Add Golden Braid Centering Gate documentation

6-Documentation/famm/GOLDEN_BRAID_CENTERING_GATE.md

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+# Golden Braid Centering Gate
+
+## Purpose
+
+Add Fibonacci/golden-ratio centering as a BraidStorm calibration primitive.
+
+This gate is not a solver. It is a stability, survivor, and coarsening test for 16D braid-strand states.
+
+```text
+BraidStorm strand states
+→ braid crossing / hypothesis recombination
+→ golden center pull
+→ collapse-rate receipt
+→ FAMM scar / coarsening decision
+→ NUVMAP Delta-DAG update
+→ Builder-Judge-Warden promotion or block
+```
+
+## Core intuition
+
+Every strand receives the same harmonic reference: the shared 16D center `c`.
+
+Healthy strands contract toward the center at the expected golden rate. Bad crossings, wrong hypotheses, missing boundaries, chirality mismatches, or overfit false centers produce measurable deviation.
+
+```text
+not useful: everything eventually reaches the center
+useful: which strands reach center cleanly, which resist, and which crossings create drift
+```
+
+## Golden contraction law
+
+Let:
+
+```math
+\varphi=\frac{1+\sqrt 5}{2}
+```
+
+For a 16D strand state:
+
+```math
+s_i(t)\in\mathbb R^{16}
+```
+
+with center:
+
+```math
+c\in\mathbb R^{16}
+```
+
+apply:
+
+```math
+s_i(t+1)=c+\varphi^{-1}(s_i(t)-c)
+```
+
+After `t` steps:
+
+```math
+s_i(t)-c=\varphi^{-t}(s_i(0)-c)
+```
+
+Ideal distance law:
+
+```math
+\|s_i(t)-c\|=\varphi^{-t}\|s_i(0)-c\|
+```
+
+## Braid crossing with centering receipt
+
+A normal BraidStorm crossing is:
+
+```math
+\beta_{ij}:(s_i,s_j)\to(s_i',s_j',r_{ij},\epsilon_{ij},\Omega_{ij})
+```
+
+Add golden-centering receipt:
+
+```math
+\beta_{ij}:(s_i,s_j)\to(s_i',s_j',r_{ij},\epsilon_{ij},\Omega_{ij},G_{ij})
+```
+
+where:
+
+```math
+G_{ij}
+=
+\left|
+\frac{\|s_i'-c\|}{\|s_i-c\|}
+-\varphi^{-1}
+\right|
++
+\left|
+\frac{\|s_j'-c\|}{\|s_j-c\|}
+-\varphi^{-1}
+\right|
+```
+
+Ideal crossing:
+
+```math
+G_{ij}=0
+```
+
+Scarred crossing:
+
+```math
+G_{ij}>\Theta_G
+```
+
+## Multi-strand collapse receipt
+
+For strand set `S={s_1,...,s_m}` define:
+
+```math
+\Omega_{\mathrm{center}}
+=
+\sum_i R_i(t)
++
+\left\|\operatorname{Cov}(S_t)-\lambda I\right\|
++
+\left\|\bar s_t-c\right\|
++
+\Omega_{\chi}
+```
+
+where:
+
+```math
+R_i(t)
+=
+\left|
+\frac{\|s_i(t)-c\|}{\|s_i(0)-c\|}
+-\varphi^{-t}
+\right|
+```
+
+and `Omega_chi` is chirality/orientation mismatch when chirality is active.
+
+## Failure modes detected
+
+| Failure | Centering symptom | FAMM action |
+|---|---|---|
+| Bad hypothesis | high collapse residual | scar / coarsen basin |
+| Toxic crossing | sudden center drift | Warden block |
+| Chirality mismatch | asymmetric contraction or sign flip | chirality scar |
+| Overfit local basin | false fast collapse into wrong center | require external Judge receipt |
+| Missing boundary | persistent residual after centering | boundary investigation |
+| Good survivor strand | clean contraction with receipt preservation | promote candidate |
+
+## Coarsening behavior
+
+If a strand or crossing repeatedly fails golden centering:
+
+```text
+failed center pull
+→ residual measured
+→ scar written
+→ basin coarsened
+→ fine search downweighted
+→ route reopened only by new evidence or changed boundary
+```
+
+## Stack placement
+
+```text
+Braid crossing
+→ hypothesis recombination
+→ GOLDEN_BRAID_CENTERING_GATE
+→ FAMM residual/scar measurement
+→ NUVMAP node update
+→ Delta-DAG crossing receipt
+→ BJW Judge/Warden decision
+```
+
+In the larger stack:
+
+```text
+BraidStorm Gossip
+→ PIST / PIST-Chaos transition
+→ Golden Braid Centering Gate
+→ FAMM Scar Ledger
+→ NUVMAP Delta-DAG
+→ Coarsening Agent
+→ Builder-Judge-Warden
+→ Exact / statistical receipt
+```
+
+## Boundary
+
+This is not a claim that all useful dynamics should collapse to a literal center. It is a calibration primitive. The endpoint is boring; the collapse receipt is useful.
+
+## Project sentence
+
+In BraidStorm, Fibonacci centering is a harmonic calibration gate: every strand is pulled toward the shared 16D center at the expected `phi^-1` rate, and any deviation becomes a FAMM scar showing which braid crossings, hypotheses, or dimensions resisted lawful collapse.