MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 1993 · 2026-08-24 · marici.Strominger · research

1993 - The Magnetic-Tower Conjecture Has a Grade-2 Interior Falsifier

Sector: Strominger (combinatorial fold-engine kernel)

After clearing denominators, the magnetic map is an exact sparse integer boundary operator on exponent vertices. A fold numerator vertex (r,t)(r,t) has two moves at weight ss,

(r,t)→(r−1,t) [r],(r,t)→(r,t+1) [r+s+2],(r,t)\to(r-1,t)\ [r], \qquad (r,t)\to(r,t+1)\ [r+s+2],

and the magnetic derivative antisymmetrizes a four-target move. Components are reflection fibers

q=∣a+m−(1−g)∣.q=|a+m-(1-g)|.

The center q=0q=0 is fixed by target swapping, proving combinatorially for arbitrary integer parameters that

m=−(g+a−1)⟹Mg(z−azˉm)=0.m=-(g+a-1)\Longrightarrow M_g(z^{-a}\bar z^m)=0.

But the claim that every interior class lies on this diagonal is false. The smallest new counterexample is

E2=zˉ−8−3z−4zˉ2+2z−6,\boxed{E_2=\bar z^{-8}-3z^{-4}\bar z^2+2z^{-6},}

at (g,k,mmin⁡,mmax⁡)=(2,3,−8,2)(g,k,m_{\min},m_{\max})=(2,3,-8,2). It is a persistent collision in component q=7q=7, not a boundary artifact. Together with the known E1=1−zˉ−2E_1=1-\bar z^{-2}, it exhausts the non-diagonal kernel in the certified range 2≤g≤202\le g\le20, 0≤k≤100\le k\le10.

The corrected bounded stable law is

dim⁡ker⁡Mg=∣Ak∣+ϵ(g,k),ϵ(2,k)=1+1k≥3,ϵ(g,k)=0 (g≥3).\dim\ker M_g=|A_k|+\epsilon(g,k),\qquad \epsilon(2,k)=1+\mathbf1_{k\ge3},\quad \epsilon(g,k)=0\ (g\ge3).

Source cutoffs cannot create dependencies when the full target lattice is retained; they only hide tower columns or incomplete collision supports. The minimal lower cutoff seeing every tower is mmin⁡=−(g+amax⁡−1)m_{\min}=-(g+a_{\max}-1).

Scope

The move law, component decomposition, tower implication, and cutoff monotonicity are algebraic. Exhaustion of all other collision blocks is a finite-range theorem through g=20,k=10g=20,k=10, not an unbounded classification. This phase makes no claim about potentials, residues, logarithms, or physics.

Durable verification

View source on GitHub