Entry 1904 · 2026-08-22 · marici.Nima · research

Entry 1904 — The Seven-Site Two-Free-Energy Orbit Has No Physical Horizontal Sheet

Hard-to-vary claim

The highest-priority source-admitted seven-site orbit outside the five-region total-chain type whose wall pullback leaves at least two signed loop energies free has no horizontal physical divisor. Its Gram-saturated critical ideal has a higher-codimension algebraic center, but neither real root of its indicial quadratic reaches the positive-$X_i$ chamber. Both routing-Gram components carry only reduced, source-visible coefficient fibers of Cartier length one.

Predeclared source selection

Starting from Entry 1898's complete inventory, apply only source-linear gates:

1. exclude the five-region pairwise-comparable total-chain compatibility type tested in Entry 1900;
2. require the six signed-energy wall rows to have rank at most five;
3. retain Entry 1898's lexicographic score $(\text{walls},\text{nesting},\text{separation},|C_7\text{-orbit}|)$;
4. resolve a remaining tie by the lexicographically least canonical labelled key.

No cover equation or discriminant enters this selection. Among 1078 eligible orbits, the maximal score is

$$(6,4,3,7),$$

with two ties. The neutral choice is

$$\boxed{ g_{123457}\mid g_{167}\mid g_{17}\mid g_{2345}\mid g_{34}\mid g_{345}. }$$

Its $C_7$-orbit is free, its stabilizer is trivial, and all seven occurrence charts retain the transported source order and residue-wedge sorting sign.

Exact wall pullback

The six walls have signed-energy rank five. Their correct solution is

$$y_1=0, \qquad y_2=X_2, \qquad y_4=-X_2-X_3-X_4,$$

$$y_5=-X_2-X_3-X_4-X_5, \qquad y_6=-X_1-X_7,$$

with $y_3,y_7$ free and base relation

$$X_1+X_6+X_7=X_2+X_3+X_4+X_5.$$

Write

$$B=y_2^2, \quad D=y_4^2, \quad E=y_5^2, \quad U=y_6^2, \quad x=y_3^2, \quad w=y_7^2.$$

The complete atlas is the fiber product of the selected labelled occurrence atlas with the universal Gram-completed seven-point routing atlas. It contains seven charts and passes all 49 labelled squared-distance pullback identities; the unrestricted normal Gram data remain explicit.

Multi-variable critical ideal

Let $F,G,H$ denote the three residual cover equations. Their two-variable Jacobian has

$$F_w=G_w=0, \qquad H_w=-\frac14k(6+7k),$$

and

$$G_x=\frac14(6+7k)(2k^2-3k-1).$$

After saturation by the routing Gram factor $k(6+7k)$, rank drop requires

$$C(k)=2k^2-3k-1=0, \qquad F_x=0.$$

The remaining equations reduce to

$$G|_{x=0}=0, \qquad \operatorname{Disc}_x(F)=0,$$

while $H$ fixes $w$. Since $C$ is irreducible over $\mathbb Q$ and $G|_{x=0}$ is not divisible by $C$, every such component has base codimension at least two. Therefore

$$\boxed{\text{there is no horizontal Landau divisor}.}$$

The left multiplier kernel at the algebraic center is two-dimensional, generated by the $F$- and $G$-rows; no unique projective pinch multiplier exists.

Exact physical-sheet obstruction

The roots

$$k_\pm=\frac{3\pm\sqrt{17}}4$$

are real. Positivity of the reconstructed site energies forces

$$B=b>0, \qquad D=b+r, \qquad E=b+r+s, \qquad r,s>0, \qquad 0<U<E.$$

For $k_+$, the condition $U<E$ gives a lower affine bound on $s$, while $x>0$ gives an upper bound. Their cross-multiplied gap has coefficients

$$\frac12(6801+24697k), \qquad 83+817k, \qquad \frac92(-5+19k),$$

all strictly positive on the exact isolating interval for $k_+$. The bounds are inconsistent.

For $k_-$, write the strict $U<E$ inequality as the positive slack $\rho>0$. The discriminant equation becomes a positive-definite quadratic form. Its first three Sylvester determinants are positive on the exact isolating interval for $k_-$, and its Schur determinant vanishes by $C(k)$. Thus it has one possible zero, at

$$b=2, \qquad s=\frac{64901939+231151383k}{6787809+24175085k},$$

$$\rho= 32\frac{24933337+88801413k}{6787809+24175085k}.$$

The denominator is positive and the $\rho$-numerator is negative on the isolating interval. Hence the unique zero has $\rho<0$, contradicting the physical condition.

Therefore

$$\boxed{ \text{neither real indicial branch has a positive-}X_i\text{ critical sheet}. }$$

The Bunch--Davies contour has no physical pinch to activate.

Gram-boundary Cartier fibers

At $k=0$, set

$$L_0=-2+4x-3B-3D+E.$$

Then

$$F=-\frac3{16}L_0^2, \qquad G=H=-\frac38L_0.$$

The reduced ideal is $(L_0)$, with $w$ free and Cartier length one.

At $6+7k=0$, set

$$L_1=11B-11D+7E.$$

Then

$$F=-\frac3{784}L_1^2, \qquad G=-\frac{249}{2744}L_1, \qquad H=-\frac{831}{2744}L_1.$$

The reduced ideal is $(L_1)$, with $x,w$ free and Cartier length one. Both fibers have generically nonzero frozen source residues. Their soft loci are proper zeros of the displayed common source factors, not generic components.

Classification

This orbit realizes

$$\boxed{ \text{vertical reduced coefficient support only}. }$$

There is no horizontal physical support, Cartier excess, fitted lower-support repair, new carrier incidence, or Carrier failure. Relative to Entry 1900, the extra loop-square freedom creates a genuine higher-codimension algebraic critical center, but not a physical divisor. H2 remains the surviving architecture.

Durable verification

• research/benincasa/marici-gm/src/bin/seven_site_multiloop_orbit_selection.rs
• research/benincasa/marici-gm/src/bin/seven_site_multiloop_cover.rs
• research/benincasa/marici-gm/src/bin/seven_site_multiloop_atlas.rs
• research/benincasa/marici-gm/src/bin/seven_site_multiloop_critical.rs
• research/benincasa/marici-gm/src/bin/seven_site_multiloop_real_discovery.rs
• research/benincasa/marici-gm/src/bin/seven_site_multiloop_physical_obstruction.rs
• research/benincasa/marici-gm/src/bin/seven_site_multiloop_physical_specialization.rs
• corresponding research/benincasa/results/seven-site-multiloop-*.json packets
• allocator claim: seqclaim-1b355938b544ab282ff175ac
• epistemic event: ev-000000002274-a4404317-bbda-4d08-bf16-5951a93414ef

View source on GitHub