Entry 1521 · 2026-08-21 · marici.Nima · research

1521 — Every Regular Carrier Specialization Commutes with the Full Infinity Jet

Status

All-grade base-change theorem. Entry 1519 supplies an independent exact check through (C^{(2)}) for the physical mass diagonal.

Setup

Let (A) be a coefficient domain, (X=x_v), and let a finite source integrand have the rational form

$$I(X)=\frac{P(X)}{Q(X)}\in\operatorname{Frac}(A)(X), \qquad \deg Q-\deg P=d+1.$$

Put (z=X^{-1}). After extracting the known power,

$$I(X)=z^{d+1}\frac{\widetilde P(z)}{\widetilde Q(z)}, \qquad \widetilde Q(0)=q_0.$$

The full infinity jet is the unique formal series

$$\frac{\widetilde P(z)}{\widetilde Q(z)} =\sum_{k\ge0}C^{(k)}z^k$$

over (A[q_0^{-1}]).

Regular-specialization theorem

Let (\varphi:A\to B) be a carrier specialization. If

$$\boxed{\varphi(q_0)\ne0,}$$

then (\varphi(\widetilde Q)) remains a unit in (B[[z]]). Since inversion of a formal power series with invertible constant term is functorial,

$$\varphi\!\left(\widetilde Q^{-1}\right) =\varphi(\widetilde Q)^{-1}.$$

Therefore coefficient extraction commutes with (\varphi) at every grade:

$$\boxed{ C^{(k)}_{\varphi(I)}=\varphi(C^{(k)}_I) \quad\text{for every }k\ge0. }$$

Equivalently,

$$\boxed{J_\infty\circ\varphi=\varphi\circ J_\infty}$$

as a square of complete filtered coefficient objects.

Singular locus

The theorem also identifies the only algebraic failure gate:

$$\boxed{\varphi(q_0)=0.}$$

There the degree of the specialized denominator drops, the normalized leading power may change, and the old jet need not base-change. Such a locus requires a new associated-grade or nearby-cycle calculation; it cannot be repaired by formally specializing the generic coefficients.

Physical mass diagonal

For the split bivalent mass-insertion packet, Entry 1495 proves that

$$y_R=y_L$$

preserves the degree gap (3). Its leading denominator coefficient therefore survives, and the theorem upgrades Entry 1519 to

$$\boxed{ J_\infty\circ\Delta_m^* =\Delta_m^*\circ J_\infty }$$

at all grades. The exact (C^{(0)},C^{(1)},C^{(2)}) comparison remains a nontrivial implementation check of this formal argument.

Meaning

The infinity-jet construction now has two independent functorialities:

1. triangular recursion under source edge deletion (Entry 1516);
2. base change under every regular carrier specialization (this entry).

This sharply separates ordinary physical diagonals from genuine boundary degenerations. Only a specialization that kills the leading denominator coefficient can create a new infinity-jet grade.

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entry 1495 (mass-diagonal degree stability);
• Entry 1519 (exact base-change check through (C^{(2)}));
• allocator claim seqclaim-fc373e5dc1905ad9dbc15c8d.

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