ABSTRACT (original article): Can an artificial intelligence (AI) generate a scientific hypothesis outside a human collaborator’s active hypothesis space (AHS), and can human-AI research be organized to make such breakthroughs more likely? We document such a case while proving a theorem that connects two basic organizing mechanisms of statistical physics: collective behavior arising in zero field from competing interactions and that induced or controlled by an external field. A zero-field -vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions (·) and (·) is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler open chain with nearest-neighbor interaction (·) and axial single-spin potential () for every integer and every system size . The homogeneous linear specialization maps the foundational frustrated - model onto the canonical field model---with being the Ising, XY, and Heisenberg classical spin models, respectively; the theorem resolved a longstanding challenge for published in 1990. Its proof was done with an AI-synthesized recursive Householder moving frame and understood via a human-recognized hidden reciprocity. An analogous theorem holds when the continuous spins are replaced by the -state Potts spins, implying a closed-form exact solution of the - standard Potts open chain for every and every . The emergence of these theorems from a human-AI co-development framework suggests that sustained AI involvement throughout a systematic research program may incubate autonomous scientific breakthroughs and make aspects of the discovery process experimentally testable. CITATION (original article): Weiguo Yin (2026), A Human-AI Theorem Connecting Spontaneous and Field-Induced Mechanisms of Collective Behavior in One Dimension, arXiv:2609.00322. https://doi.org/10.48550/arXiv.2609.00322
O(n)
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n=1,2,3
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This Wolfram notebook illustrates how Wolfram Mathematica programming was used within the human--AI co-development framework described in the manuscript A Human--AI Theorem Connecting Spontaneous and Field-Induced Mechanisms of Collective Behavior in One Dimension (https://doi.org/10.48550/arXiv.2609.00322) and its accompanying human--AI conversation transcript (https://doi.org/10.5281/zenodo.22718925). The resulting research trajectory underwent a “big bang” expansion from a recent AI-bootstrapped single point (https://community.wolfram.com/t/25512) to an organizationally four-axis but mathematically infinite-dimensional active hypothesis space (AHS), far beyond the human’s initial one-axis AHS deliberately designed to bootstrap the AI’s scientific development. Wolfram Mathematica played two specific supporting roles in this process: the human used it to design the initial AHS, while the AI used it to perform an independent numerical stress test of the nonlinear U-V mapping.
1. Human-Designed Initial Type-III AHS
1. Human-Designed Initial Type-III AHS
See Part I: Messages 0015 and 0016 of the the human - AI conversation transcript (I : 0015 and I : 0016) https://doi.org/10.5281/zenodo.22718925
v1 .0 2026 - 08 - 06 The q thermodynamic mapping between the J1 - J2 and J - h chains described by the q - state standard Potts and clock models, respectively, with the periodic boundary condition (PBC) .
2
q
Matrices (order)
1
.J1 - J2 chain in the overlapping - pairs formulation : TMatOP ), TblockOP (q);
2
q
2
.J - h chain with h = J1 and J = J2 : TMatB (q);
For the thermodynamic mapping we proved :
TblockOP . R = TMatB;
where R is the q x q reveral matrix : R_ {r, s} = delta (r, -s) .
TblockOP . R = TMatB;
where R is the q x q reveral matrix : R_ {r, s} = delta (r, -s) .
(*ParametersandBasicFunctions*)ClearAll[delta];(*Kroneckerdelta*)delta[x_,y_]:=If[x===y,1,0];
The q-state standard Potts model
The q-state standard Potts model
1. Model
1. Model
ClearAll[q,exchangeForm,fieldForm];exchangeForm[x_,y_]:=delta[x,y];fieldForm[x_]:=delta[x,1];
The Potts states are labeled as {1, 2, ..., q}. No need to explicitly represent the Potts states as {0, 1, ..., q - 1} in : and symmetries are coded in Uq2xq and R, respectively .
Z
q
Z
q
Z
2
2. The model with nearest-neighbor interaction J in a magnetic field h
2. The model with nearest-neighbor interaction J in a magnetic field h
a. Build the q x q transfer matrix Tmat
In[]:=
ClearAll[q,x,B,β,J,h,TMatB];x=Exp[βJ];B=Exp[βh/2];TMatB[x_,B_]:=Table[Module[{bondTerm,fieldTerm},bondTerm=x^exchangeForm[i,j];fieldTerm=B^(fieldForm[i]+fieldForm[j]);PowerExpand[bondTerm*fieldTerm]],{i,1,q},{j,1,q}];q=2;Print["TMatB[q=",q,"]=",TMatB[x,B]//MatrixForm];q=3;Print["TMatB[q=",q,"]=",TMatB[x,B]//MatrixForm];q=4;Print["TMatB[q=",q,"]=",TMatB[x,B]//MatrixForm];ClearAll[x,B];q=2;Print["TMatB[q=",q,"]=",TMatB[x,B]//MatrixForm];q=3;Print["TMatB[q=",q,"]=",TMatB[x,B]//MatrixForm];q=4;Print["TMatB[q=",q,"]=",TMatB[x,B]//MatrixForm];ClearAll[q];
TMatB[q=2]=
βh+βJ | βh/2 |
βh/2 | βJ |
TMatB[q=3]=
βh+βJ | βh/2 | βh/2 |
βh/2 | βJ | 1 |
βh/2 | 1 | βJ |
TMatB[q=4]=
βh+βJ | βh/2 | βh/2 | βh/2 |
βh/2 | βJ | 1 | 1 |
βh/2 | 1 | βJ | 1 |
βh/2 | 1 | 1 | βJ |
TMatB[q=2]=
2 B | B |
B | x |
TMatB[q=3]=
2 B | B | B |
B | x | 1 |
B | 1 | x |
TMatB[q=4]=
2 B | B | B | B |
B | x | 1 | 1 |
B | 1 | x | 1 |
B | 1 | 1 | x |
3. The J1-J2 model
3. The J1-J2 model
The “overlapping pairs” formulation.
One spin per unit cell
One spin per unit cell
a. Build the q^2 x q^2 transfer matrix TMatOP
In[]:=
ClearAll[q,x,y,z,β,J1,J2,TMatOP]x=Exp[βJ2];y=Exp[βJ1/2];z=Exp[βJ1/2];TMatOP[x_,y_,z_]:=Table[Module[{states,onRungLeft,onRungRight,diagBond,legBonds},(*1.Constructthecomposite(rung)statespace:states={(a,b)|a,bin{1,...,q}}*)states=Flatten[Table[{a,b},{a,1,q},{b,1,q}],1];onRungLeft=y^exchangeForm[states[[i,1]],states[[i,2]]];onRungRight=z^exchangeForm[states[[j,1]],states[[j,2]]];diagBond=delta[states[[i,2]],states[[j,1]]];legBonds=x^exchangeForm[states[[i,1]],states[[j,2]]];PowerExpand[onRungLeft*onRungRight*diagBond*legBonds]],{i,1,q^2},{j,1,q^2}];q=2;Print["TMatOP[q=",q,"]=",TMatOP[x,y,z]//MatrixForm];q=3;Print["TMatOP[q=",q,"]=",TMatOP[x,y,z]//MatrixForm];ClearAll[x,y,z];q=2;Print["TMatOP[q=",q,"]=",TMatOP[x,y,z]//MatrixForm];q=3;Print["TMatOP[q=",q,"]=",TMatOP[x,y,z]//MatrixForm];ClearAll[q];
TMatOP[q=2]=
βJ1+βJ2 | βJ1/2 | 0 | 0 |
0 | 0 | βJ2 | βJ1/2 |
βJ1/2 | βJ2 | 0 | 0 |
0 | 0 | βJ1/2 | βJ1+βJ2 |
TMatOP[q=3]=
βJ1+βJ2 | βJ1/2 | βJ1/2 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | βJ2 | βJ1/2 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | βJ2 | 1 | βJ1/2 |
βJ1/2 | βJ2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | βJ1/2 | βJ1+βJ2 | βJ1/2 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | βJ2 | βJ1/2 |
βJ1/2 | 1 | βJ2 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | βJ1/2 | βJ2 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | βJ1/2 | βJ1/2 | βJ1+βJ2 |
TMatOP[q=2]=
xyz | y | 0 | 0 |
0 | 0 | x | z |
z | x | 0 | 0 |
0 | 0 | y | xyz |
TMatOP[q=3]=
xyz | y | y | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | x | z | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | x | 1 | z |
z | x | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | y | xyz | y | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | x | z |
z | 1 | x | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | z | x | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | y | y | xyz |
b. Create q q^2x1 basis vectors that span the maximally symmetric subspace
ClearAll[q,Uq2xq,Uqxq2,R](*Thefollowinglineusessymmetrywithalladditionsandsubtractionsunderstoodmoduloq*)Uq2xq:=Transpose[Table[Normalize[Flatten[Table[If[j==If[i+k<=q,i+k,i+k-q],1,0],{i,1,q},{j,1,q}]]],{k,0,q-1}]];Uqxq2=Transpose[Uq2xq];(*Thefollowinglineuses.Risthereversalmatrix:R_{r,s}=delta(r,-s)*)R:=Module[{a},a=Table[If[i+j==q+2,1,0],{i,1,q},{j,1,q}];a[[1,1]]=1;a];q=2;R//MatrixFormUqxq2//MatrixFormq=3;R//MatrixFormUqxq2//MatrixFormClearAll[q];
Z
q
Z
2
Out[]//MatrixForm=
1 | 0 |
0 | 1 |
Out[]//MatrixForm=
1 2 | 0 | 0 | 1 2 |
0 | 1 2 | 1 2 | 0 |
c. The resulting qxq matrix TblockOP
The q-state clock/planar/vector Potts model
The q-state clock/planar/vector Potts model
1. Model
1. Model
2. The model with nearest-neighbor interaction J in a magnetic field h
2. The model with nearest-neighbor interaction J in a magnetic field h
a. Build the q x q transfer matrix Tmat
3. The J1-J2 model
3. The J1-J2 model
The “overlapping pairs” formulation.
One spin per unit cell
One spin per unit cell
a. Build the q^2 x q^2 transfer matrix TMatOP
b. Create q q^2x1 basis vectors that span the maximally symmetric subspace
c. The resulting qxq matrix TblockOP and the mapping
2. AI-Designed Numerical Test
2. AI-Designed Numerical Test
Independent numerical stress test of the nonlinear U-V mapping (see the end of I:0054, i.e., Part I, Message 0054 of the human-AI conversation transcript https://doi.org/10.5281/zenodo.22718925). See I:0055 and I:0056 for the discussion about Tzz = Tov.Tov in the code.
This Wolfram Mathematica 14.3 code was fully generated by OpenAI’s GPT 5.6 in Extra High mode.
This Wolfram Mathematica 14.3 code was fully generated by OpenAI’s GPT 5.6 in Extra High mode.
This is a self-contained Mathematica translation of the independent numerical test used in the manuscript. It does NOT use the reduced continuous O(n) kernel. Instead, for q-state clock spins it independently constructs:
1
.the full q^2 x q^2 overlapping-pair transfer matrix Tov of the range-two model with nonlinear isotropic functions U and V;
2
.the q x q transfer matrix A of the mapped range-one model;
3
.Tzz = Tov.Tov, the two-step zigzag-ladder transfer matrix.
It then compares
rho(Tov) with rho(A)
and rho(Tzz) with rho(A)^2.
and rho(Tzz) with rho(A)^2.
The nonlinear functions below are intentionally non-polynomial/mixed so that the test is not a disguised bilinear calculation.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
A human-AI theorem connecting spontaneous and field-induced mechanisms of collective behavior in one dimension
by Weiguo Yin
Wolfram Community, STAFF PICKS, September 30, 2026
https://community.wolfram.com/t/28223
by Weiguo Yin
Wolfram Community, STAFF PICKS, September 30, 2026
https://community.wolfram.com/t/28223