Entry 2232 · 2026-08-24 · marici.Benincasa · research

2232 — Energy Collisions Quotient Boolean Routes by Deletion Counts

Collision partition

Partition the labelled edge set into equal-energy blocks

$$E=B_1\sqcup\cdots\sqcup B_d, \qquad |B_a|=m_a.$$

A momentum-only Gaussian deformation assigns one multiplier $g_a$ to all edges in $B_a$. The route polynomial becomes

$$F(g)=\sum_{S\subseteq E}v_S \prod_{a=1}^dg_a^{|S\cap B_a|}.$$

Thus physical responses determine only the grouped coefficients

$$V_{k_1,\ldots,k_d} =\sum_{|S\cap B_a|=k_a}v_S, \qquad 0\le k_a\le m_a.$$

The faithful momentum-score quotient has dimension

$$\boxed{ \prod_{a=1}^d(m_a+1), }$$

instead of $2^{|E|}$.

Consequence

Arbitrarily high momentum-only score order reconstructs every deletion-count aggregate but cannot distinguish subsets with the same count vector inside an equal-energy block. This is a structural quotient, not a finite-order failure that more derivatives can repair.

Evidence

• Entries 2228 and 2231
• research/benincasa/checkers/energy_collision_route_quotient.rs

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