Research-Stack/docs/research/ERDOS_FAMM_EIGENSOLID_NOTES_2026_05_20.md

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.