Moving the Boundary: The Scalar Displacement Kernel Is Indefinite

Date: 16 October 2026 Branches: B06 and B08 Claim status: exact model result; BCFT displacement-operator identification remains open

1. Why vary the interval length?

The boundary branch needs more than a static response matrix. A displacement operator measures the response to moving a boundary or defect. In the interval model, the elementary geometric deformation is (L\mapsto L+\delta L). At fixed boundary sources,

The matrix

is therefore the fixed-source displacement-response kernel of this reduced model.

It is not yet the full local BCFT displacement two-point function. That stronger identification requires a codimension-one relativistic theory, Ward-identity normalization and contact-term control.

2. Exact eigenresponses

The static even and odd face responses are

Their length derivatives are

and

Consequently

The positive static Jordan Hessian has an indefinite geometric derivative.

3. Exact rational example

For (m=1) and (L=\log3),

so in the left/right face basis

Its determinant is negative. This is not a numerical accident; the opposite monotonicities hold for all (m,L>0).

4. The uncomfortable lesson

Three notions must not be conflated:

1. (X>0): convexity of the on-shell action in boundary values;
2. (D=\partial_LX): response at fixed sources to a geometric displacement;
3. positivity of a properly normalized displacement-operator two-point function in a unitary boundary QFT.

Only the first has been proved here. The second is indefinite. The third lives in a different positivity statement and cannot be inferred from either sign of (D) without deriving the Ward identity and removing scheme-dependent contact terms.

This blocks a tempting shortcut in B08: the positive Jordan cone cannot simply be relabeled as a positive displacement sector.

5. Interpretation through Hamilton–Jacobi evolution

The on-shell action is Hamilton’s principal function for evolution in the interval direction. Its length derivative is controlled by the radial Hamiltonian. The opposing even/odd signs reflect the fact that increasing separation softens the antisymmetric gradient-dominated mode while increasing the symmetric massive response toward its asymptotic value.

This supports the phase-space interpretation: geometric evolution is a Riccati/Hamilton–Jacobi flow on positive response matrices, but its tangent vector need not remain inside the positive cone.

6. Branch consequence

B08 remains viable, but its next gate is now precise. A relativistic boundary model must derive:

• the local displacement operator from broken normal translations;
• its Ward identity and normalization;
• the relation, if any, between its integrated kernel and (partial_LX);
• the appropriate spectral/reflection positivity statement.

If that calculation forces the same (H_2(\mathbb R)) carrier while explaining the indefinite fixed-source derivative, the branch strengthens. If it requires an unrelated structure, the present interface model remains an effective analogy.

7. Stop-everything and falsification tests

Stop-everything positive: the normalized displacement Ward identity reconstructs the same response matrix and fixes one of its coefficients independently.

Stop-everything negative: unitarity requires a displacement kernel whose representation content is incompatible with every (H_2(\mathbb R)) response interpretation.

8. Certificate

0 verifies the exact derivative matrix, its even/odd eigenvalues and its ((1,1)) signature at (m=1,L=\log3).

Companion documents

• Boundary and defect selector
• Scalar interval response
• Quadratic Hamiltonians and Riccati flow