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+# Erdős FAMM / Eigensolid Research Notes
+
+Date: 2026-05-20
+Status: BEAUTIFUL_PROVISIONAL
+
+## Objective
+
+Treat solved Erdős structures and finite witness packets as FAMM scar priors.
+
+Use:
+
+- NUVMAP projection
+- braid aggregation
+- rope collapse
+- eigensolid closure
+- selective residual memory
+
+to reduce combinatorial explosion.
+
+## Core Pipeline
+
+solved/scar packets
+-> NUVMAP projection
+-> braid traversal
+-> chirality filtering
+-> rope aggregation
+-> eigensolid closure
+-> exact receipt verification
+
+## Exact Gate Boundary
+
+Active proof-spine paths required:
+
+- gate = G:111
+- min_omega = 0
+
+All other routes are treated as diagnostic scars rather than active proof paths.
+
+## Color Taxonomy
+
+Green:
+
+core exact receipt spine
+
+Blue:
+
+active zero-residual reserve
+
+Cyan:
+
+wide-alpha cold reserve
+
+Slate:
+
+large-alpha cold reserve
+
+Amber:
+
+failed gate / negative scar only
+
+## Eigensolid Families
+
+Leaf eigensolids:
+
+exact residue-alpha leaves
+
+Residue eigensolids:
+
+collapse multiple leaves by residue class
+
+Alpha eigensolids:
+
+cross-residue alpha families
+
+Phase eigensolids:
+
+chirality / phase closure families
+
+Global eigensolid:
+
+run-now admissible closure body
+
+## Alpha Family Collapse
+
+Observed useful alpha family:
+
+{3, 7, 11, 15, 19, 23}
+
+Expanded rescue family:
+
+{27, 31, 35, 39, 43, 47, 51, 55, 59}
+
+Unused search window:
+
+63..127
+
+## Structural Collapse
+
+For fixed alpha:
+
+x = (p + alpha)/4
+b = p x
+
+Need:
+
+e divides b^2
+
+e == -b mod alpha
+
+Then:
+
+y = (b^2/e + b)/alpha
+z = (e + b)/alpha
+
+The swarm collapses from:
+
+millions of braid candidates
+
+to:
+
+fixed-alpha divisor-residue channels.
+
+## Inclusion-Exclusion Interpretation
+
+Eigensolid closure behaves as topological inclusion-exclusion:
+
+Count lawful braid routes.
+Subtract overlapping duplicates.
+Add back higher-order closure overlaps.
+Continue until only the stable reconstruction body remains.
+
+## Attention Residuals Interpretation
+
+Attention Residuals map directly to FAMM selective reuse.
+
+Instead of all prior braid states contributing equally:
+
+alpha_(t,i) = softmax(M_i - Omega_i + S_i)
+
+Only high-mass, low-obstruction, scar-supported states dominate closure.
+
+## Core Insight
+
+The architecture is not searching proofs naively.
+
+It is constructing admissible reconstruction topology under finite scar guidance.