Spectral Positivity of the Dressed Two-Face Chiral Response
Date: 17 October 2026 Branches: B05, B08 and B11 Claim status: exact matrix-valued spectral theorem and finite certificate for the CP-even spin-projected sector; full fermionic reflection proof remains open
1. Spectral response
Let the two dressed faces be
After projecting spinor indices against admissible Euclidean test spinors, a positive spectral representation has residues
in the CP-even one-generation sector.
The Euclidean response is the matrix Stieltjes function
Every coefficient is positive, so
2. Pole residues and the Jordan cone
Each isolated one-particle residue is rank one:
It therefore lies on an extreme ray of the Lorentz/Jordan cone.
If at least two residue vectors are linearly independent, then their positive sum has
and lies in the cone interior.
This supplies a direct spectral interpretation:
- pure stable channels are boundary rays;
- mixtures of independent chiral overlaps populate the interior;
- positivity of spectral weight is positivity in the Jordan carrier.
3. Lorentzian continuation
Define
For
one has
Thus \(R\) is a matrix-valued Herglotz/Nevanlinna response. It has no negative-residue pole and no upper-half-plane violation of passivity in the certified sector.
4. Reflection-positive time kernel
For positive Euclidean times,
The Osterwalder–Schrader candidate kernel is
It factorizes as
Therefore \(K\) is a Gram matrix and
This establishes reflection positivity for the reduced scalar/chiral-channel kernel represented by the spectral residues.
5. What remains for fermions
Fermionic Osterwalder–Schrader positivity is not identical to scalar Gram positivity. A full proof must restore:
- Euclidean spinor indices;
- the reflection matrix involving the Euclidean time gamma matrix;
- Grassmann ordering signs;
- contact terms;
- gauge-fixed versus gauge-invariant composite definitions;
- continuum spectral integrals and renormalization.
The present theorem says that once the correct spinor projection yields a positive two-face spectral density, its carrier lives naturally in \(H_2(\mathbb R)_+\). It does not prove that every proposed dressed-fermion kernel has that density.
6. Complex spectral phases
Generic residue vectors may be complex:
Their real projections remain positive, but the imaginary off-diagonal channel may encode absorptive or CP-sensitive information. A one-generation real Yukawa parameter does not automatically force every interacting spectral residue to share one real phase.
Accordingly:
- the \(H_2(\mathbb R)\) carrier describes the CP-even reciprocal response;
- the discarded imaginary channel must either vanish by an antiunitary symmetry or be retained as a separate observable;
- the three-generation flavor sector cannot be made entrywise real.
7. Programme consequence
The dressed carrier now passes three successive gates:
It also matches the earlier scalar-interval result, where every lossless pole had a positive rank-one two-face residue. The common structure is a positive matrix-valued response, not the microscopic statistics of the field producing it.
8. Stop-everything and falsification criteria
Positive: the complete dressed spinor kernel admits this positive spectral representation, and a protected relation ties its pole/residue data to the external interval or full-lock parameters.
Negative: the correct fermionic reflection matrix produces an indefinite face residue, or renormalization requires a response outside the \(H_2(\mathbb R)\) cone.
Current status: reduced spectral/reflection positivity established; full fermionic OS gate open.
Unsafe shortcut: cite scalar Gram factorization as a complete proof of fermionic reflection positivity.
9. Certificate
The dressed spectral-response certificate checks positive rank-one residues, interior multipole response, Euclidean Stieltjes positivity, upper-half-plane Herglotz positivity and time-reflection Gram factorization.