C04.2 · Computations

Declarative object and assertion

The primary object under test is the explicitly printed linear map from (t,u,v,w,z) to four Hessian weights, together with the four Phi_diag equations and the five source stability inequalities in (t,t2,u,v,w,z). Compute exact ranks, nullspaces, row-deletion ranks, strict slacks and intersections with the two source-reported Casimir systems. Independently expand the actual scalar even-exponential character to check its quadratic and quartic coefficients; the matrix version is a conditional functional-calculus argument, not an unshipped-matrix test. The small nonlinear sum-of-squares example tests only the stated chain rule. No full parent action, source map, H2 complex, carrier projector or detector matrix is reconstructed.

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_C04_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: `a8a5005757ecc8cbd4193c31c9e6237130c5589fb222894f0d76885454f07a1c`. Actual stdout SHA-256: `c2bafffee556a20bebe64aa46d724ab8aa9cf7f71134178048f48a449a6ebd57`. 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")

R=S.Rational t,t2,u,v,w,z=S.symbols("t t2 u v w z",real=True) c=S.Matrix([t,u,v,w,z]) # This tests the displayed coefficient-to-two-jet map, NOT the missing # 70-dimensional projectors or their frozen E7 frame. h=S.Matrix([u,4*t/9+v,16*t/27+w-z,16*t/27+w+z]) J=h.jacobian(c) ker=J.nullspace() assert J.rank()==4 and len(ker)==1 k=S.Matrix([1,0,-R(4,9),-R(16,27),0]) assert J*k==S.zeros(4,1) report("object","published coefficient-to-Hessian-weight map, conditional on its carrier character") report("two_jet_matrix",J.tolist()) report("two_jet_rank",J.rank()) report("two_jet_kernel_vector",list(k)) phi=S.Matrix([v,w,u-t,27*z-t]).jacobian(c) selected=S.Matrix([1,1,0,0,R(1,27)]) assert phi.rank()==4 and phi*selected==S.zeros(4,1) drop=[phi.extract([i for i in range(4) if i!=j],list(range(5))).rank() for j in range(4)] assert drop==[3]*4 report("Phi_diag_matrix",phi.tolist()) report("Phi_diag_rank",phi.rank()) report("Phi_diag_selected_ray",list(selected)) report("Phi_diag_row_deletion_ranks",drop) report("selected_Hessian_weights",list(J*selected)) assert list(J*selected)==[1,R(4,9),R(5,9),R(17,27)] # Exact scalar expansion of the actual trace-character function, before # contraction with the unavailable loaded projector. s=S.symbols("s",real=True) char_s=R(12,54)*(S.exp(2*s)+S.exp(-2*s)-2) series=S.series(char_s,s,0,8).removeO() defect=char_s-R(8,9)*s*s report("scalar_character_series_through_degree_6",series) assert S.expand(series).coeff(s,4)==R(8,27) assert [S.diff(defect,s,i).subs(s,0) for i in range(3)]==[0,0,0] report("defect_value_first_second_derivatives",[0,0,0]) report("defect_fourth_derivative",S.diff(defect,s,4).subs(s,0)) # Strict interior witness for the source's non-degenerate six-variable class. forms=list(h)+[t2] point={t:1,t2:1,u:1,v:1,w:1,z:0} slacks=[f.subs(point) for f in forms] assert all(x>0 for x in slacks) report("stability_point_order","(t,t2,u,v,w,z)") report("stability_point",(1,1,1,1,1,0)) report("strict_stability_slacks",slacks) report("ambient_coefficient_dimension",6) report("positive_scale_quotient_dimension",5) # Compute intersections of the source-reported Casimir conditions with Phi. literal=S.Matrix([u,v,w,z]).jacobian(c) support=S.Matrix([u-v,v-w,z]).jacobian(c) rl=phi.col_join(literal).rank(); rs=phi.col_join(support).rank() assert rl==rs==5 report("Phi_plus_literal_Casimir_rank",rl) report("Phi_plus_support_Casimir_rank",rs) report("nonzero_adopted_ray_survives_either_Casimir_system",False) # The general zero-level sum-of-squares Hessian rule follows by differentiation. x,y=S.symbols("x y",real=True) mu=S.Matrix([x+x*y,y+x*x]); objective=(mu.T*mu)[0] at={x:0,y:0} H=S.hessian(objective,(x,y)).subs(at) D=mu.jacobian((x,y)).subs(at) assert H==2*D.T*D report("sum_of_squares_chain_rule_control",True) report("sum_of_squares_control_is_frozen_Kempf_Ness_test",False) report("frozen_detector_minimum_character_and_H2_certificates","NOT VERIFIED: original objects absent") report("result","PASS: displayed coefficient, jet and source-condition implications only") ```

Actual stdout

```text environment = Python 3.13.5; SymPy 1.14.0 arithmetic = exact rationals / symbolic polynomials / algebraic radicals object = published coefficient-to-Hessian-weight map, conditional on its carrier character two_jet_matrix = [[0, 1, 0, 0, 0], [4/9, 0, 1, 0, 0], [16/27, 0, 0, 1, -1], [16/27, 0, 0, 1, 1]] two_jet_rank = 4 two_jet_kernel_vector = [1, 0, -4/9, -16/27, 0] Phi_diag_matrix = [[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [-1, 1, 0, 0, 0], [-1, 0, 0, 0, 27]] Phi_diag_rank = 4 Phi_diag_selected_ray = [1, 1, 0, 0, 1/27] Phi_diag_row_deletion_ranks = [3, 3, 3, 3] selected_Hessian_weights = [1, 4/9, 5/9, 17/27] scalar_character_series_through_degree_6 = 16*s**6/405 + 8*s**4/27 + 8*s**2/9 defect_value_first_second_derivatives = [0, 0, 0] defect_fourth_derivative = 64/9 stability_point_order = (t,t2,u,v,w,z) stability_point = (1, 1, 1, 1, 1, 0) strict_stability_slacks = [1, 13/9, 43/27, 43/27, 1] ambient_coefficient_dimension = 6 positive_scale_quotient_dimension = 5 Phi_plus_literal_Casimir_rank = 5 Phi_plus_support_Casimir_rank = 5 nonzero_adopted_ray_survives_either_Casimir_system = False sum_of_squares_chain_rule_control = True sum_of_squares_control_is_frozen_Kempf_Ness_test = False frozen_detector_minimum_character_and_H2_certificates = NOT VERIFIED: original objects absent result = PASS: displayed coefficient, jet and source-condition implications only ```