"""v415 -- The square CM-norm dictionary is operator-realised, and the mu4 deck is an intrinsic diamond object. The Gaussian arithmetic of v222/v230 (the square seam: 13 = N_Z[i](3+2i) = Delta_Q, 41 = N_Z[i](5+4i) = 10 b1) is lifted from NUMBERS to OPERATORS via the single complex structure born from the commutator of the binary (V) and ternary (U) compilers (v410/v95). [E] algebra, [C] the geometric-deck identification (v140 residue, sharpened not closed). [E] 1. THE OPERATOR 'i'. With U = Q diag(1,0,0) (winding, Spec {3,0,0}) and V = Q diag(0,1,1) (sheet, Spec {0,1,2}) the integer matrix J := [U,V]/3 = [[-1,1,0],[-2,1,0],[-3,1,0]] has Spec(J) = {i, -i, 0}, J^2 = -I on the rank-2 image and J^3 + J = 0 (order 4 on the plane) -- a genuine mu4 quarter-turn complex structure, born from the non-commutativity of the two internal compilers. [E] 2. THE GAUSSIAN INTEGERS ARE EIGENVALUES. 3I + 2J has spectrum {3+2i, 3-2i, 3} and 5I + 4J has {5+4i, 5-4i, 5}: the abstract Gaussian integers 3+2i (norm 13 = Delta_Q) and 5+4i (norm 41 = 10 b1) are LITERAL eigenvalues of concrete diamond operators -- lifting v222/v230 from norm to spectrum. [E] 3. THE NORM DICTIONARY AS OPERATOR IDENTITIES. (3I+2J)(3I-2J) = 9I - 4J^2, Spec = {13, 13, 9}, (5I+4J)(5I-4J) = 25I - 16J^2, Spec = {41, 41, 25}; on the mu4-plane these are 13*I = Delta_Q and 41*I = 10 b1, while the kernel slot reads the real part squared (9 = N_fam^2, 25 = g_car^2). [E] 4. THE INTRINSIC mu4 DECK. Pi2 := I + J^2 is the projector onto ker J (the self-conjugate chi2 line), and D := J - Pi2 = -I + J - J^2 = [[-1,1,0],[-2,1,0],[-4,3,-1]] is a genuine order-4 deck: D^4 = I, chi_D = (x+1)(x^2+1), Spec(D) = {i, -1, -i} (the seam-deck spectrum of v146). The diamond canonically singles out the self-conjugate line as ker[U,V] = (0,0,1) = a - 1 = the V-fixed line (Spec-1 eigenvector). [C] 5. DECK IDENTIFICATION (v140 residue, SHARPENED not closed). D has the same characteristic polynomial as the geometric seam deck delta (v146, z->iz on H^1), and the diamond fixes the chi2 line internally as a-1. Whether a-1 is the GEOMETRIC self-conjugate line is exactly the one Z_3 family-rotation residue of v140 -- so this gives the deck as an intrinsic diamond object and reduces the residue to a single concrete identification, but does NOT by itself close it. The exact identities (1-4) are mirrored in wolfram/tfpt_readouts_extension.wl; the deck-identification (5) stays the v140 [C]/[P] residue. """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) ONE = sp.Matrix([1, 1, 1]) A = sp.Matrix([1, 1, 2]) I3 = sp.eye(3) U = Q * sp.diag(1, 0, 0) V = Q * sp.diag(0, 1, 1) J = (U * V - V * U) / 3 DELTA_Q = 13 # = N_fam^2 + |Z2|^2 = 3^2 + 2^2 (v14) EM_INDEX = 41 # = 10 b1 = g_car^2 + |mu4|^2 = 5^2 + 4^2 (v222) i = sp.I def run(): reset() print("v415 Gaussian operator: J=[U,V]/3 is the mu4 quarter-turn; " "3+2i, 5+4i are eigenvalues; the deck is intrinsic") # ---- 1. the operator i ---- check("THE OPERATOR i [E]: J = [U,V]/3 = [[-1,1,0],[-2,1,0],[-3,1,0]] is " "integer, Spec(J)={i,-i,0}, J^2=-I on the rank-2 image, J^3+J=0 " "(order 4 on the plane) -- a complex structure from [binary V, " "ternary U]", J == sp.Matrix([[-1, 1, 0], [-2, 1, 0], [-3, 1, 0]]) and set(J.eigenvals().keys()) == {i, -i, sp.Integer(0)} and sp.simplify(J**3 + J) == sp.zeros(3) and J.rank() == 2) # ---- 2. the Gaussian integers are eigenvalues ---- check("GAUSSIAN INTEGERS ARE EIGENVALUES [E]: 3I+2J has spectrum " "{3+2i,3-2i,3} (|3+2i|^2=13=Delta_Q) and 5I+4J has {5+4i,5-4i,5} " "(|5+4i|^2=41=10 b1) -- the abstract Gaussian integers are literal " "operator eigenvalues", set((3 * I3 + 2 * J).eigenvals().keys()) == {3 + 2 * i, 3 - 2 * i, sp.Integer(3)} and set((5 * I3 + 4 * J).eigenvals().keys()) == {5 + 4 * i, 5 - 4 * i, sp.Integer(5)} and (3**2 + 2**2) == DELTA_Q and (5**2 + 4**2) == EM_INDEX) # ---- 3. the norm dictionary as operator identities ---- N32 = sp.simplify((3 * I3 + 2 * J) * (3 * I3 - 2 * J)) N54 = sp.simplify((5 * I3 + 4 * J) * (5 * I3 - 4 * J)) check("NORM DICTIONARY AS OPERATORS [E]: (3I+2J)(3I-2J)=9I-4J^2 has Spec " "{13,13,9} (13=Delta_Q on the mu4-plane, 9=N_fam^2 on the kernel); " "(5I+4J)(5I-4J)=25I-16J^2 has Spec {41,41,25} (41=10 b1, " "25=g_car^2) -- the v222/v230 norms lifted to operators", N32 == sp.simplify(9 * I3 - 4 * J * J) and N32.eigenvals() == {DELTA_Q: 2, N_fam**2: 1} and N54 == sp.simplify(25 * I3 - 16 * J * J) and N54.eigenvals() == {EM_INDEX: 2, g_car**2: 1}) # ---- 4. the intrinsic mu4 deck ---- Pi2 = sp.simplify(I3 + J * J) D = sp.simplify(J - Pi2) x = sp.symbols('x') check("INTRINSIC mu4 DECK [E]: Pi2=I+J^2 projects onto ker J; " "D=J-Pi2=-I+J-J^2=[[-1,1,0],[-2,1,0],[-4,3,-1]] is order 4 (D^4=I), " "chi_D=(x+1)(x^2+1), Spec(D)={i,-1,-i} (the seam-deck spectrum); " "the chi2 line is canonically ker[U,V]=(0,0,1)=a-1=the V-fixed line", Pi2.eigenvals() == {0: 2, 1: 1} and D == sp.Matrix([[-1, 1, 0], [-2, 1, 0], [-4, 3, -1]]) and sp.simplify(D**4) == I3 and sp.factor(D.charpoly(x).as_expr()) == (x + 1) * (x**2 + 1) and set(D.eigenvals().keys()) == {i, -i, sp.Integer(-1)} and J.nullspace() == [A - ONE] and (A - ONE) == sp.Matrix([0, 0, 1])) # ---- 5. deck identification: SHARPENED, not closed (v140 residue) ---- delta = sp.diag(i, -1, -i) # the v146 geometric deck check("DECK IDENTIFICATION [C] (v140 residue, sharpened): D and the " "geometric seam deck delta=diag(i,-1,-i) (v146) share chi=(x+1)" "(x^2+1) and order 4; the diamond fixes the chi2 line as a-1 " "INTERNALLY, reducing the v140 Z3 choice to the single concrete " "identification 'is a-1 the geometric self-conjugate line?' -- " "intrinsic deck [E], geometric match still [C]/[P]", sp.factor(delta.charpoly(x).as_expr()) == sp.factor(D.charpoly(x).as_expr()) and delta**4 == I3) return summary("v415 Gaussian operator (J=mu4 quarter-turn; 3+2i/5+4i " "eigenvalues; intrinsic deck D)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)