claimroutineinferred_dep

Symbol prescription gives K·1_B = 0

Prescribed Legendre symbols give

K\mathbf1_B=0

kernel vector property for the block argument

Claim·§3.4 One parameter for both twists·inside lemOne squarefree n serving both twists

Statement (paper)

With the above prescription of the symbols

[q_i/q_b]=[q_b/q_i]=C_{bi}

for

b\in B

i\in I

[q_0/q_b]=\sum_{i\in I}C_{bi}

for

b\in B

(with reverse symbols given by reciprocity), all other off-diagonal symbols incident with

I

equal to zero, and the

q_0

J

symbols free, the matrix

K

satisfies

K\mathbf1_B=0

on both supports

Q=P

and

Q=P\cup J

In plain English

With the prescribed Legendre-symbol values between the auxiliary primes indexed by

I

, the primes of

B

, and the prime

q_0

(symbols between

q_0

and

J

left free), the matrix

K

kills the all-ones vector on

B

, i.e.

K\mathbf1_B=0

, on both supports

Q=P

and

Q=P\cup J

Why it's here

This checks that the chosen symbols make the row sums over

B

K

vanish. The cancellation uses the fixed sums over

B

and

J

from the previous claim, and the choice

[q_0/q_b]=\sum_i C_{bi}

compensates the contribution of the

I

-primes. It gives the kernel-vector property used in the block-matrix argument for

M_0

in the next claim and thus toward the matrix target of the lemma.

Classified routine:

A direct verification: substituting the prescribed Legendre symbols and using the fixed B- and J-sums from the earlier claim (with quadratic reciprocity) shows that the row sums over B cancel. It is linear-algebra bookkeeping inside the lemma's construction, not a new idea.

Uses

claimFixed symbol sums for B and J

Used by

claimDeleted matrix on smaller support is nonsingular

Sources

PaperOpen

Section: §3.4 One parameter for both twists
Page: —
Label: —
Line: sections/03-parity.tex:627

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Symbol prescription ensures K 1_B = 0" rests on claim-par-simultaneous-matrices-1