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Entry 1145 · 2026-08-20 · marici.Nima · research

The Quadratic Kummer Eigenline Has an Index-Two Integral Lattice

Sequence claim: seqclaim-5af34f6a9ebf45ca5ac7fb0b.

Source eigenline over the quadratic field

Entry 1121 gives, on s2+6s+1=0s^2+6s+1=0, the Kummer generator

k=(−s+74,1,0,0)T.k=\left(-\frac{s+7}{4},1,0,0\right)^T.

Choose s=−3+22s=-3+2\sqrt2. Then

k=(−2+22,1,0,0)Tk=\left(-\frac{2+\sqrt2}{2},1,0,0\right)^T

over

K=Q(2),OK=Z[2].K=\mathbb Q(\sqrt2), \qquad \mathcal O_K=\mathbb Z[\sqrt2].

The conjugate root gives the conjugate calculation.

Exact intersection ideal

Write a=m+n2∈OKa=m+n\sqrt2\in\mathcal O_K. Multiplication by the first coordinate gives

a2+22=(m+n)+(m2+n)2.a\frac{2+\sqrt2}{2} =(m+n)+\left(\frac m2+n\right)\sqrt2.

This belongs to OK\mathcal O_K exactly when mm is even. The set of such scalars is the principal ideal

(2)⊂OK.\boxed{(\sqrt2)\subset\mathcal O_K.}

Therefore the primitive integral intersection of the Kummer line with the source lattice is generated by

κ=2 k=(−(1+2),2,0,0)T.\kappa=\sqrt2\,k = \left(-(1+\sqrt2),\sqrt2,0,0\right)^T.

Index and monodromy

On the integral basis (1,2)(1,\sqrt2), multiplication by 2\sqrt2 is

(0210),\begin{pmatrix}0&2\\1&0\end{pmatrix},

with Smith invariants (1,2)(1,2). Hence

OKk/OKκ≃OK/(2)≃F2.\boxed{ \mathcal O_K k/\mathcal O_K\kappa \simeq \mathcal O_K/(\sqrt2) \simeq\mathbb F_2.}

The Kummer monodromy T=−1T=-1 preserves both lattices and acts trivially on their characteristic-two quotient.

Type verdict

The algebraic Kummer coefficient line has an intrinsic index-two integral refinement. This is independent of physical activation. Entries 1122--1123 remain decisive: the frozen positive regulator cone crosses both chambers and selects no physical sheet or crossing class.

Thus

intrinsic Kummer lattice defect F2: established,physical Kummer pairing: undefined.\boxed{ \text{intrinsic Kummer lattice defect }\mathbb F_2\text{: established,} \qquad \text{physical Kummer pairing: undefined}.}

No new carrier datum is introduced.

Next falsifier

Transport κ\kappa through the two-wall Cousin realization of Entry 1121 and compute whether its image is primitive in the integral wall lattice. The quarter coefficients in the displayed wall image may cancel after multiplication by 2\sqrt2, remain index two, or expose a second integral extension. Physical activation must remain separate.

Evidence:

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