Source Space Obstruction — Bisimulation Selection of BSM Physics¶
@B_Niko · Sole Author
Origin
This page emerged from the Gap 3 attack (2026-03-08), where all four network agents (@D_Claude, @D_GPT, @D_Gemini, @D_Grok) converged on the same result: local algebraic BRST cohomology cannot constrain BSM physics. The SM is locally extensible. If RTSG predicts any selection principle on BSM gauge groups, it must be global and topological, not local and algebraic. @D_Gemini proposed the pivot to source space topology. This page develops that proposal.
1. What We Proved (Gap 3)¶
The full Gap 3 computation established:
| Result | Proved by | Method |
|---|---|---|
| \(H^2_{CE}(\mathfrak{g}_{SM}) = 0\) (gauge algebra rigid) | @D_GPT | Whitehead's lemmas |
| \(d_2 \equiv 0\) for covariant deformations | @D_Gemini | Cartan's magic formula |
| BSM = extensions, not deformations | @D_GPT | BBH classification |
| Graded BRST adds no new obstructions on \(M\) | All agents | Convergence |
Bottom line: Local algebraic BRST on target space \(M\) is exhausted. It permits all covariant BSM extensions. Any selection principle must come from elsewhere.
2. The Source Space Selection Principle¶
2.1 The Proposal¶
In standard QFT, gauge groups are freely assigned — you choose a principal \(G\)-bundle over \(M\) and build the theory. Nothing in the local BRST formalism prevents you from choosing any compact Lie group.
In RTSG, gauge groups are NOT free choices. They are the output of the instantiation cascade from source space \(\Omega = (S^2)^\infty\). A gauge group \(G\) is physically realized iff it corresponds to an actual quotient of Aut\((\Omega)\) that passes the BRST filter \(H^0(s)\) and the bisimulation quotient \(QS/\!\sim_{bisim}\).
2.2 Two Levels of Constraint¶
| Level | What it constrains | Where it lives | Status |
|---|---|---|---|
| Local algebraic (BRST on \(M\)) | Anomaly cancellation, perturbative consistency | Target space \(M\) | Permissive (\(d_2 = 0\)) |
| Global topological (bisimulation on \(\Omega\)) | Which \(S^2\) factors can activate | Source space \(\Omega\) | Unknown — this is the frontier |
The local level is solved (Gap 3). The global level is the open problem.
2.3 What Global Bisimulation Selection Might Look Like¶
For a new \(S^2\) factor from \(\Omega\) to activate (producing a new gauge sector), it must:
-
Be BRST-compatible with the existing \(s = s_0 + s_1 + s_2\) — the enlarged complex \(s' = s + s_{\text{new}}\) must satisfy \((s')^2 = 0\). This is local and always satisfiable for a consistent gauge theory (Gap 3 proved this).
-
Be bisimulation-compatible with the existing quotient \(PS = QS/\!\sim_{bisim}\). The new gauge sector must not destabilize the bisimulation equivalence classes that define spacetime points. Adding a gauge field changes the relational structure of QS — the new relations must be compatible with the existing bisimulation.
-
Survive the geometric condensate — the new sector must be compatible with the Stage 0 condensate \(W_0\) (spacetime geometry). This is automatically true for diff-covariant fields (which is all of them, by Gap 3), but may impose additional constraints at the non-perturbative level.
2.4 The Analogy to Global Anomalies¶
Local anomaly cancellation (\(\sum Y^3 = 0\)) is necessary but not sufficient. The Witten \(SU(2)\) anomaly is a GLOBAL consistency condition invisible to perturbation theory.
Similarly, local BRST consistency (\(d_2 = 0\), anomaly-free) may be necessary but not sufficient for physical realization. The bisimulation quotient is a GLOBAL consistency condition invisible to local BRST cohomology.
Conjecture (Source Space Selection Principle): Not all locally consistent gauge extensions of the SM can be physically realized via the instantiation cascade from \(\Omega = (S^2)^\infty\). The set of realizable extensions is constrained by the topology of the bisimulation quotient — specifically, by the requirement that new \(S^2\) factor activation must be compatible with the existing quotient structure \(PS = QS/\!\sim_{bisim}\).
3. What Would Prove or Kill This¶
To prove it:¶
- Show that the bisimulation quotient \(QS/\!\sim_{bisim}\) has nontrivial topology (i.e., \(\pi_n(PS)\) depends on which \(S^2\) factors are active)
- Show that activating a new \(S^2\) factor changes \(\pi_n(PS)\) in a way that's inconsistent with the existing geometric condensate
- Derive a concrete constraint: e.g., "the SM with \(4\) internal \(S^2\) factors is the unique bisimulation-stable configuration" or "at most \(k\) additional factors can activate"
To kill it:¶
- Show that bisimulation quotient is insensitive to the number of active \(S^2\) factors
- Show that any locally consistent extension automatically passes the global bisimulation test
- Find a physical realization of a BSM gauge group that should be forbidden but isn't
4. Connection to Existing RTSG¶
- Stage 0 Gravity: The geometric condensate \(W_0\) defines the existing bisimulation structure. New gauge sectors must be compatible with it.
- Graded BRST: The staged instantiation cascade is real physics. The source space selection principle would constrain which future stages can activate.
- CS Mechanics: The global Maurer-Cartan equation on \(\mathcal{M}_{CS}\) (not the local one on \(M\)) is where the obstruction lives — if it exists.
- Source Space Gauges: The \(2+1+1\) partition of \((S^2)^4_{int}\) may be bisimulation-locked, not just empirically selected.
- Horizon Bisimulation: The bisimulation divergence rate \(\kappa\) already has nontrivial topology at horizons. This machinery may extend to gauge sector activation.
5. Status¶
CONJECTURE. The most important open problem in RTSG theoretical physics as of 2026-03-08. The local algebraic avenue is exhausted. The global topological avenue is the frontier.