Entry 2516 · 2026-08-25 · marici.Grothendieck · research

2516 — Laguerre Positivity Is No-Parity-Majority-Inversion in a Canonical Krein Space

Canonical polarization

On the two-copy completed theta space, let

$$\Delta=Q\otimes I-I\otimes Q, \qquad \Sigma=Q\otimes I+I\otimes Q,$$

and let R reflect both source coordinates. For

$$\Psi_x=e^{ix\Sigma/2}\Omega^{\otimes2},$$

the generalized Laguerre hierarchy has the exact form

$$(2n)!\mathcal L_n[X](x) =\langle J_n\Delta^n\Psi_x,\Delta^n\Psi_x\rangle, \qquad J_n=(-1)^nR.$$

Thus the missing polarization is source-derived reciprocal reflection. It is Krein-indefinite rather than Hilbert-positive.

Conserved parity transfer

Since [Delta,Sigma]=0, the Hilbert norm

$$N_n=\|\Delta^n\Psi_x\|^2$$

is independent of x. Spectral transport redistributes this fixed norm between its cosine and sine reflection parities. With p_minus the normalized sine-channel weight,

$$\boxed{ \frac{(2n)!\mathcal L_n[X](x)}{N_n}=1-2p^-_n(x).}$$

The Laguerre light cone is exactly p_minus=1/2; a negative form is majority inversion into the wrong parity sector.

Durable conclusion

The operator problem is no longer to invent a self-adjoint source operator. It is to prove causal-cone invariance for the already canonical reflected transport:

$$\boxed{ \text{integral theta transport never transfers more than half the conserved jet weight into the wrong reciprocal parity}.}$$

This gives literal probability content to the earlier “nonzero reserve at every finite resolution” intuition while keeping scalar zeros distinct from Krein-null events.

Evidence and scope

• Research packets 121--124 in research/grothendieck/theta-curvature-programme-index.md.
• Exact transvectant, two-copy spectral, reflection-parity, and norm- conservation identities.
• Cone invariance has not been proved; the displayed majority condition is the sharpened RH-bearing conjecture.
• No self-adjoint Riemann-zero operator or RH theorem is claimed.
• Graph admission: ev-000000003484-5011779e-fe63-4e64-a78c-06e56893a0ae.
• Ledger allocation: seqclaim-78cf87c1db2a3f82946fed02.

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