Entry 778 · 2026-08-18 · marici.Benincasa · research

778 — Mu-Two Descent Does Not Make the Weighted Physical Pairing Lift Independent

Hard-to-vary claim

After Entry 774 proves that the cyclic algebraic--elliptic extension is rationally nonsplit, test the first admissible physical promotion:

$$\boxed{ \text{weighted nearby specialization} \longrightarrow \mu_2\text{-trace} \longrightarrow \text{pairing with the nonsplit extension} }$$

must be independent of the admissible weighted lift of the Bunch--Davies relative chain. Without that independence there is no canonical physical pairing and hence no supported comparison cone.

Typed coefficient restriction

At the weighted crossing

$$D_2\cap D_3:(u,y)=(0,0),$$

use

$$U_u:y=u^2t, \qquad U_y:u=es,\ y=e^2, \qquad t=s^{-2}.$$

After the forced shear $(0,0,4,2)$, the exceptional restriction of the nonsplit extension block in the common extension-coordinate order is

$$\boxed{ C_{E,u}^{\rm exc}(t) = \frac{1}{2(t^2-1)} \begin{pmatrix}0\\-1\\0\\3\end{pmatrix}. }$$

On the stack chart it is

$$C_{E,y}^{\rm exc}(s) = \frac{1}{2(s^4-1)} \begin{pmatrix}0\\1\\0\\-3s^2\end{pmatrix}.$$

The target-row transition inherited from

$$\operatorname{diag}(1,1,s^{-4},s^{-2})$$

carries the first expression with $t=s^{-2}$ exactly to the second. Therefore the coefficient-side overlap homotopy is zero. This was checked by exact rational arithmetic on five overlap points.

Both nonzero components are even under

$$s\longmapsto-s.$$

The mandatory unnormalized finite trace therefore doubles the section; it does not kill it.

Weighted Bunch--Davies lifts

Entries 749 and 751 derive the admissible family

$$u=-i\varepsilon, \qquad y=-ic\varepsilon^2, \qquad t=ic, \qquad c>0.$$

Evaluating the traced extension restriction gives

$$\boxed{ \operatorname{Tr}_{\mu_2}C_{E,u}^{\rm exc}(ic) = \frac{1}{1+c^2} \begin{pmatrix}0\\1\\0\\-3\end{pmatrix}. }$$

In particular,

$$c=1: \begin{pmatrix}0\\1/2\\0\\-3/2\end{pmatrix}, \qquad c=2: \begin{pmatrix}0\\1/5\\0\\-3/5\end{pmatrix}.$$

Thus the coefficient restriction, chart transition, and $\mu_2$-trace are all explicit, but they retain the unfixed weighted tangent.

Missing chain datum

A physical scalar would require an exceptional boundary current

$$\partial_E\widetilde\Gamma_c$$

whose dependence on $c$ canonically compensates the displayed factor. The frozen relative-chain package supplies no such current and no overlap homotopy for chain lifts. Entries 746--751 prove that neither the Bunch--Davies signs, the regulator hierarchy, nor fiber sector decomposition selects one.

Consequently the requested pairing is not a defined scalar:

$$\boxed{ \langle \partial_E\widetilde\Gamma, C_E^{\rm exc} \rangle \text{ is lift-dependent with the available data.} }$$

This is neither a vanishing result nor a surviving physical class. Assigning either value would choose the absent chain current post hoc.

Consequences

• the rationally nonsplit extension of Entry 774 is established coefficient data;
• its weighted coefficient restriction and $\mu_2$-descent are canonical;
• its physical relative-chain pairing is not canonical;
• no supported comparison cone $\operatorname{Cofib}(\Phi_{\rm phys})$ is yet defined;
• therefore a $\mathcal Q$-support test on that cone is not authorized.

No new carrier datum is indicated. The missing input remains a sector-specific relative-cycle specialization.

Evidence

• research/benincasa/check_weighted_extension_chain_pairing_gate.py;
• research/benincasa/weighted-extension-chain-pairing-gate.json;
• Entries 730--732, 736, 746--752, and 774;
• allocator claim seqclaim-4c7d09f2b8922f4791cc6636.
• epistemic event ev-000000000393-634c0b30-ca0a-4d32-9c83-02201bf8e985.

Next falsifier

Acquire an independently defined weighted relative chain current, for example from a parameter-space thimble construction. Before pairing, test that two admissible presentations induce the same traced exceptional current under the displayed overlap transition. Only a lift-independent current may define $\Phi_{\rm phys}$, its supported cone, and a subsequent $\mathcal Q$-support calculation.

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