C03.2 · Computations

Declarative object and assertion

This script tests an identification diagnostic, not the registered Lie-triple tensor. Inputs are the displayed finite-type Cartan matrices for D4 and G2 direct-sum G2, plus the standard matrix definition so(4,4)={M:M^T eta+eta M=0}. Generate each root set by closing the simple roots under its simple reflections. Compute rank, root count and the split compact/noncompact count, then distinguish the Dynkin components. In a separate explicit orthogonal matrix model compute compact/noncompact basis dimensions and the trace-form signs. A successful run supplies no projector, leakage tensor or frozen-frame identification for LIB2-007 through LIB2-010.

Reproduction settings and input contract

Command, from the directory holding the delivered files:

```sh PYTHONHASHSEED=0 OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python CHATGPT_D1259_C03_2_CHECKS_S310.py ```

The script is standalone. It imports only the Python standard library and SymPy; it reads no input file, downloads nothing, uses no random generator, and uses no floating-point tolerance. All scientific inputs are the displayed definitions encoded in the script. The article's source inputs are pinned in Sources/receipt; they are not silently consumed as scientific arrays. Run without `-O`. Software versions are printed, not assumed.

Script SHA-256: `c477c6e4f3f5bd809364796ac95a8e30ee2a8f39fcd42995e9bd50cb21be9ad8`. Actual stdout SHA-256: `fdae457a19558884a4dde042cb4768b340a2f5aaa0d47fbc52e07d6fd35c16d0`. Re-execution of this code is not an independent house implementation.

Complete executable code

```python """D1259 exact verification. All arithmetic is symbolic or rational. Run without -O. No external data, network calls, floating-point tolerance, or claim of equivalence to unavailable frozen programme arrays is used. """ import sympy as S import platform if not __debug__: raise SystemExit("Assertions require a run without -O.") def report(key, value): print(key + " = " + str(value)) report("environment", "Python " + platform.python_version() + "; SymPy " + S.__version__) report("arithmetic", "exact rationals / symbolic polynomials / algebraic radicals")

from itertools import combinations,product # Standard root-system identification control. These are NOT the programme's # tangent-plane matrices or a re-expression of their output. D=S.Matrix([[2,-1,0,0],[-1,2,-1,-1],[0,-1,2,0],[0,-1,0,2]]) G=S.Matrix([[2,-1],[-3,2]]) GG=S.diag(G,G) def roots(C): n=C.rows seen={tuple(int(i==j) for i in range(n)) for j in range(n)} todo=list(seen) while todo: v=todo.pop() for i in range(n): w=list(v) w[i]-=sum(C[i,j]*v[j] for j in range(n)) w=tuple(w) if w not in seen: seen.add(w);todo.append(w) if len(seen)>10000: raise RuntimeError("Non-finite root closure") return seen def components(C): unseen=set(range(C.rows));out=[] while unseen: seed=min(unseen);q=[seed];got=set() while q: j=q.pop() if j in got: continue got.add(j) q.extend(i for i in range(C.rows) if i not in got and (C[i,j] or C[j,i])) unseen-=got;out.append(sorted(got)) return out RD=roots(D); RG=roots(GG) assert len(RD)==len(RG)==24 assert len(components(D))==1 and len(components(GG))==2 # For a split semisimple real form: one compact direction per positive # root, and a split Cartan plus one noncompact direction per positive root. def fingerprint(C,rr): rank=C.rows;positive=len(rr)//2 return {"rank":rank,"dimension":rank+len(rr), "compact_dimension":positive,"noncompact_dimension":rank+positive, "Cartan_character":rank,"center_dimension":0} fd=fingerprint(D,RD);fg=fingerprint(GG,RG) assert fd==fg report("object","root-data control for identification fingerprints, not frozen complement") report("D4_Cartan_matrix",D.tolist()) report("G2_plus_G2_Cartan_matrix",GG.tolist()) report("root_counts",(len(RD),len(RG))) report("identical_split_fingerprints",fd) report("Dynkin_component_counts",(len(components(D)),len(components(GG)))) report("Cartan_determinants",(D.det(),GG.det())) # The explicitly specified 8x8 so(4,4) Cartan decomposition gives an # independent elementary dimension and trace-form sign check for this model. eta=S.diag(1,1,1,1,-1,-1,-1,-1) B=[] for i,j in combinations(range(8),2): M=S.zeros(8);M[i,j]=1;M[j,i]=-eta[i,i]/eta[j,j];B.append(M) assert all(M.T*eta+eta*M==S.zeros(8) for M in B) compact=[M for M in B if M.T==-M] noncompact=[M for M in B if M.T==M] assert len(compact)==12 and len(noncompact)==16 T=S.Matrix(28,28,lambda i,j:S.trace(B[i]*B[j])) assert T.is_diagonal() positive=sum(bool(T[i,i]>0) for i in range(28)) negative=sum(bool(T[i,i]<0) for i in range(28)) report("explicit_so44_model_k_p_dimensions",(len(compact),len(noncompact))) report("explicit_so44_trace_form_inertia",(positive,negative,0)) report("fingerprint_is_complete_type_certificate",False) report("registered_54_failure_16_closure_and_generated_algebra","NOT VERIFIED: exact frozen arrays absent") report("result","PASS: standard diagnostic only; no registered Lie-triple result reexecuted") ```

Actual stdout

```text environment = Python 3.13.5; SymPy 1.14.0 arithmetic = exact rationals / symbolic polynomials / algebraic radicals object = root-data control for identification fingerprints, not frozen complement D4_Cartan_matrix = [[2, -1, 0, 0], [-1, 2, -1, -1], [0, -1, 2, 0], [0, -1, 0, 2]] G2_plus_G2_Cartan_matrix = [[2, -1, 0, 0], [-3, 2, 0, 0], [0, 0, 2, -1], [0, 0, -3, 2]] root_counts = (24, 24) identical_split_fingerprints = {'rank': 4, 'dimension': 28, 'compact_dimension': 12, 'noncompact_dimension': 16, 'Cartan_character': 4, 'center_dimension': 0} Dynkin_component_counts = (1, 2) Cartan_determinants = (4, 1) explicit_so44_model_k_p_dimensions = (12, 16) explicit_so44_trace_form_inertia = (16, 12, 0) fingerprint_is_complete_type_certificate = False registered_54_failure_16_closure_and_generated_algebra = NOT VERIFIED: exact frozen arrays absent result = PASS: standard diagnostic only; no registered Lie-triple result reexecuted ```