The Question
What modification of the world would preserve the appearance of this result while making its interpretation false?
Findings
The claim asserts that block-positivity transplants to the pinned upstream zero side. The stated assumptions reveal that the upstream zero side analytically requires the un-conjugated transpose u u^T.
If an implementation or numerical control mistakenly computes the conjugate transpose u u^* (as the stated assumption hints), it constructs a Gram matrix that is positive semi-definite by definition. This modification of the world perfectly preserves the appearance of the result, any block-positivity scan will pass unconditionally.
However, this makes the interpretation strictly false. The actual upstream zero side requires u u^T. For an off-line root $\gamma = \alpha + i\beta$, the vector $u = \hat{\phi}(\gamma - \tau)$ is complex ($u = x + iy$). The pairing of such a root with its conjugate yields:
$u u^T + \bar{u} \bar{u}^T = (x+iy)(x^T+iy^T) + (x-iy)(x^T-iy^T) = 2(xx^T - yy^T)$
This is a hyperbolic block. A matrix of the form $xx^T - yy^T$ has a negative eigenvalue (unless $y=0$, which is only true for on-line roots). Therefore, the true block is not positive semi-definite, and its interaction with other blocks (like the on-line part) can be strictly negative.
Evidence Generated
We ran a dedicated scan (hunts/frontier_math/blind_attack.py) calculating the exact interaction between an on-line pair and an off-line pair using the correct u u^T construction.
Configuration:
- On-line part: weight 2 at center
- Off-line pair: weight 3 at center + 18.58, depth 0.452
Result: The cross-block interaction evaluates to approximately -0.000435, strictly falsifying the assumption that "off-line pair blocks interact non-negatively with on-line part."
Conclusion
The claim blockpos-0.672529 fails. The numerical evidence supporting it was likely gathered on the modified $u u^*$ world, which does not transplant to the required analytic structure.