TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/CLOCK-Q-PACKET-COMPATIBILITY.txt
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000001 | CLOCK AND NEGATIVE SIGN PACKET
000002 | Exact compatibility of the geometric quarter lift with the v117 coefficient clock
000003 | 2026-10-26
000004 | 
000005 | VERDICT
000006 | 
000007 | PROVED at the flat coefficient level.  The geometric quarter map preserves the
000008 | selected negative Weierstrass orbit.  It does not exchange that orbit with the
000009 | positive one.  Its unique order-four lift at the t=0 branch acts on the three
000010 | negative sign lines with square (+,-,-), up to their ordering, and is conjugate
000011 | to the original v117 clock U.  With the old source convention, the positive
000012 | t=0 monodromy is U^(-1).
000013 | 
000014 | This proves actual C4 equivariance of the coefficient local system.  It does
000015 | not yet construct the action on charged CAR fields, their state, or local
000016 | operator support.
000017 | 
000018 | 
000019 | 1. THE CORRECT GEOMETRIC LIFT
000020 | 
000021 | Set
000022 | 
000023 |     f(z)=(z^2-1)/(z^2+1),
000024 |     w^3=f(z),
000025 |     eta=z(1-w^3),
000026 |     C: eta^2=1-w^6.                                                (1)
000027 | 
000028 | A direct calculation gives
000029 | 
000030 |     f(i z)=1/f(z).                                                  (2)
000031 | 
000032 | Therefore a lift of the quarter map z->i z has w' equal to a cube root of
000033 | w^(-3).  The marking used in the September source is
000034 | 
000035 |     Q:(z,w,eta) |-> (i z, 1/w, -i eta/w^3).                       (3)
000036 | 
000037 | The eta factor is not optional.  It follows from eta=z(1-w^3):
000038 | 
000039 |     eta' = i z(1-w^(-3)) = -i eta/w^3.                             (4)
000040 | 
000041 | For q_a=-omega^a one has
000042 | 
000043 |     1/q_a = -omega^(-a) = q_(-a).                                 (5)
000044 | 
000045 | Thus Q preserves the negative orbit {q_0,q_1,q_2}.  It also preserves the
000046 | positive orbit separately.  The apparent q to p exchange came from dropping
000047 | the minus sign in the reciprocal of q_a.
000048 | 
000049 | Applying (3) twice gives
000050 | 
000051 |     Q^2:(z,w,eta) |-> (-z,w,-eta),                                 (6)
000052 | 
000053 | which is the hyperelliptic involution of C over the w-sphere.  In particular
000054 | Q has order four on C.
000055 | 
000056 | 
000057 | 2. ACTION ON THE SELECTED TWO-TORSION LINES
000058 | 
000059 | The exact coefficient packet is
000060 | 
000061 |     L_a^q=O_C(q_a-q_(a+1)),
000062 |     h_a^2=H_a(w):=(w+omega^a)/(w+omega^(a+1)),
000063 |     h_0 h_1 h_2=1.                                                 (7)
000064 | 
000065 | Pullback of the divisor gives
000066 | 
000067 |     Q^* L_a^q
000068 |       = O_C(q_(-a)-q_(-a-1))
000069 |       = (L_(-a-1)^q)^(-1)
000070 |       = L_(-a-1)^q.                                                (8)
000071 | 
000072 | The last equality uses two-torsion.  Hence Q fixes L_1^q and exchanges
000073 | L_0^q with L_2^q.  This already proves invariance of the unordered coefficient
000074 | packet.
000075 | 
000076 | The Kummer functions give the sharper lift.  From (7),
000077 | 
000078 |     H_a(1/w) = omega^2 / H_(-a-1)(w).                              (9)
000079 | 
000080 | Consequently every lift has the form
000081 | 
000082 |     Q^* h_a = epsilon_a omega / h_(-a-1),
000083 |     epsilon_0 epsilon_1 epsilon_2=1.                               (10)
000084 | 
000085 | The product condition is forced by h_0 h_1 h_2=1.  Formula (10) also shows why
000086 | the bare permutation of the three line classes does not yet distinguish the
000087 | order-two and order-four lifts.
000088 | 
000089 | 
000090 | 3. LOCAL t=0 INERTIA SELECTS THE ORDER-FOUR LIFT
000091 | 
000092 | The quotient coordinate is
000093 | 
000094 |     t=z^4=((1+w^3)/(1-w^3))^2.                                    (11)
000095 | 
000096 | At q_0=-1, write delta=w+1.  Then t is a nonzero constant times delta^2.
000097 | The cover Z_q -> P1_w defined by (7) is ramified at q_0, so delta is a nonzero
000098 | constant times a local square s^2 on Z_q.  Therefore
000099 | 
000100 |     t = constant times s^4.                                       (12)
000101 | 
000102 | The positive t=0 inertia must act by s->i s.  Its square acts by s->-s and is
000103 | the nontrivial V4 inertia above q_0.
000104 | 
000105 | For a=0, equation (10) gives
000106 | 
000107 |     Q^2 h_0 = (epsilon_0/epsilon_2) h_0.                           (13)
000108 | 
000109 | The local condition (12) therefore forces
000110 | 
000111 |     epsilon_0/epsilon_2=-1.                                       (14)
000112 | 
000113 | Together with the product condition, this gives epsilon_1=-1.  Up to the
000114 | single diagonal V4 change of the exchanged h_0,h_2 frames, take
000115 | 
000116 |     (epsilon_0,epsilon_1,epsilon_2)=(1,-1,-1).                     (15)
000117 | 
000118 | The alternative with epsilon_0=epsilon_2 has order two above q_0 and cannot
000119 | be the inertia of the marked degree-four branch.  Thus the order-four lift is
000120 | selected by the actual local geometry, without importing a target matrix.
000121 | 
000122 | 
000123 | 4. EXACT COEFFICIENT CLOCK
000124 | 
000125 | Choose flat character frames e_a for L_a^q after the canonical identifications
000126 | with their duals.  With the order-four lift (15), one may take
000127 | 
000128 |     Q e_0=e_2,
000129 |     Q e_1=e_1,
000130 |     Q e_2=-e_0.                                                    (16)
000131 | 
000132 | Its matrix is
000133 | 
000134 |     A = [[0,0,-1],
000135 |          [0,1, 0],
000136 |          [1,0, 0]],
000137 | 
000138 |     A^2=diag(-1,1,-1),
000139 |     A^4=I,
000140 |     det(A)=1,
000141 |     spec(A)={1,i,-i}.                                              (17)
000142 | 
000143 | The fixed-line sign in (16) is selected by the determinant-one coefficient
000144 | representation already proved in the original triangle source.  The other
000145 | choice is the ordinary standard S4 representation with determinant equal to
000146 | the sign character; the old source excludes it.
000147 | 
000148 | In the F basis printed in the September source,
000149 | 
000150 |     R=F^dagger U F
000151 |      = [[1,0,0],
000152 |         [0,0,1],
000153 |         [0,-1,0]].                                                 (18)
000154 | 
000155 | The constant signed permutation
000156 | 
000157 |     K = [[0,1,0],
000158 |          [1,0,0],
000159 |          [0,0,-1]]                                                 (19)
000160 | 
000161 | satisfies
000162 | 
000163 |     K A K^(-1)=R.                                                  (20)
000164 | 
000165 | This is a comparison after the lift has already been selected by (12), not a
000166 | definition of A from R.  Since pullback transports monodromy contravariantly,
000167 | the positive loop around t=0 acts on flat solutions by A^(-1).  Equation (20)
000168 | therefore gives exactly
000169 | 
000170 |     K A^(-1) K^(-1)=R^(-1)=F^dagger U^(-1) F,                     (21)
000171 | 
000172 | which is the positive-loop convention stated in the old source.
000173 | 
000174 | On C, Q^2 is the hyperelliptic involution (6).  On the coefficient cover Z_q,
000175 | which lies over the w-sphere, the same square is the nontrivial V4 deck element
000176 | shown in (17).  On their fiber product Y_q it carries both pieces.  This is why
000177 | Q^2 can be geometrically nontrivial even though each two-torsion line class is
000178 | fixed after two applications.
000179 | 
000180 |