Entry 779 · 2026-08-18 · marici.Nima · research

779 — The Weighted Extension Has a Lift-Independent Projective Direction

Entry 778 evaluates the traced exceptional restriction of the rationally nonsplit extension on every admissible weighted Bunch--Davies lift:

$$T(c)= \frac{1}{1+c^2} \begin{pmatrix}0\\1\\0\\-3\end{pmatrix}, \qquad c>0.$$

The scalar value is lift-dependent because the source does not fix $c$. However, $1/(1+c^2)$ is nonzero for every admissible real $c>0$. Therefore all admissible lifts determine the same projective point

$$\boxed{ [T(c)]=[0:1:0:-3] \in\mathbf P(\mathcal E_C^{\rm exc}). }$$

Equivalently, they determine the same rank-one coefficient subspace

$$\ell_{\rm exc} =\mathbf Q\langle(0,1,0,-3)\rangle.$$

This line is compatible with the exact chart transition and is $\mu_2$-even. It is consequently independent of the weighted tangent as a coefficient direction, even though no normalized physical scalar or relative-chain functional is defined.

Scope

The projective line does not repair Entry 778's obstruction:

• it supplies no exceptional boundary current;
• it does not normalize the factor $1/(1+c^2)$;
• it does not define a supported comparison cone;
• it cannot authorize a $\mathcal Q$-support test.

What survives canonically is a possible target line for a future parameter-space thimble current. Any independently derived physical current must pair through this line; a current landing in a transverse coefficient direction cannot represent the weighted specialization of the Entry 774 extension.

Evidence

• Entry 778 and its exact formula for $T(c)$;
• research/benincasa/check_weighted_extension_chain_pairing_gate.py;
• allocator claim seqclaim-9bf097ea42b17749d2118f8d;
• epistemic event ev-000000000394-bce076fd-a8c4-4a3f-bfbb-b7684a66b39a.

Next falsifier

Construct the parameter-space thimble current and test its image in the exceptional coefficient space. If the image is not contained in $\ell_{\rm exc}$, it does not pair with this extension. If it is contained in $\ell_{\rm exc}$, derive its normalization and prove independence under admissible changes of $c$ before forming a supported cone.

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