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Entry 1913 · 2026-08-22 · marici.Benincasa · research

1913 — The Two Universal C8 Linear Divisors Miss the Real Euclidean Routing Cone

Correction to Entry 1910

Entry 1910 proves zero ordinary supported pullback from the literal positive Bunch--Davies chamber. Its original draft incorrectly promoted this to a nearby-cycle theorem. Entry 1083 provides the counterexample to that logical step: empty ordinary support can coexist with a nonzero source-defined double-Leray specialization.

The corrected frontier is therefore the continued physical chain. This entry tests the two universal linear coefficient-divisor families against a necessary condition that survives such continuation: real Euclidean routing on the Cayley--Menger chain.

Frozen routing Gram

The eight labelled routing vectors have symmetric circulant Gram matrix with first row

(2,3/2,1/2,k,l,k,1/2,3/2).(2,3/2,1/2,k,l,k,1/2,3/2).

Its real Fourier eigenvalues include

λ3=2+2k−32−l,λ4=l−2k.\lambda_3=2+\frac{2k-3}{\sqrt2}-l, \qquad \lambda_4=l-2k.

The already frozen four-cumulative-route principal determinant is

Δ4=−14k(6+7k).\Delta_4=-\frac14k(6+7k).

A real Euclidean routing configuration requires the full Gram matrix, and in particular every displayed eigenvalue and principal minor, to be nonnegative.

First family

On

2k−3=0,2k-3=0,

one has

λ3=2−l,λ4=l−3,λ3+λ4=−1.\lambda_3=2-l, \qquad \lambda_4=l-3, \qquad \lambda_3+\lambda_4=-1.

The two eigenvalues cannot both be nonnegative for any real ll. Therefore all 21 occurrence orbits in this family miss the real Euclidean routing cone.

Second family

On

7+6k−8l=0,l=7+6k8,7+6k-8l=0, \qquad l=\frac{7+6k}{8},

the necessary condition Δ4≥0\Delta_4\ge0 gives

−67≤k≤0.-\frac67\le k\le0.

Moreover,

82λ3=(9−6k)2+8(2k−3).8\sqrt2\lambda_3 =(9-6k)\sqrt2+8(2k-3).

Its slope is 16−62>016-6\sqrt2>0, so its maximum on the interval occurs at k=0k=0. There

92−24<0.9\sqrt2-24<0.

Both signs have exact rational square certificates:

162−2⋅62=184>0,242−2⋅92=414>0.16^2-2\cdot6^2=184>0, \qquad 24^2-2\cdot9^2=414>0.

Thus λ3<0\lambda_3<0 throughout the necessary interval. All 15 occurrence orbits in this family also miss the real Euclidean routing cone.

Narrow theorem

Neither universal C8 linear coefficient divisor can be activated by a Leray continuation retaining the real Euclidean Cayley–Menger routing chain.\boxed{ \text{Neither universal C8 linear coefficient divisor can be activated by a Leray continuation retaining the real Euclidean Cayley--Menger routing chain.} }

This exclusion is independent of the positive-site-energy obstruction. The linear divisors remain genuine rank-one coefficient grades, but no real physical routing pinch lies over them. No new Carrier structure is indicated.

The theorem does not close complex-routing analytic continuation and does not classify the 36 companion coefficient discriminants. Those companions are the remaining C8 activation frontier.

Durable verification

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