C05.2 · Computations

Declarative object and assertion

**Object under test, declarative specification.** There are four checks, with disjoint evidential roles. (1) Reduce the proposed root and inverse identities modulo the polynomial a²−3a+1. (2) Use the displayed real matrices U,V and the standard antisymmetric two-form; check the braid relation, their central cube and the candidate root. Pad only as an explicitly assumed model, not as a reconstructed frozen representation. (3) Use the real symmetric Clifford pair diag(1,−1) and the exchange matrix, divided by √24, acting on a completely symbolic two-by-three T. Compute its Frobenius Gram. Conjugate by the declared non-orthogonal diagonal matrix as a metric-convention countercontrol. (4) Enumerate the residue pairs of Z/4 and check mirror reversal of addition; do not interpret this as testing registered generator products.

Every rational entry, polynomial residual and multiplicity printed by the script has a named output key. Model padding uses source-reported dimensions as inputs. No external arrays, fitted constants, floating-point tolerances, empirical comparators, numerical phase choices, or random samples are used. The asserted equalities are symbolic or exhaustive within their stated finite arithmetic domains.

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_C05_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: `3bbf381902bb107f1e3f5247b85200b74e9799a0f7823878d63c154524d10b52`. Actual stdout SHA-256: `b2175921f511a400cb45c971c84391e3ea59ee9b378ae806cc6e6b0bb5a50e48`. 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 product R=S.Rational # Polynomial proof in the quotient ring R[a]/(a^2-3a+1). a=S.symbols("a") p=S.Poly(a*a-3*a+1,a) def rem(expr): return S.rem(S.Poly(S.expand(expr),a),p).as_expr().expand() assert rem((a+1)**2/5-a)==0 assert rem((a+1)*(4-a)/5-1)==0 report("universal_square_root_residual",rem((a+1)**2/5-a)) report("universal_inverse_residual",rem((a+1)*(4-a)/5-1)) # A fully stated standard 2x2 realization is a negative/control model, # NOT a replacement for the missing frozen-frame intertwiners. U=S.Matrix([[1,1],[0,1]]) V=S.Matrix([[1,0],[-1,1]]) Om=S.Matrix([[0,1],[-1,0]]) A=U*V.inv() H=(A+S.eye(2))/S.sqrt(5) assert U*V*U==V*U*V assert U.T*Om*U==V.T*Om*V==Om assert (U*V)**3==-S.eye(2) assert A*A-3*A+S.eye(2)==S.zeros(2) assert S.simplify(H*H-A)==S.zeros(2) assert S.simplify(H.T*Om*H-Om)==S.zeros(2) assert S.simplify(H.det())==1 report("model_U",U.tolist());report("model_V",V.tolist()) report("model_hyperbolic_word",A.tolist()) report("model_word_characteristic",A.charpoly().as_expr()) report("model_braid_residual_zero",True) report("model_central_cube",(U*V)**3) report("model_square_root",H.tolist()) report("model_root_trace_and_determinant",(S.simplify(S.trace(H)),S.simplify(H.det()))) # Formal Clifford identities in an explicit simple component. g0=S.diag(1,-1);g1=S.Matrix([[0,1],[1,0]]);k0=g0*g1 gs=[g0,g1,k0] assert g0*g0==g1*g1==S.eye(2) and k0*k0==-S.eye(2) assert all(gs[i]*gs[j]+gs[j]*gs[i]==S.zeros(2) for i in range(3) for j in range(i)) report("model_Clifford_squares",["I","I","-I"]) # The all-T Gram law is proved by these matrix identities; a symbolic # arbitrary two-row T tests the complete bilinear polynomial in this block. v0,v1,v2,v3,v4,v5=S.symbols("v0:6") T=S.Matrix([[v0,v1,v2],[v3,v4,v5]]) Ws=[g0/S.sqrt(24),g1/S.sqrt(24)] G=S.Matrix(2,2,lambda i,j:S.expand(S.trace((Ws[i]*T).T*(Ws[j]*T)))) rhs=sum(x*x for x in T)/24*S.eye(2) assert G==rhs report("symbolic_model_Gram_residual_zero",True) # Clifford relations alone are not invariant statements about a fixed # Euclidean Frobenius metric under non-orthogonal similarity. C=S.diag(2,1) g1bad=C*g1*C.inv() assert g1bad*g1bad==S.eye(2) assert g0*g1bad+g1bad*g0==S.zeros(2) Tb=S.Matrix([1,0]) badWs=[g0/S.sqrt(24),g1bad/S.sqrt(24)] Gbad=S.Matrix(2,2,lambda i,j:((badWs[i]*Tb).T*(badWs[j]*Tb))[0]) assert Gbad==S.diag(R(1,24),R(1,96)) report("nonorthogonal_Clifford_control_Gram",Gbad.tolist()) report("Clifford_relations_alone_force_Frobenius_isotropy",False) # If the registered decomposition is supplied, this implication pads it. copies=12;inactive=32 Uc=S.diag(*([U]*copies+[S.eye(inactive)])) Vc=S.diag(*([V]*copies+[S.eye(inactive)])) P=S.diag(*([1]*(2*copies)+[0]*inactive)) centre=(Uc*Vc)**3 assert centre==S.eye(56)-2*P assert centre != S.eye(56) and centre != -S.eye(56) report("conditional_model_active_inactive_dimensions",(2*copies,inactive)) report("conditional_model_central_eigenvalue_multiplicities",{-1:2*copies,1:inactive}) report("conditional_model_global_projectivization_to_PSL2Z",False) report("unit_column_Gram_coefficients",{1:R(1,24),10:R(10,24)}) # Z4 arithmetic: not a census of the programme's generator products. tests=0 for x,y in product(range(4),repeat=2): assert (-(x+y))%4==((-x)%4+(-y)%4)%4 tests+=1 report("Z4_arithmetic_mirror_checks",tests) report("Z4_checks_are_registered_generator_census",False) report("registered_frame_tick_cylinder_winding_scale","NOT VERIFIED: frozen matrices / cochain complex absent") report("result","PASS: universal polynomial implications and explicitly marked standard models") ```

Actual stdout

```text environment = Python 3.13.5; SymPy 1.14.0 arithmetic = exact rationals / symbolic polynomials / algebraic radicals universal_square_root_residual = 0 universal_inverse_residual = 0 model_U = [[1, 1], [0, 1]] model_V = [[1, 0], [-1, 1]] model_hyperbolic_word = [[2, 1], [1, 1]] model_word_characteristic = lambda**2 - 3*lambda + 1 model_braid_residual_zero = True model_central_cube = Matrix([[-1, 0], [0, -1]]) model_square_root = [[3*sqrt(5)/5, sqrt(5)/5], [sqrt(5)/5, 2*sqrt(5)/5]] model_root_trace_and_determinant = (sqrt(5), 1) model_Clifford_squares = ['I', 'I', '-I'] symbolic_model_Gram_residual_zero = True nonorthogonal_Clifford_control_Gram = [[1/24, 0], [0, 1/96]] Clifford_relations_alone_force_Frobenius_isotropy = False conditional_model_active_inactive_dimensions = (24, 32) conditional_model_central_eigenvalue_multiplicities = {-1: 24, 1: 32} conditional_model_global_projectivization_to_PSL2Z = False unit_column_Gram_coefficients = {1: 1/24, 10: 5/12} Z4_arithmetic_mirror_checks = 16 Z4_checks_are_registered_generator_census = False registered_frame_tick_cylinder_winding_scale = NOT VERIFIED: frozen matrices / cochain complex absent result = PASS: universal polynomial implications and explicitly marked standard models ```