# n175 — C1, first construction: the dark-pair propagator at bubble
#         level — positive norm, no tachyon, both channels
# (2026-07-11. The first of the six constructions n174 demanded.)

## The three positivity legs
1. KALLEN-LEHMANN, MANIFEST: the dark channels' spectral density is
   |M(k,q)|^2/(4 w1 w2) - a sum of squares. Non-negative by
   construction, in both {+} and {x}.
2. NO GHOST, MEASURED: dPi_E/dOm^2 < 0 at every tested (q, channel)
   - equivalently the Minkowski kinetic coefficient A > 0. The
   collective mode's kinetic term has the right sign everywhere:
   {+}: -0.0051..-0.0039 · {x}: -0.0022..-0.0019 (Euclidean slopes).
3. RESTRICTION PRESERVES POSITIVITY: the dark modes live in the
   constraint kernel; a positive form restricted to a subspace stays
   positive. [T-trivial but load-bearing]

## No tachyon
dPi_E/dq^2 < 0 at every tested point: the induced action ADDS
positive spatial stiffness (G^-1 = U'' - Pi gains +|dPi/dq^2| q^2),
and the bare elasticity's own gradient term is non-negative. So
   G(q, omega) = 1/(A omega^2 - B q^2 + ...),  A > 0, B > 0
in both TT channels: real dispersion, no runaway.

## On-shell stability
The pair continuum's edge weight vanishes at the cone ((w-cq)^2.7,
n140): the collective pole has no open decay channel on shell - no
complex pole, zero width. The same zero that once looked like a
curse completes the propagator's stability.

## Status and boundary
C1 at bubble level: DONE [A-toy for the numbers, T-structural for
positivity]. The boundary, named: this is the free-matter bubble
around the symmetric point; the same three checks must be re-run in
the PINNED (sandpile) state once it is constructed - the
interacting version remains open, as does the speed (B/A vs c^2:
the n149 trigger condition, unchanged). Queue: C2 (the helicity-
(+-2) representation proof under the emergent Lorentz action).