Hello,

I have to admit that I wasn't able to fully grasp the depth of the problem. My answer will therefore be somewhat general. In JAG3D, the a priori dispersion matrix of the observations must be positive definite. Thus, a zero variance (or zero standard deviation) is not allowed. To enforce a specific configuration, small variances (high weights) must be introduced to the stochastic model. To fix a specific measurement, its variance must be small compared to the variances of the other measurements - what matters is the relative ratio, not the absolute value. For example, if you want to fix a distance and all distances have an uncertainty of 10 mm, a standard deviation of 0.1 mm is sufficient to fix that distance adequately.

1) σ of the gyro direction observation → 0 (essentially a "hard constraint"):
- Does this effectively only fix the azimuth relationship between the start and end point, while both points otherwise remain free (translation, connection to the rest of the network)?

Yes, the points remain free (as long as the point type is not set as a e.g. reference point).

- Is there a known numerical lower bound for σ below which the normal equation system becomes unstable/singular?

No. Not only the magnitude of the value, but also the network configuration is critical to numerical stability. Think of a so-called danger circle during a three-point resection.

2) σ as a real, finite value (derived from an uncertainty budget combining several components):
- During the adjustment, how is the "tension" from this one direction observation distributed across the other observations connected to the start/end points? Does it depend purely on the weight/redundancy of the overall configuration, or is there any preferential treatment of the start point vs. the end point?

In general, the start and end points of a measurement are treated as equivalent; no distinction is made during the adjustment. Therefore, the configuration, the quality of the observations, and the weighting strategy are important for estimating the position of a point (regardless of whether the point is a station or a target point).

3) For the specific use case (gyro azimuth lines along a tunnel drive, some with and some without a forced connection to known points): from your perspective, what would be the methodically cleanest way to introduce such an observation – a highly weighted pseudo-observation with a very small σ, a real observation with a realistic σ, or rather a true constraint equation/datum point construction instead of a weighted observation?

I find it difficult to visualize this configuration well enough to make a precise statement. There may be exceptions (for example, if I need an orthogonality condition specified by an object), but generally speaking, I tend to assign a suitable measurement uncertainty to measurements. Your case may be an exception, but at this time I cannot reliably evaluate it.

Kind regards
Micha