FINITE NEUTRAL COMPLEX: STATEMENT AND PROOF
Version 1, 26 October 2026. Author: Carles Marin.
This is a certificate of a specified finite complex, not a new theorem.

Let c,s be any complex scalars. Matrices A:C^7 -> C^2 and M:C^5 -> C^7
are exactly those in certificate.json. No circle relation is needed.
Coordinates in C^7: (alpha1,alpha2,beta1,beta2,chiO,chiP1,chiP2).
Use zero-based row and column indices throughout.

1. AM=0, so im M is contained in ker A. The A minor on columns (0,2)
is 1, hence rank A=2 and dim ker A=5 for every c,s.
2. The M minor on rows (0,1,3,4,6), all columns, is -s.
Thus for s!=0, rank M=5, and ker A/im M has dimension zero.
3. The M minor on rows (0,1,4,6), columns (1,2,3,4), is 1.
Thus rank M>=4 everywhere. q=(1,0,0,-1/2,0) is nonzero and Mq=-s h,
where h=(0,0,0,1,0,0,0). At s=0 this gives rank M<=4.
Consequently rank M=4 and dim(ker A/im M)=1 at s=0.
4. Ah=0. At s=0 the row w=(0,-c,0,1,0,0,0) annihilates M,
whereas wh=1. Therefore h represents a nonzero quotient class, and
it spans this one-dimensional quotient. At s!=0, h=M(-q/s).
The disappearance of this class is exact, not a numerical tolerance.
5. rank-nullity gives dim ker M=0 (s!=0) or 1 (s=0).
These are dimensions of different spaces, not additional physical states.

The verifier checks polynomial identities over Q[c,s], not sampled angles.
The proof supplies both upper and lower rank bounds; tests or hashes alone
would not do so. Computation uses existing SymPy exact linear algebra.
Standard reference: rank-nullity and determinant rank criterion.
No mathematical originality is claimed.

PHYSICAL APPLICATION (NOT PART OF THIS CERTIFICATE)
Setting c=cos(theta),s=sin(theta) is allowed. To identify the quotient with
continuous scalar zero modes one must independently prove the reduction
of the chosen weak operator domain to these matrices and control the gauge
range. Matching that realization to the full action, corner conditions and
orbifold remains external. A Wilson-Higgs direction outside this block,
a Higgs mass, an experimental fit, and the vacuum are not certified.
Relevant continuum background: Costabel-McIntosh, arXiv:0808.2614;
boundary fields: Csaki-Hubisz-Meade, hep-ph/0510275, section 2.3.
These references do not assert this particular matrix result.