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+# Builder-Judge-Warden Geodesic Cleanup Filter
+
+## Purpose
+
+Add a triadic cleanup layer for the math forest.
+
+The filter takes a messy equation, proof route, transform stack, NP search state, or FAMM packet and finds the shortest lawful cleanup path that preserves invariants and ends in a receipt.
+
+```text
+messy math object
+→ Builder proposals
+→ Judge receipts
+→ Warden scars/blocks
+→ accepted cleanup geodesic
+→ NUVMAP Delta-DAG
+→ exact receipt
+```
+
+## Core idea
+
+```text
+Builder = proposes cleanup moves.
+Judge   = verifies law/invariant/receipt preservation.
+Warden  = blocks unsafe collapse, drift, false merges, and scarred routes.
+```
+
+## Cleanup loss
+
+```math
+\mathcal L_{\mathrm{BJW}}(E)
+=
+\alpha C(E)
++
+\beta R(E)
++
+\gamma\Omega(E)
+-
+\delta I(E)
++
+\lambda P(E)
+```
+
+Where:
+
+| Term | Meaning |
+|---|---|
+| `C(E)` | complexity / byte / operator cost |
+| `R(E)` | residual or unresolved tail |
+| `Ω(E)` | FAMM scar / bad route pressure |
+| `I(E)` | preserved invariant strength |
+| `P(E)` | proof / receipt burden |
+
+## Builder set
+
+The Builder proposes constructive transformations:
+
+```text
+compactify
+factor
+project
+transform
+bind
+delta-compress
+merge DAG states
+replace history with recurrence
+replace search tree with exact gate
+```
+
+Formal sketch:
+
+```math
+B(E_t)
+=
+\operatorname{argmin}_{E'}
+[
+C(E')+R(E')
+]
+```
+
+subject to a lawful transition candidate.
+
+## Judge set
+
+The Judge checks whether the move preserved the relevant object.
+
+```math
+J(E_t,E')=1
+```
+
+iff:
+
+```math
+\operatorname{Inv}(E_t)=\operatorname{Inv}(E')
+```
+
+and:
+
+```math
+\operatorname{Receipt}(E_t\to E')=\mathrm{PASS}
+```
+
+Judge tests include:
+
+```text
+same theorem
+same payload
+same exact verifier
+same invariant
+same boundary condition
+same residual policy
+same active chirality if chirality matters
+```
+
+## Warden set
+
+The Warden blocks unsafe shortcuts:
+
+```text
+overcompression
+false equivalence
+NUVMAP false merge
+wrong-handed chirality
+missing boundary
+unsealed residual
+unsupported theorem claim
+recurrence outside ROC
+scarred transition repeated too often
+```
+
+Warden penalty:
+
+```math
+W(E_t,E')
+=
+\Omega_{\mathrm{scar}}
++
+\Omega_{\mathrm{drift}}
++
+\Omega_{\mathrm{boundary}}
++
+\Omega_{\mathrm{receipt}}
+```
+
+Reject if:
+
+```math
+W(E_t,E')>\Theta_W
+```
+
+## Transition rule
+
+```math
+E_{t+1}
+=
+\begin{cases}
+E' & \text{if } J(E_t,E')=1 \text{ and } W(E_t,E')\le \Theta_W \\
+E_t+\operatorname{Scar}(E') & \text{otherwise}
+\end{cases}
+```
+
+where:
+
+```math
+E'=B(E_t)
+```
+
+## Accepted geodesic
+
+```math
+\mathcal G^*
+=
+\operatorname{argmin}_{\mathcal G_{\mathrm{clean}}}
+\sum_t
+[
+C(E_t)
++
+R(E_t)
++
+\Omega(E_t)
++
+P(E_t)
+]
+```
+
+subject to every transition passing Judge and Warden.
+
+## Erdős-Szekeres calibration
+
+Original route:
+
+```text
+scan all subsequences
+→ check increasing/decreasing
+```
+
+Builder proposes:
+
+```text
+replace subsequence forest with (I_i, D_i) labels
+```
+
+Judge verifies:
+
+```text
+if a_i < a_j, then I_j > I_i
+if a_i > a_j, then D_j > D_i
+therefore labels are injective
+```
+
+Warden blocks:
+
+```text
+non-distinct sequence
+wrong r/s bounds
+infinite-domain overclaim
+duplicate labels
+```
+
+Cleaned receipt:
+
+```text
+avoidance box has (r-1)(s-1) cells
+sequence has one more point
+injective labels impossible
+PASS
+```
+
+## Stack placement
+
+```text
+Equation forest
+→ Builder proposals
+→ Judge receipts
+→ Warden scars
+→ BJW geodesic
+→ NUVMAP Delta-DAG
+→ cleaned equation
+→ exact receipt
+```
+
+## No-drift boundary
+
+This filter is a cleanup/search governor, not a proof generator by itself. Exactness still comes from the Judge and the downstream verifier. The Warden must block any cleanup path that hides assumptions, changes theorem strength, loses active chirality, or leaves residuals unsealed.
+
+## Project sentence
+
+The Builder-Judge-Warden filter turns mathematical cleanup into a lawful shortest-path problem: Builder proposes compression moves, Judge verifies invariant preservation, Warden blocks drift and scarred routes, and the accepted path becomes a delta-compressed NUVMAP DAG ending in an exact receipt.