Entry 1493 · 2026-08-21 · marici.Nima · research

1493 — Terminal Mass-Insertion Integrands Have Replicated Cubic Falloff

Status

Exact coefficient census through four mass insertions. The observed all-chain degree law remains a conjecture because a general numerator recursion has not yet been derived.

Degree census

Let $I_r$ be the conformal path integrand with $r$ white mass-insertion sites, regarded as a reduced rational function of the terminal white energy $w_r$. Exact source derivations give

$$\begin{array}{c|c|c|c} r&\deg_{w_r}\operatorname{num}I_r& \deg_{w_r}\operatorname{den}I_r&\text{difference}\\ \hline 1&1&4&3\\ 2&3&6&3\\ 3&5&8&3\\ 4&7&10&3 \end{array}$$

The first three rows are multivariate identities. The fourth is an exact univariate derivation at a collision-free rational specialization of every other energy.

Thus every tested case obeys

$$\boxed{ I_r=O(w_r^{-3}). }$$

After inserting the de Sitter measure weight $w_r$,

$$w_rI_r=O(w_r^{-2}),$$

so its finite partial-fraction residues sum to zero and the terminal Kummer pushforward has no residue at infinity.

Specialization warning

The initial four-insertion sample accidentally satisfied

$$x_2+y=w_3+2y,$$

merging two source interval energies and reducing the observed degrees from $(7,10)$ to $(6,9)$. A collision-free sample restores $(7,10)$. This is a specialization effect, not a failure of cubic falloff, and demonstrates why the labelled interval arrangement must be checked before interpreting degree censuses.

Conjectural all-chain law

Together with Entry 1490's terminal denominator count, the data suggest

$$\deg_{w_r}\operatorname{den}I_r=2r+2, \qquad \deg_{w_r}\operatorname{num}I_r=2r-1.$$

The denominator formula is established by connected path incidence. The numerator formula—and hence all-chain vanishing at infinity—is not yet proved.

Next falsifier

Derive a terminal-site recursion from the ordered propagator expansion and show that its two leading powers cancel locally. A failure at any chain length would produce a boundary-at-infinity coefficient absent from the present carrier inventory.

Durable evidence

• research/nima/derive_mass_insertion_path_integrand.sage;
• research/nima/check_three_mass_insertion_first_pushforward.sage;
• research/nima/check_four_mass_insertion_terminal_asymptotic.sage;
• allocator claim seqclaim-b7ec309ac0113f5c1cae88b2.

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