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Entry 1532 · 2026-08-21 · marici.Nima · research

1532 — Unequal Mass First Appears at Second Normal Grade

Status

Exact normal form of the source-to-readout map around the physical mass diagonal.

Normal coordinates

Set

yˉ=y1+y22,δ=y1−y2.\bar y=\frac{y_1+y_2}{2}, \qquad \delta=y_1-y_2.

The physical equal-edge locus is δ=0\delta=0. The two channel variables are

A=4yˉ2,B=δ2.A=4\bar y^2, \qquad B=\delta^2.

Therefore the invariant readout coordinates have the exact normal form

s=4yˉ2+δ2,p=4yˉ2δ2.\boxed{ s=4\bar y^2+\delta^2, \qquad p=4\bar y^2\delta^2. }

Readout jet

The first two sufficient scalar coefficients become

C(2)=8yˉ2+2δ2,\boxed{ C^{(2)}=8\bar y^2+2\delta^2, }

and

C(4)=32yˉ4+8yˉ2δ2+2δ4.\boxed{ C^{(4)} =32\bar y^4 +8\bar y^2\delta^2 +2\delta^4. }

There is no term linear in δ\delta. More generally, deck invariance δ↦−δ\delta\mapsto-\delta forces every scalar coefficient to be an even function of the normal coordinate.

Hence

unequal-mass departure is invisible at first normal grade and begins at grade two.\boxed{ \text{unequal-mass departure is invisible at first normal grade and begins at grade two.} }

Meaning

The doubled normal behavior is derived from the source involution and the invariant readout; it is not a fitted filtration weight. The natural transverse coordinate seen by physical scalar data is

δ2,\delta^2,

not δ\delta.

This provides a precise local model for why weighted or doubled normal geometry can be required even when the underlying carrier divisor is ordinary: the coefficient/readout functor sees only the invariant second normal grade.

Durable evidence

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