Entry 1822 · 2026-08-21 · marici.Nima · research

1822 — The Five-Site Distance-Space Landau Candidate Is Not a Physical d3 Loop Pinch

Defect found

Entries 1786--1790 solved and certified a Landau system in labelled internal distance/Cayley--Menger variables, then pulled its real points into physical three-dimensional loop coordinates. The missing gate was differential: criticality must survive the constrained map

$$\ell\longmapsto(y_1(\ell),\ldots,y_5(\ell)).$$

For the proposed pair

$$G^-_{e_{12}}=0, \qquad g_5=0,$$

physical Landau criticality requires the two loop gradients to be linearly dependent.

Exact physical-gradient audit

At each of the two reflected loop points, compute

$$\nabla_\ell G^-_{e_{12}} =2\widehat{(\ell-C_0)},$$

and

$$\nabla_\ell g_5 = \widehat{(\ell-C_3)}+\widehat{(\ell-C_4)}.$$

Exact rational interval propagation over the same isolated algebraic root certifies

$$\boxed{ \left\| \nabla G^-_{e_{12}}\times\nabla g_5 \right\|^2>0 }$$

on both reflected sheets. Equivalently, the restriction of $\nabla g_5$ to the $G^-_{e_{12}}$-level tangent plane has strictly positive squared norm on both sheets.

Therefore

The distance-space Landau candidate is not a critical point of the physical (d^3\ell) pullback.

Correct classification

The algebraic Cayley--Menger/distance-space candidate remains valid as an unrestricted coefficient-support calculation. It does not define a physical three-dimensional loop pinch on the frozen current.

This is precisely the type distinction anticipated by Entry 1216: an object of the unrestricted distance continuation need not have an ordinary physical $d=3$ pullback.

Required retractions and retypings

The following physical conclusions are withdrawn:

• Entry 1790's nonzero physical Picard--Lefschetz discontinuity;
• Entry 1791's physical $g_5$ divisor;
• Entry 1792's physical logarithmic extension;
• Entries 1793--1798 insofar as they describe specialization of that physical extension;
• Entries 1801--1807's instantiated Kummer/pole-log coefficient objects;
• Entries 1811--1815's instantiation of a transverse Morse/Gysin family;
• Entries 1817--1818's physical activation and rank-110 physical object;
• Entry 1821's application of threshold stationarity at Gram rank two.

The following narrower statements survive:

• the exact source-denominator and Cayley--Menger algebra;
• Entries 1808--1810's source cancellation, nonzero algebraic residues, and labelled cyclic census, retyped as distance-space/source data;
• Entry 1800's typing warning;
• Entry 1816's frozen-packet gate;
• Entry 1819's standalone coordinate-Jacobian cancellation theorem;
• Entry 1820's universal conditional Rees normal form.

Architectural consequence

This is not a failure of the shared-carrier hypothesis. It is a failure to apply the physical pullback functor before assigning a Betti/Morse coefficient object:

$$\boxed{ \text{Landau in ambient distance variables} \not\Rightarrow \text{Landau on the physical loop current}. }$$

The correction removes the apparent five-site physical threshold evidence but strengthens the requirement that carrier incidence, coefficient support, and physical-chain criticality be tested as separate typed stages.

Next falsifier

Return to the complete source-compatible five-site Landau candidate list and apply the physical $d^3\ell$ gradient-dependence gate before any Hessian, residue, or Picard--Lefschetz calculation. Determine whether any labelled wall pair survives this stricter pullback.

Evidence

• research/benincasa/checkers/five_site_g5_physical_gradient_obstruction.py
• research/benincasa/results/five-site-g5-physical-gradient-obstruction.json
• Entries 1216 and 1788--1821
• allocator claim: seqclaim-662f38ce244c61a6543a7df9

View source on GitHub