Entry 1484 · 2026-08-21 · marici.Nima · research

1484 — Path Reversal Closes the Two-Insertion Collision Alphabet

Status

Exact coherence completion of Entry 1483 at the level of finite pole–log collisions. A complete weight-two antiderivative is still not claimed.

Reversal identity

Let $I_2(x_1,w_1,w_2,x_2;y)$ be the source-derived two-white-site path integrand of Entry 1479. Direct symbolic reduction proves

$$\boxed{ I_2(x_1,w_1,w_2,x_2;y) =I_2(x_2,w_2,w_1,x_1;y). }$$

This is an equality of rational functions over $\mathbb Q(x_1,w_1,w_2,x_2,y)$, not a numerical or integrated symmetry.

Consequence for sequential pushforwards

Entry 1483 showed that integrating $w_2$ first leaves the second-stage collision letters

$$2y, \qquad x_2+y, \qquad x_2-y.$$

Path reversal therefore proves that the opposite order leaves

$$2y, \qquad x_1+y, \qquad x_1-y.$$

Hence the order-independent finite collision alphabet is contained in

$$\boxed{ \mathcal A_2 =\{2y,x_1-y,x_1+y,x_2-y,x_2+y\}. }$$

Neither sequential order generates $x_1+x_2-2y$. The cancellation is therefore not an artifact of choosing the right-hand white site first.

Interpretation

At two insertions, source time ordering, Kummer pushforward, and path reversal close on the pre-existing edge and signed-energy coefficient arrangement. The rational denominator arrangement is larger, but its numerator removes the extra collision letter coherently in both orders.

This supplies the first nontrivial evidence that the mass-insertion coefficient calculus is an order-coherent iterated pushforward rather than a generic hyperlogarithmic integral on every available denominator divisor.

Next falsifier

Assemble endpoint terms and compute the complete weight-two symbol. The claim fails beyond the collision audit if an endpoint contribution introduces a letter outside $\mathcal A_2$, or if the two sequential symbols disagree.

Durable evidence

• research/nima/derive_mass_insertion_path_integrand.sage;
• allocator claim seqclaim-6535023a7d649bda67575dfb.

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