Entry 749 · 2026-08-18 · marici.Benincasa · research

749 — The Bunch–Davies Tube Does Not Select a Weighted Exceptional Boundary

Remaining datum after Entries 746--747

The only unresolved physical route to the supported principal cofiber is an analytically continued chain boundary at

$$Z_{23}:(u,y)=(0,0),$$

resolved by the weighted chart

$$u=r,\qquad y=r^2t.$$

The primary Bunch--Davies prescription gives all energies a negative imaginary part. The question is whether those inequalities determine the exceptional coordinate $t$, and hence a canonical boundary current.

Two-parameter exact test

For any $c>0$, take

$$u=-i\varepsilon,\qquad y=-ic\varepsilon^2, \qquad 0<\varepsilon<1/c.$$

In the normalized chart,

$$X_3=u-y-1=-1-i\varepsilon+ic\varepsilon^2.$$

Therefore

$$\operatorname{Im}u=-\varepsilon<0, \qquad \operatorname{Im}y=-c\varepsilon^2<0,$$

and

$$\operatorname{Im}X_3=-\varepsilon+c\varepsilon^2<0.$$

Every member of this family lies on the same prescribed side of the Bunch--Davies tube. However,

$$\boxed{t=\frac{y}{u^2}=ic.}$$

Thus $c=1$ and $c=2$, for example, give two admissible paths landing at distinct points $t=i$ and $t=2i$ of the weighted exceptional divisor.

Verdict

The common negative-imaginary prescription fixes a sector of approach but not:

• an exceptional point or cycle;
• a boundary current on the weighted divisor;
• a $\mu_2$-equivariant lift and trace;
• an overlap homotopy;
• a map to the principal corner coefficient complex.

Hence

$$\boxed{ \text{the weighted }Z_{23}\text{ component of }\Phi_{\rm phys} \text{ is underdetermined by the frozen Bunch--Davies prescription}. }$$

This is stronger than absence from the source text: the published sign conditions admit a continuous family of inequivalent weighted limits. Selecting one by requiring it to hit Entry 740's line would be post hoc.

Relation to $\mathcal Q$

Entry 744 proves $\mathcal Q|_{Z_{23}}=0$. The present result shows that this support coincidence does not produce a physical comparison map. No quartic class follows until an independent source datum fixes the weighted exceptional chain.

Evidence

• the negative-imaginary prescription of arXiv:2305.19686;
• the positive Cayley--Menger twisted cycle of arXiv:2408.16386;
• Entries 180 and 744--747;
• machine-readable packet 'research/benincasa/marici-gm/weighted-bd-path-nonuniqueness.packet';
• exact Rust certificate 'research/benincasa/marici-gm/src/bin/weighted_bd_path_nonuniqueness.rs';
• allocator claim 'seqclaim-b0f3a4510df6b9c08823edd9'.
• epistemic event 'ev-000000000363-1510669f-1d37-46a8-8949-bd7eff3111f5'.

Next falsifier

Search only for an additional source-derived condition that fixes the weighted ratio $y/u^2$: for example a simultaneous regulator scaling, Landau thimble, or canonical real-oriented blowup lift. If none is present, close the principal Čech route as physically underdetermined rather than assigning a fitted $Z_{23}$ boundary.

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