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Entry 1906 · 2026-08-22 · marici.Benincasa · research

Entry 1906 — The Complete Seven-Site Inventory Has No Horizontal Physical Divisor

Hard-to-vary claim

No orbit in the complete frozen seven-site source inventory carries a horizontal physical divisor in the strictly positive site-energy Bunch--Davies chamber.

This is a finite exhaustion theorem over 2,104 source-admitted C7C_7-orbits, not an all-site or all-loop theorem.

Frozen partition

Before constructing any cover discriminant, partition every orbit by:

  1. signed-energy wall rank;
  2. compatibility-poset isomorphism type with occurrence type and region cardinality retained;
  3. number of generic free signed energies;
  4. augmented homogeneous-diagonal rank.

Within each class, retain Entry 1898's source score and use the least canonical labelled key only as a neutral tie-break. No discriminant enters the ranking.

The exact census is

wall rankfree energiesorbitsclasses431044252974338611026348.\begin{array}{c|c|c|c} \text{wall rank}&\text{free energies}&\text{orbits}&\text{classes}\\ \hline 4&3&104&42\\ 5&2&974&338\\ 6&1&1026&348. \end{array}

Every rank-four and rank-five orbit meets the homogeneous diagonal only at the origin. Every rank-six orbit retains a generic homogeneous line.

Universal Jacobian criterion

Solve each labelled six-wall system over Q\mathbb Q, preserving all occurrences and induced base relations. Pull its seven signed energies into the same universal Gram-completed routing atlas as Entries 1900 and 1904. For the residual cover equations

F=(F1,F6,F7),F=(F_1,F_6,F_7),

form the maximal-minor ideal of the Jacobian in the actual free signed energies. Remove only the declared routing-Gram factor

k(6+7k)k(6+7k)

and source-derived soft factors before testing for a common codimension-one factor.

The orbit-level result is

r=5:974/974 residual common GCDs are units,r=6:1026/1026 residual common GCDs are units.\boxed{ \begin{aligned} r=5:&\quad 974/974\text{ residual common GCDs are units},\\ r=6:&\quad 1026/1026\text{ residual common GCDs are units}. \end{aligned} }

Thus neither positive-cone-feasible lane contains a horizontal Jacobian divisor. Any remaining critical locus has codimension at least two.

Rank-four algebraic candidates

Every rank-four Jacobian determinant is nonzero. After saturating by routing Gram, all seven pulled soft forms, and every remaining frozen connected-region or fused-facet wall, all 104 determinants retain a nonconstant factor. These are genuine algebraic coefficient discriminants, not fitted Carrier strata or inherited lower incidence.

For each candidate, a source-row-normalized projective multiplier is supplied by a nonzero row of adj⁡(J)\operatorname{adj}(J). It obeys the exact identity

λJ=det⁡(J)ei,\lambda J=\det(J)e_i,

so λJ=0\lambda J=0 on the discriminant without choosing a solver section.

Exact physical exclusion

The six wall equations induce two labelled homogeneous relations among the XiX_i in every rank-four orbit. Gordan's alternative gives, for each of the 104 row spaces, a nonzero nonnegative vector. Hence

ker⁡(AX)∩R>07=∅\ker(A_X)\cap\mathbb R_{>0}^7=\varnothing

for every rank-four orbit.

Therefore none of the 104 algebraic discriminants has a positive-XiX_i real sheet, and the Bunch--Davies contour has no physical pinch to activate.

Labelled atlases and Gram fibers

The certificate materializes all 104 candidate occurrence atlases. Each has seven transported C7C_7-charts with source order, canonical display order, residue-wedge sorting sign, orbit size, and stabilizer retained. They are fibered with the universal routing atlas certified by 49 exact labelled squared-distance pullback identities.

At both routing-Gram boundaries

k=0,6+7k=0,k=0, \qquad 6+7k=0,

all 208 candidate fibers are principal and squarefree. Each has Cartier length one. No hidden Gram-boundary excess appears.

Classification

The complete frozen inventory decomposes as

104genuine algebraic discriminants, physically empty974higher-codimension critical centers only1026higher-codimension critical centers (possibly empty).\boxed{ \begin{array}{c|c} 104&\text{genuine algebraic discriminants, physically empty}\\ 974&\text{higher-codimension critical centers only}\\ 1026&\text{higher-codimension critical centers (possibly empty)}. \end{array} }

There is no horizontal physical support, fitted lower-support repair, new Carrier incidence, or Carrier failure. H2 survives this seven-site exhaustion.

Scope

This entry does not assert that seven-site coefficient discriminants are absent: 104 are present algebraically. It asserts that the only divisorial lane is excluded from the positive physical chamber by source-linear base relations. It does not extrapolate beyond the frozen seven-site inventory.

Durable verification

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