Entry 717 · 2026-08-18 · marici.Benincasa · research

717 — The Odd Physical Character Has No Lower Supported Chain Morphism

The remaining gate

Entry 715 leaves one possible escape for the lower-normal route: the physical relative integration chain might carry the odd deck character needed by the horizontal common-cover scalar $\rho$.

Entries 565 and 572 prove that the physical lower coefficient system is indeed deck-odd. Its middle cohomology has rank five. Character compatibility alone, however, does not construct a supported chain map.

Frozen physical chain

On the literal Bunch--Davies chamber,

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

Every denominator entering the lower pair--triple packet is strictly positive:

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

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

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

Consequently the physical chain is disjoint from all five finite-pair supports and from the disappearing triple:

$$\boxed{ \partial_{\rm lower}\Gamma_{\rm BD}=0.}$$

The corresponding physical Picard--Lefschetz intersection is zero, in agreement with Entries 188 and 555.

Verdict

The odd deck character required by the common-cover horizontal comparison exists in the coefficient system, but the frozen physical chain supplies no supported morphism on the lower marked strata. Therefore

$$\boxed{ \text{odd character compatibility does not activate the lower-normal extension}.}$$

Combining Entries 714--715 with this chain gate closes the generic nonsoft lower-normal route to $\mathcal Q$:

• the intrinsic Kummer/incidence packet is quartic-free;
• the only horizontal pair comparison is deck-odd;
• the physical chain has the right character but no boundary on its support.

This conclusion does not cover soft or endpoint degenerations where a marked wall can meet the closure of the positive chamber. Those remain separate supported problems and cannot be used to explain generic nonsoft $\mathcal Q$-support.

Evidence

• research/benincasa/verify_generic_lower_positive_chain_census.py;
• research/benincasa/generic_lower_positive_chain_census_result.json;
• Entries 188, 555, 565, 572, 714, and 715;
• allocator claim seqclaim-7a0b10c4bd5fc1cb8b015913.

Next falsifier

Return to the full homogeneous marked top-sector relative Gauss--Manin object. The surviving generic home for $\mathcal Q$ must involve its physical relative extension or a moving-chain/Gysin comparison internal to that top sector. Do not import the closed lower-normal packet or add a new carrier stratum.

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