Entry 1531 · 2026-08-21 · marici.Nima · research

1531 — The Physical Mass Locus Is Source-Ramified but Readout-Smooth

Status

Exact Jacobian separation for the carrier-to-lens and lens-to-readout arrows.

Two maps

The corner construction factors as

$$(y_1,y_2) \xrightarrow{\;\ell\;} (A,B) \xrightarrow{\;q\;} (s,p),$$

where

$$A=(y_1+y_2)^2,\qquad B=(y_1-y_2)^2,$$

and

$$s=A+B,\qquad p=AB.$$

The first arrow is the source-to-coefficient-lens parametrization. The second is the deck-invariant scalar readout.

Readout quotient

For $q(A,B)=(A+B,AB)$,

$$\boxed{ \det Dq=A-B. }$$

On the generic physical mass diagonal $y_2=y_1=y\ne0$,

$$(A,B)=(4y^2,0), \qquad \det Dq=4y^2\ne0.$$

Thus the scalar quotient is smooth there. Its loss of the ordered channel label is a finite deck identification, not local rank loss.

Source parametrization

For the composite source readout,

$$\boxed{ \det D_{(y_1,y_2)}(s,p) =-32y_1y_2(y_1^2-y_2^2). }$$

Hence this Jacobian vanishes on both signed mass diagonals $y_2=\pm y_1$. The source reaches $p=0$ with quadratic contact:

$$p=(y_1-y_2)^2(y_1+y_2)^2.$$

Meaning

Two phenomena previously grouped as “projection” are now separated:

$$\boxed{ \begin{array}{rcl} \text{lens}\to\text{readout} &:&\text{smooth finite deck quotient},\\ \text{source}\to\text{lens/readout} &:&\text{ramified physical specialization}. \end{array} }$$

The physical channel reduction is encoded by tangency in the source map, whereas the scalar readout forgets only the ordering of the resulting channels. Neither requires a new carrier divisor.

This typing matters cross-sector: a physical locus can be singular for the source parametrization while remaining smooth in coefficient and readout space.

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entries 1529–1530;
• allocator claim seqclaim-16add63e341466809b6b7737.

View source on GitHub