandt-Henneaux, 1994): For Yang-Mills + matter in 4D, the local BRST cohomology \(H^{g,n}(s|d)\) (ghost number \(g\), form degree \(n\)) is fully classified:

• \(H^{0,4}(s|d)\) = gauge-invariant Lagrangians (counterterms, deformations)
• \(H^{1,4}(s|d)\) = candidate anomalies
• \(H^{-1,4}(s|d)\) = global symmetries (Noether currents)

2.3 Deformations of BRST¶

The deformation problem for BRST (Barnich-Henneaux, 1993) asks: given \(s\) with \(s^2 = 0\), what are the consistent deformations \(s \to s + g s_1 + g^2 s_2 + ...\)?

The first-order deformation \(s_1\) must satisfy:

The obstruction to extending to second order:

If \([s_1, s_1] = 0\) in \(H^2\), the deformation extends. If not, it's obstructed.

3. Agent Assignments¶

@D_Claude — Algebraic Structure¶

Compute \(H^1(s_{SM})\) using the Barnich-Brandt-Henneaux classification. Specifically: 1. What is \(H^{0,4}(s|d)\) for the SM field content? This gives the space of consistent deformations of the SM Lagrangian. 2. Does the graded structure \(s = s_0 + s_1 + s_2\) introduce additional cohomology beyond the ungraded case? 3. What is the relationship between \(H^1(s_{SM})\) and the space of consistent gauge group extensions \(G_{SM} \to G_{SM} \times G'\)?

@D_GPT — Obstruction Computation¶

Compute \(H^2(s_{SM})\) or bound it. Specifically: 1. For each candidate \([s_1] \in H^1\), compute the obstruction \([s_1, s_1] \in H^2\). 2. Use the Barnich-Henneaux deformation theory to determine which BSM extensions are obstructed. 3. Is GUT unification (\(SU(3) \times SU(2) \times U(1) \to SU(5)\)) an unobstructed deformation? Is \(SO(10)\)? 4. What about adding a \(U(1)'\) (dark photon)? Extra \(SU(2)\)?

@D_Gemini — Deep Think: Spectral Sequence¶

Use the spectral sequence from graded_brst.md Prop 2: 1. The filtration \(F^p\Gamma\) gives a spectral sequence \(E_r \Rightarrow H^*(s)\) that degenerates at \(E_3\). 2. Compute \(E_1, E_2, E_3\) pages explicitly for the SM. 3. Does the spectral sequence structure constrain \(H^1\) and \(H^2\) beyond what the ungraded computation gives? 4. Key question: Does the graded BRST decomposition introduce new obstructions not present in the standard (ungraded) BRST cohomology?

@D_Grok — Literature Search + Numerical¶

1. Search for existing computations of BRST deformation cohomology for the SM. Key authors: Barnich, Brandt, Henneaux, Boulanger, Bekaert.
2. Search for any computation of \(H^2\) for \(SU(3) \times SU(2) \times U(1)\) specifically.
3. Compile a table of all known consistent deformations of the SM BRST complex from the literature.
4. Flag any results that contradict or confirm the RTSG graded structure.

⚠ @D_Grok: Do not fabricate citations. Provide URLs/DOIs for every claim. If you cannot find a source, say so.

4. Expected Outputs¶

Each agent posts results to agents/ai_notes.md under:

## 2026-03-08 · @D_{agent} · Gap 3 Attack — {subtopic}

Desired format: - State the result clearly (theorem, computation, or literature finding) - Mark confidence: PROVED / COMPUTED / LITERATURE / CONJECTURE - Flag contradictions with other agents' results - Flag any result that "proves too much" (if \(H^1 = 0\) trivially, something is wrong)

5. What Victory Looks Like¶

Best case: Explicit computation of \(\dim H^1(s_{SM})\) and \(\dim H^2(s_{SM})\), with the graded BRST spectral sequence showing whether the staging introduces new constraints.

Good case: Bounds on \(H^1\) and \(H^2\) sufficient to make a prediction (e.g., "the SM admits exactly \(k\) independent deformation directions" or "the SM is rigid").

Minimum viable: Literature compilation showing the current state of knowledge, with the RTSG graded structure identified as the novel contribution that could change the answer.

6. Connection to RTSG¶

If \(H^1(s_{SM}) = 0\): The SM instantiation rule is the unique consistent one. No new gauge forces. No new stages. The partition \(2+1+1\) of \((S^2)^4_{\text{int}}\) is rigid. This would be a prediction of RTSG — the framework predicts that no BSM gauge physics exists.

If \(H^1(s_{SM}) \neq 0\): There exist BSM directions. These correspond to activating additional \(S^2\) factors from the infinite tail of \(\Omega = (S^2)^\infty\). The directions in \(H^1\) tell us exactly which extensions are consistent. This constrains BSM model-building from RTSG principles.

Either way, it's a result.