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Entry 1900 · 2026-08-22 · marici.Nima · research

Entry 1900 — The Maximal Seven-Site Flag Has Only a Reduced Vertical Gram Layer

Hard-to-vary claim

For the source-admitted maximal seven-site flag selected before inspecting its coefficient geometry, the generic critical projection is empty after routing- Gram saturation. Its two Gram-boundary components yield reduced lines of Cartier length one: one is source-visible, while the other lies entirely in the inherited all-soft locus. Hence this orbit carries neither a horizontal physical divisor nor a vertical Cartier excess.

Frozen source orbit

Entry 1898 selected the neutral representative

G−e12∣g134567∣g14567∣g1567∣g167∣g17,G-e_{12}\mid g_{134567}\mid g_{14567}\mid g_{1567}\mid g_{167}\mid g_{17},

from the 88 orbits tied at the predeclared maximal score (6,5,0,7)(6,5,0,7). It has affine face rank seven, denominator rank fourteen, trivial stabilizer, and a free labelled C7C_7-orbit. No discriminant information entered its selection.

Generic cover and wall pullback

Use the C7C_7-invariant increment pairing with distance classes

2,32,12,k.2,\qquad \frac32,\qquad \frac12,\qquad k.

The routing Gram determinant is

det⁡Groute=−14k(6+7k).\det G_{\rm route}=-\frac14 k(6+7k).

On the homogeneous site-energy slice Xi=tX_i=t, the six walls solve as

(y1,…,y6)=t2(−7,−5,−3,−1,1,3),(y_1,\ldots,y_6)=\frac t2(-7,-5,-3,-1,1,3),

with y7y_7 free. Set z=t2z=t^2 and x=y72x=y_7^2. The complete labelled routing atlas contains seven charts and passes all 49 labelled squared-distance pullback identities. Unrestricted normal Gram data remain explicit.

Generic critical ideal

Let F,G,HF,G,H be the pulled-back cover equations. Only HH depends on the free loop square xx, and

∂xH=4k(6+7k).\boxed{\partial_xH=4k(6+7k).}

This is a unit away from the frozen routing-Gram divisor. Therefore

(F,G,H,∂xH):(k(6+7k))∞=(1).(F,G,H,\partial_xH):\bigl(k(6+7k)\bigr)^\infty=(1).

There is no generic critical locus and hence no horizontal Landau divisor, real critical sheet, multiplier, or Bunch--Davies pinch to activate. Ordinary solutions of F=G=H=0F=G=H=0 away from the Gram divisor are transverse in xx and are not critical support.

Gram specializations

At k=0k=0, the critical equations reduce to

F=−12(z−1)2,G=6(z−1),H=12(z−1),∂xH=0.F=-12(z-1)^2,\qquad G=6(z-1),\qquad H=12(z-1),\qquad \partial_xH=0.

Thus the scheme-theoretic ideal is (z−1)\boxed{(z-1)}, with xx free. The simple factors in GG and HH kill the apparent square in FF, so the vertical object is reduced and has Cartier length 1\boxed{1}.

On the source slice t=1t=1, with signed free loop energy r=y7r=y_7, the seven singleton residue factors are

r−52, −5, −3, −1, 1, 3, r+52.r-\frac52,\ -5,\ -3,\ -1,\ 1,\ 3,\ r+\frac52.

Their product is generically nonzero away from the inherited soft points r=±5/2r=\pm5/2. The vertical Gram line is source-visible, but it is not the specialization of a generic vanishing cycle.

The second routing-Gram component is 6+7k=06+7k=0. At k=−6/7k=-6/7,

F=−97249z2,G=−4482343z,H=−29916343z.F=-\frac{972}{49}z^2, \qquad G=-\frac{4482}{343}z, \qquad H=-\frac{29916}{343}z.

Its critical scheme is therefore the reduced line (z)(z), again with xx free and Cartier length one. But z=t2=0z=t^2=0 forces

t=y1=⋯=y6=0t=y_1=\cdots=y_6=0

under the frozen wall solution. The universal and singleton source factors vanish there. This entire second line is inherited all-soft support and is not a new source-visible vertical sector.

Classification

The selected highest-priority source orbit realizes

one source-visible reduced vertical Gram line+ one inherited all-soft reduced line.\boxed{ \begin{gathered} \text{one source-visible reduced vertical Gram line}\\ +\ \text{one inherited all-soft reduced line}. \end{gathered} }

It has no horizontal physical support, vertical Cartier excess, new carrier incidence, or Carrier failure. This refutes direct inheritance of the six-site horizontal-physical-plus-vertical-Cartier architecture for this orbit while supporting the narrower H2 architecture: the existing Carrier remains sufficient and the survivor is coefficient data on an existing Gram boundary.

Next falsifier

Select, from the same frozen inventory, a maximal orbit with a different compatibility type and test whether more than one free loop square survives its wall pullback. A horizontal divisor can occur only when the critical Jacobian is not forced to be a routing-Gram unit as it is here.

Durable verification

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