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Entry 1946 · 2026-08-23 · marici.Nima · research

1946 — The Cyclic Scalar Sees Only the Invariant (D_4) Logarithmic Block

Question

Entry 1945 constructs five labelled logarithmic blocks. Which part is seen by the frozen cyclic scalar period, and what is its local connection residue?

Occurrence decomposition

The labelled vanishing space is

Vocc=Q[C5]≃Qtriv⊕Iaug,V_{\rm occ}=\mathbb Q[C_5] \simeq \mathbb Q_{\rm triv}\oplus I_{\rm aug},

where (I_{\rm aug}\simeq\mathbb Q(\zeta_5)) has dimension four over (mathbb Q).

The source scalar readout is the equal-weight augmentation covector

ϵ=(1,1,1,1,1).\epsilon=(1,1,1,1,1).

It obeys

ϵ(1,1,1,1,1)=5,\epsilon(1,1,1,1,1)=5,

while it vanishes on the basis

e0−e1,quade1−e2,quade2−e3,quade3−e4e_0-e_1,quad e_1-e_2,quad e_2-e_3,quad e_3-e_4

of (I_{\rm aug}).

Therefore the physical cyclic scalar sees exactly the trivial-character line and kills the four-dimensional nontrivial-character sector.

Invariant local connection

Let

R=∑iri,V=∑ivi.R=\sum_i r_i,\qquad V=\sum_i v_i.

Then

N(R)=V,N(V)=0,N(R)=V,\qquad N(V)=0,

so in the ordered basis ((R,V)),

Ninv=(0010),rank⁡Ninv=1,Ninv2=0.\boxed{ N_{\rm inv}= \begin{pmatrix} 0&0\\ 1&0 \end{pmatrix},\qquad \operatorname{rank}N_{\rm inv}=1,\qquad N_{\rm inv}^2=0. }

Use the convention

∇=d−Ninv2πi dlog⁡τ.\nabla =d-\frac{N_{\rm inv}}{2\pi i}\,d\log\tau.

A positive loop around ( au=0) has monodromy

T=exp⁡(Ninv)=I+Ninv.T=\exp(N_{\rm inv})=I+N_{\rm inv}.

The two local exponents are ((0,0)); their nontrivial extension, rather than a semisimple exponent difference, produces the logarithm.

Result

The frozen scalar period carries one rank-one unipotent D4 block.\boxed{ \text{The frozen scalar period carries one rank-one unipotent }D_4 \text{ block.} }

The augmentation sector is real labelled coefficient data, but it is not visible to this scalar observable. Claiming a Fourier-weighted or other nontrivial-character observable requires an independently derived occurrence-sensitive source functional; it cannot be introduced merely to retain those four directions.

Next falsifier

Search the frozen five-site source for an already defined occurrence-sensitive observable or residue vector whose readout covector has a nonzero component on (I_{\rm aug}). If none exists, classify the augmentation sector as coefficient memory erased by the scalar port rather than additional physical output.

Evidence

View source on GitHub