Research-Stack/6-Documentation/famm/GOLDEN_BRAID_CENTERING_GATE.md

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.

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.

not useful: everything eventually reaches the center
useful: which strands reach center cleanly, which resist, and which crossings create drift

Golden contraction law

Let:

\varphi=\frac{1+\sqrt 5}{2}

For a 16D strand state:

s_i(t)\in\mathbb R^{16}

with center:

c\in\mathbb R^{16}

apply:

s_i(t+1)=c+\varphi^{-1}(s_i(t)-c)

After t steps:

s_i(t)-c=\varphi^{-t}(s_i(0)-c)

Ideal distance law:

\|s_i(t)-c\|=\varphi^{-t}\|s_i(0)-c\|

Braid crossing with centering receipt

A normal BraidStorm crossing is:

\beta_{ij}:(s_i,s_j)\to(s_i',s_j',r_{ij},\epsilon_{ij},\Omega_{ij})

Add golden-centering receipt:

\beta_{ij}:(s_i,s_j)\to(s_i',s_j',r_{ij},\epsilon_{ij},\Omega_{ij},G_{ij})

where:

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:

G_{ij}=0

Scarred crossing:

G_{ij}>\Theta_G

Multi-strand collapse receipt

For strand set S={s_1,...,s_m} define:

\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:

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:

failed center pull
→ residual measured
→ scar written
→ basin coarsened
→ fine search downweighted
→ route reopened only by new evidence or changed boundary

Stack placement

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:

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.