A report archived by the project, published as it was written. All reports · Markdown source

P1 proved for every cell: a curl-free frozen state has a slab-type type field (linear algebra in the chain-weight picture, the kernel step proved by the stratified Fourier method), and the bond state is then a slab sum because the profiles must close on the torus

Claude (Fable 5.1), main loop, for Lyndon Drake

2026-09-17

1. Target statement

P1 (curl-reduction note, “checked, not proved”, certified on six cells): every ice state with even curl on every hexagon is a slab sum, \(s_e = X_{x(e)}+Y_{y(e)}+Z_{z(e)} \pmod 2\) for profiles on the three families of slabs. With the profile lemma (proved) it gives Theorem P for every cell, and P1 is the only unproved step on the P side of the periodicity theorem. The analogy reading of today (2026-09-17-analogies.md) pointed at it as the one-hot statement to attack first; it turned out to split into a linear statement and an elementary bridge.

2. Outcome

P1 is proved for all cells. The linear step (L) is verified numerically on eleven cells and proved for every cell by the stratified Fourier method (probe K, 2026-09-17-sc-linear.md: the refined per-sublattice match system is block diagonal, its kernel on every stratum is spanned by the slab modes, with the two sublattice-antisymmetric constants added at the sixteen lifts of wavevector zero and nowhere else; main loop re-ran both scripts). The proof:

  1. (Lemma C) even curl implies frozen, so the type field is \(\tau=e_x+\sum_C f_C\,m_C+\text{const}\) (completeness, proved today).
  2. The A-match count of a hexagon \(h\) is \(1+L_h(f)\) with \(L_h(f)=\sum_{v\in h\cap A}\sum_{C\ni v}(w_{E(C)})_{\alpha_h(v)}\); the constants contribute \(0\) because the three A vertices of \(h\) carry the three distinct axes. By Lemma L a hexagon is even iff it has one match on each sublattice. So curl-free \(\iff L f=0\).
  3. (L) \(\ker L = S+G\): the slab-constant weights plus the gauge. Verified numerically (ranks, r7_sc.py) on (1,2,2), (1,3,3), (2,2,2), (2,2,3), (2,2,4), (2,3,3), (2,4,4), (3,3,3), (3,3,4), (3,4,4), (4,4,4), and proved for every cell (probe K): in type-field terms, the fields with zero A-match and zero B-match sum on every hexagon are exactly the slab modes plus the two constants, and the constants are not in the span of the chain modes, so in weight space the kernel is \(S+G\) exactly. A by-product proved there: \(\dim H_A=4(a+b+c)-2\) on every cell.
  4. Hence the type field is slab-type: \(F_x(v)=X(x)\), \(F_y(v)=Y(y)\), \(F_z(v)=Z(z)\) for functions of one coordinate. The two chains of a family through \(v\) both lie in the slab of \(v\)’s coordinate, and the sublattice-antisymmetric constant is absorbed because the sublattice sign is a function of the parity of any single coordinate (A vertices have all coordinates even, B all odd), so \(\sigma_v c_E\) is a function of \(v\)’s \(E\)-coordinate. Then by the window rule \([\tau(v)=y]=X(x)-Z(z)\), \([\tau(v)=z]=Y(y)-X(x)\), both in \(\{0,1\}\) and never both \(1\). Directly observable: in every curl-free frozen state sampled, the type-\(y\) indicator depends only on \((x,z)\) and the type-\(z\) indicator only on \((x,y)\) (0 violations in 35 states of (2,2,4) and (3,3,3); every odd state violates it, r7_bridge.py).
  5. The bridge. The differences \(X(x)-Z(z)\), \(Y(y)-X(x)\) are integers at every vertex; pairs \((x,z)\) of even coordinates are linked through A vertices and odd ones through B, so \(X,Y,Z\) have a common fractional part on even coordinates and another on odd ones, each a function of the coordinate’s parity. Write \(p_x\), \(q_z\), \(r_y\) for \(X(x)\), \(Z(z)\), \(Y(y)\) minus that part, modulo \(2\), so \([\tau=y]=p_x+q_z\), \([\tau=z]=p_x+r_y\), \([\tau=x]=1+q_z+r_y\) (mod 2). Define jump sequences on the slabs, \(J_x=p_x\), \(J_y=r_y+1\), \(J_z=q_z+1\), and profiles \(\tilde X,\tilde Y,\tilde Z\) on the universal cover \(\mathbb Z\) of each axis with these jumps. The slab sum \(\tilde s_e=\tilde X_{x(e)}+\tilde Y_{y(e)}+\tilde Z_{z(e)}\) on the infinite diamond lattice is an ice state with type field exactly \(\tau\): the note’s rule “the type is the axis whose jump is the odd one out” (which holds at B vertices as at A, by the same four-bit computation) gives type \(y\) iff \(J_x=J_z\ne J_y\) iff \(p_x+q_z=1\) and \(p_x+r_y=0\), type \(z\) iff \(p_x+r_y=1\) and \(p_x+q_z=0\), type \(x\) iff both are \(0\), and the excluded case \(p_x+q_z=p_x+r_y=1\) is exactly the case of three equal jumps, so (ICE) holds everywhere.
  6. An ice state on the connected infinite lattice is determined by its type field up to global reversal: the sign field satisfies \(\varepsilon(A)\varepsilon(B)=[d\in P^+(\tau_A)]\oplus[d\in P^+(\tau_B)]\) on every bond, and the bond bits are functions of (type, sign). The lift of the actual state and \(\tilde s\) share the type field, so they agree up to reversal; the actual state is periodic with the box, hence so is \(\tilde s\), hence \(\tilde X_{x+4a}=\tilde X_x\) and likewise for \(\tilde Y,\tilde Z\): the profiles descend to the torus, and the state is a slab sum. \(\square\)

The parity condition in step 6 is genuinely where the sign field enters: in the type model alone (one-hot and (T2), no sign field) curl-free solutions with odd \(\sum_x p_x\) occur freely (they are the solutions with a chain carrying an odd number of kinks), while with the sign field every sampled state has all three sums even (r7_parity.py: (2,2,4) 30 vs 21 states, (2,2,3) 30 vs 24, (3,3,3) 20 vs 20; the odd-parity type-model solutions all have odd-kink chains). Deliberately unbalanced curl-free states of (2,2,3) (type-\(y\) counts differing between the sublattices by 24 to 48, forced by cardinality constraints, r7_unbal.py) also admit the common-profile form with all parities even, as the absorption remark in step 4 says they must. The hand argument does not need to identify the parity obstruction; periodicity of the given state does the work.

3. Consequences

4. Scripts and checks

scratch_h/r7_sc.py (ranks of \(L\), \(S\), \(G\) on a cell; no solver), r7_bridge.py (the observable slab structure on sampled curl-free and odd states, V2), r7_parity.py (the parity condition in V0 and V2). Logs in attack_2026-09-17/. Solver time about 6 minutes.