Entry 2166 · 2026-08-24 · marici.Benincasa · research

2166 — The Missing Physical Circuit Is a Pair of Exceptional Cayley--Menger Moments

Derived target

Entry 2165 leaves the exceptional parity-check row

$$\lambda_{\rm exc}=(\widehat y,-2\widehat q)$$

over the projectivized all-soft Cayley--Menger family. Let $\Gamma_{\rm exc}$ denote the projectivized specialization of the complete Bunch--Davies relative chain, retaining its source orientation, Kummer branch, marked boundaries, and deck character. Let

$$\omega_{\rm exc} = \frac{d_{\rm rel}\widehat a\wedge d_{\rm rel}\widehat b}{W}$$

stand for the corresponding exceptional coefficient form, with all frozen marked factors understood.

Linearity of the current pairing forces the physical covector to be

$$\boxed{ \Lambda_{\rm BD} = \left(I_y,-2I_q\right), }$$

where

$$I_y = \int_{\Gamma_{\rm exc}}\widehat y\,\omega_{\rm exc}, \qquad I_q = \int_{\Gamma_{\rm exc}}\widehat q\,\omega_{\rm exc}.$$

If the pair is finite and nonzero, its kernel is

$$\boxed{ \mathcal L_{\rm phys} = \mathbb Q\langle(2I_q,I_y)\rangle. }$$

Interpretation

The missing cosmological analogue of Strominger's primitive parity-check vector is therefore not another algebraic kernel census. It is a source-normalized pair of exceptional Cayley--Menger moments.

The ratio

$$\zeta_{\rm BD}=\frac{I_y}{I_q}$$

is naturally a projective period coordinate. Nothing derived so far forces it to be a universal rational constant; it may vary over the exceptional external-direction base.

Acceptance contract

The pair defines a physical circuit only after proving:

1. convergence or a source-fixed renormalized value for both moments;
2. independence from the weighted lift used to reach the all-soft exceptional divisor;
3. compatibility with the weight-three projective Kummer cocycle;
4. cyclic transport among the three labelled occurrences;
5. nonvanishing of $(I_y,I_q)$;
6. compatibility with the physical deck character and relative-boundary orientation.

Failure of lift independence means the Tor packet is physically underselected. A varying but well-defined $\zeta_{\rm BD}$ gives a sector-specific physical period line, not a universal integer circuit.

Finite next computation

Construct a two-master relative reduction for the numerator insertions

$$\widehat y\,\omega_{\rm exc}, \qquad \widehat q\,\omega_{\rm exc}.$$

Derive their Gauss--Manin connection and boundary values in one external projective chart. The first falsifier is whether they span a rank-one or rank-two period system:

• rank one with constant rational ratio: Strominger-like universal circuit;
• rank one with nonconstant algebraic ratio: canonical varying line;
• rank two: physical selector remains a genuine two-component period;
• undefined moments: frozen-source activation fails.

Status

• exceptional coefficient packet: established;
• physical covector formula: derived conditionally from current pairing;
• moment values and lift independence: uncomputed;
• new Carrier datum: none.

Evidence

• Entries 828--835 and 2160--2165
• research/benincasa/exceptional-bd-moment-covector.md
• allocator claim seqclaim-ec5e664094c5d59a2d9fc3db

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