From 0c1f4112a10fde0ee64fd5253a172c982db24f49 Mon Sep 17 00:00:00 2001
From: Allaun Silverfox <28494262+allaunthefox@users.noreply.github.com>
Date: Sat, 16 May 2026 15:04:56 -0500
Subject: [PATCH] Add common-noise MFG Riccati gate documentation

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+# Common-Noise Mean-Field Game Riccati Gate
+
+## Purpose
+
+Integrate **Linear-Quadratic Mean Field Games with Common Noise: A Direct Approach** into the FAMM/Semantic Mass shortcut stack.
+
+The paper is useful because it gives a mathematically mature version of a move the project keeps making:
+
+```text
+many coupled local actors
+→ shared global field / common noise
+→ population-limit law
+→ reduced Riccati control kernel
+→ decentralized strategy
+→ bounded residual receipt
+```
+
+## Source
+
+- Wenyu Cong, Jingtao Shi, Bingchang Wang.
+- `Linear-Quadratic Mean Field Games with Common Noise: A Direct Approach`.
+- arXiv:2508.07271.
+
+## Why it matters
+
+The paper studies a linear-quadratic mean-field game with common noise where drift and diffusion terms are coupled with state, control, and mean-field state terms. It starts from a finite `N`-player game, derives FBSDEs by variational analysis, then takes the limit as `N → ∞` using law-of-large-numbers reasoning. In the limiting system, existence/uniqueness of BSDEs makes some variables identically zero, reducing the analysis enough to construct decentralized strategies with two Riccati equations. The paper also proves the constructed decentralized strategies form an `epsilon`-Nash equilibrium.
+
+## FAMM interpretation
+
+Common noise is a closure warning:
+
+```text
+if all agents share a shock,
+the shock is part of the system boundary.
+```
+
+The route object is not one local agent. It is the agent plus the population field plus the common-noise channel.
+
+## FAMM object
+
+```math
+\mathfrak C_{\mathrm{MFG}}
+=
+A_{16}(u_{\mathrm{mfg}})
+\otimes
+[
+\Sigma_{\mathrm{agent}}
++
+\Sigma_{\mathrm{mean}}
++
+\Sigma_{\mathrm{common}}
++
+\Sigma_{\mathrm{FBSDE}}
++
+\Sigma_{\mathrm{Riccati}}
++
+\Sigma_{\epsilon\mathrm{Nash}}
++
+\epsilon_{\mathrm{solv}}
+]
+```
+
+## Semantic Mass lanes
+
+```math
+\mu_{\mathrm{MFG}}[k]
+=
+w_m\|m_k\|
++
+w_0\|W^0_k\|
++
+w_u\|u_k\|
++
+w_q J_k
++
+w_r\|R_k\|
++
+w_e\epsilon_{\mathrm{Nash},k}
++
+w_s S_k
+```
+
+Where:
+
+- `m_k` = population / mean-field state.
+- `W0_k` = common-noise shock lane.
+- `u_k` = decentralized control intensity.
+- `J_k` = cost/value lane.
+- `R_k` = FBSDE/Riccati residual.
+- `epsilon_Nash,k` = bounded equilibrium error.
+- `S_k` = solvability/stability status.
+
+## 16D anchor addition
+
+```text
+COMMON_NOISE_MFG_RICCATI_GATE
+```
+
+Recommended axis placement:
+
+```text
+2 semantic mass           population-state pressure
+3 Z pole                  mean-field trajectory recurrence
+4 curvature               Riccati/value curvature
+5 delta-memory            population state as compact history
+6 closure                 common-noise boundary inclusion
+7 residual seal           epsilon-Nash residual
+11 scar                   failed coupling / unsolved FBSDE route
+13 invariant              equilibrium consistency
+14 receipt                Riccati/epsilon-Nash receipt
+```
+
+## Shortcut doctrine
+
+```text
+do not enumerate every strategic interaction
+when the population-limit field plus Riccati kernel carries the lawful structure.
+```
+
+## No-drift boundary
+
+This is a mathematical control/routing witness. It does not prove project-level claims by itself. It gives a rigorous shortcut pattern: common-noise closure, population-limit collapse, Riccati kernel, decentralized strategy, and bounded equilibrium residual.