Entry 716 · 2026-08-18 · marici.Nima · research

716 — The Positive Chain Does Not Activate the Common Lower Kummer Cover

Question left by Entry 715

Entry 715 proves that the horizontal minus-to-plus comparison on Entry 713's common cover is

$$f_{\rm hor}=\rho, \qquad \rho^2=\frac{\Delta^-_{23}}{\Delta^+_{23}},$$

and is deck-odd. It asks whether the physical relative integration chain carries the compensating odd character.

The answer at generic positive kinematics is not “deck-even.” The physical chain does not activate this supported coefficient packet at all.

Positive-chain separation

On the literal Bunch--Davies chamber,

$$a,b,c\geq0,\qquad X_1,X_2,X_3>0.$$

Every lower pole is strictly positive:

$$q_{g_1}=X_1+b+c>0,$$

$$q_{g_2}=X_2+c+a>0, \qquad q_{g_3}=X_3+a+b>0,$$

$$q_{g_{23}}=X_2+X_3+b+c>0.$$

Thus the chain and its closure at generic positive energies meet none of the four lower divisors. In particular they meet no pair or triple support:

$$\boxed{ \Gamma_{\rm BD}\cap V(q_{g_i},q_{g_j})=\varnothing, \qquad \Gamma_{\rm BD}\cap V(q_{g_2},q_{g_3},q_{g_{23}})=\varnothing. }$$

Activation verdict

The supported boundary pairing with the pair--triple Kummer packet is zero. Consequently the physical chain selects neither the even nor the odd deck grade of the common cover:

$$\boxed{ \text{physical deck character on this packet is unselected, not even}.}$$

Choosing one lift of $\Gamma_{\rm BD}$ to $\rho^2=R$, or taking the difference of its two lifts, would introduce an unsourced choice because no lower boundary or Picard--Lefschetz intersection fixes that lift.

Consequence

Entry 715's deck-odd horizontal comparison cannot descend through the generic physical chain because there is no source-derived lower-supported chain class with which to pair it. Combined with Entry 714, this closes the intrinsic generic lower-normal route to $\mathcal Q$:

$$\boxed{ \text{common Kummer coefficient geometry exists} \quad\text{but has zero generic physical activation}.}$$

This does not close analytic continuation, a soft or endpoint degeneration, or a distinct relative-chain class that actually meets the lower support.

Consequence for $\mathcal Q$

Neither the common cover nor its generic physical pairing carries $\mathcal Q$. The remaining typed frontier is the full marked top-sector relative Gauss--Manin/integration-chain extension identified in Entry 714, not another operation internal to the lower packet.

Evidence

• Entries 555 and 706--715;
• research/benincasa/check_generic_lower_physical_kummer_selection.py;
• research/benincasa/check_positive_chain_common_cover_activation.py;
• allocator claim seqclaim-fbbe1b3514c23640a0d469cc.

Next falsifier

Return to the full marked top-sector relative object. Construct the first source-derived integration-chain/Gysin extension coefficient not contained in the intrinsic lower Kummer packet, and test its invariant divisor. Do not import the unactivated lower deck-odd line as a substitute.

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