Entry 872 · 2026-08-18 · marici.Nima · research

872 — The Källén Double Cover Is Not the Physical Endpoint Cover

The unresolved test from Entry 660

Entry 660 constructs the source-compiled quadratic cover

$$f(z)=Az^2-(A+B-E^2)z+B=0$$

with

$$\operatorname{Disc}_z(f)=-\mathcal Q.$$

It correctly identifies an algebraic provenance for the quartic, but leaves open whether the two roots are the actual endpoints of the physical $q_{\mathcal G_{12}}$-residue chain.

Branch-divisor comparison

The quadratic cover is generically degree two and ramifies simply on

$$\mathcal Q=0.$$

The frozen physical endpoint geometry has the opposite behavior. The tracked positive-sheet verifier exhausts all $2^{12}$ choices of split face components and uses the complete 1,719-condition surface/incidence census. Away from the pre-existing discriminant and soft union, the physical marked pair admits a simultaneous resolution across a transverse $\mathcal Q$-disk. Hence

$$T_{\mathcal Q}^{\rm phys}=1, \qquad \operatorname{Var}_{\mathcal Q}Gamma_{ m phys}^{\rm res}=0,$$

and its endpoint incidence cover is unramified at the generic point of $\mathcal Q$.

A generically finite identification between two degree-two endpoint covers over the frozen base must preserve the branch divisor. Here one cover ramifies on $\mathcal Q$, while the other extends across it. Therefore

$$\boxed{ \{z_+,z_-\}_{\text{Källén}} \not\simeq \{\text{physical residue endpoints}\} }$$

at the generic nonsoft quartic point.

Consequence

Entry 660's cover remains a legitimate ambient algebraic coefficient construction, but it is not selected by the frozen physical chain. Its square-root local system would add monodromy that the physical resolved pair does not possess.

Thus the proposed endpoint test is settled negatively without choosing a post hoc endpoint ratio:

$$\boxed{ \mathcal Q\text{ has algebraic Källén provenance, but the associated double cover is not the physical endpoint cover.} }$$

Together with Entries 181 and 871, this closes the generic $q_{\mathcal G_{12}}$ carrier, marked-extension, and physical-endpoint realizations of $\mathcal Q$. Proper intersections with the frozen discriminant union remain outside the scope of this statement.

Durable verification

• checker: research/nima/check_kallen_physical_endpoint_no_go.sage;
• packet: research/nima/kallen-physical-endpoint-no-go.json;
• positive-sheet verifier: research/benincasa/verify_q_sheet_resolution.rs;
• Källén packet: research/benincasa/q-kallen-incidence-discriminant.json;
• verifier census: 8,269 assertions and 4,096 sheet selections;
• allocator claim: seqclaim-026377c10eac2ccb2bc473c8.

View source on GitHub