TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SPIN-TRACE-LOCAL-KERNEL.txt
Original file SHA-256: 591051be956f8bb3c464d039dca502ba66b1274b8faf84c73c411be10c84d0f7
Inclusive original lines: 1-190
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000001 | SPIN TRACE OF THE ELLIPTIC SZEGO KERNEL AT THE FOUR BRANCH MARKS
000002 | 26 October 2026
000003 | 
000004 | Question and verdict
000005 | ====================
000006 | 
000007 | Let
000008 | 
000009 |   p:E -> P1,  E:y^2=x^4-1,
000010 | 
000011 | and retain the actual Q-invariant nontrivial spin line alpha and its NS--NS
000012 | Szego Green kernel. The coefficient pushforward p_*alpha has the familiar
000013 | local deck exponents 0 and 1/2. The ramified spin trace contains an additional
000014 | half-density Jacobian. At every simple branch mark its two exponents are
000015 | 
000016 |   -1/4 and +1/4,
000017 | 
000018 | with spin-frame monodromy -i and +i. Thus the old quarter pair occurs
000019 | naturally as the LOCAL conjugacy class of the spin-trace kernel in the
000020 | central spin lift of the branch meridian. It was invisible in the raw
000021 | coefficient pushforward because that calculation deliberately computed the
000022 | ordinary rank-two coefficient bundle, not the ramified spin frame.
000023 | 
000024 | This does not yet identify the old smooth quarter circle. The actual marked
000025 | equator pullback is the ramified V4/E8 graph, and the spin trace has four
000026 | regular-singular branch points. The old B_1/4 direct-sum B_3/4 system is a
000027 | smooth circle domain with one global quarter holonomy. The two systems are
000028 | conjugate on a punctured neighborhood of one mark, but no bounded smooth
000029 | conjugacy extends through all four marked vertices. The missing datum is now
000030 | precise: an original trace or vertex-domain map selecting a smooth carrier,
000031 | not a missing quarter phase.
000032 | 
000033 | 1. Raw coefficient trace versus spin trace
000034 | ===========================================
000035 | 
000036 | Near one branch point a in mu4 choose local coordinates
000037 | 
000038 |   X=x-a=q^2.                                             (1)
000039 | 
000040 | The flat line alpha is locally trivial on E because a small loop in the q
000041 | disk contracts. A local coefficient section has the unique form
000042 | 
000043 |   f(q)=f_0(X)+q f_1(X).                                  (2)
000044 | 
000045 | For raw coefficient functions, the two summands in (2) have exponents
000046 | 
000047 |   0, 1/2
000048 | 
000049 | and monodromy +1,-1 around X=0. This is the order-two connection previously
000050 | computed for p_*alpha.
000051 | 
000052 | A chiral spinor on E is instead
000053 | 
000054 |   f(q)(dq)^(1/2).
000055 | 
000056 | In the base half-density frame,
000057 | 
000058 |   (dq)^(1/2)=(dq/dX)^(1/2)(dX)^(1/2)
000059 |              =(2q)^(-1/2)(dX)^(1/2).                   (3)
000060 | 
000061 | Equations (2)-(3) give the two spin-trace frames
000062 | 
000063 |   e_-(X)=q^(-1/2)(dX)^(1/2),
000064 |   e_+(X)=q^(+1/2)(dX)^(1/2).                            (4)
000065 | 
000066 | Up to the harmless constant 2^(-1/2), their exponents in X are -1/4 and
000067 | +1/4. Analytic continuation once around X=0 sends q to -q and therefore
000068 | 
000069 |   e_- -> -i e_-,
000070 |   e_+ -> +i e_+.                                        (5)
000071 | 
000072 | No scalar phase was inserted. The quarter phases are forced by the simple
000073 | ramification and the spin half-density.
000074 | 
000075 | There is an essential type distinction here. The raw pushforward has
000076 | 
000077 |   M_coeff=diag(1,-1),       M_coeff^2=I,
000078 | 
000079 | whereas the frames (4) have
000080 | 
000081 |   M_spin=diag(-i,+i),       M_spin^2=-I.                 (5a)
000082 | 
000083 | Thus M_spin is not an ordinary representation of the order-two branch
000084 | stabilizer. It is its central order-four spin lift, equivalently a projective
000085 | or ramified spin-frame action. On the punctured q disk the factor q^(-1/2)
000086 | itself changes sign around q=0, even though alpha is locally trivial there.
000087 | It must not be described as a single-valued gauge of the ordinary flat bundle
000088 | p_*alpha.
000089 | 
000090 | The underlying ordinary holomorphic bundle statement remains unchanged:
000091 | 
000092 |   p_*alpha=O(-1) direct-sum O(-1).
000093 | 
000094 | Factoring the base spin line K_P1^(1/2)=O(-1) is a local calculation on the
000095 | punctured base. Equations (4)-(5a) describe the ramified spin trace in the
000096 | central extension of the branch group. They do not replace the ordinary flat
000097 | connection on p_*alpha by a new connection with order-four holonomy.
000098 | 
000099 | 2. Exact pushed-forward local Cauchy kernel
000100 | ===========================================
000101 | 
000102 | Write Y=r^2 at the second point. The singular part of the cover Szego kernel
000103 | is
000104 | 
000105 |   S_E(q,r)=c (dq)^(1/2)(dr)^(1/2)/(q-r)+regular,         (6)
000106 | 
000107 | where c=i/(2pi) in the normalization of ELLIPTIC-GREEN-RP.txt. Choose the two
000108 | sheet half-density factors
000109 | 
000110 |   h_+(q)=(2q)^(-1/2),
000111 |   h_-(q)=i(2q)^(-1/2).                                  (7)
000112 | 
000113 | The factor i is not fitted: it is the analytic square root of
000114 | d(-q)/dX=-1/(2q). In the ordered sheet basis (+,-), pushing (6) to the base
000115 | is an exact operation on the actual elliptic covariance. More explicitly,
000116 | write
000117 | 
000118 |   S_E(u,v)=s_E(u,v)(du dv)^(1/2).
000119 | 
000120 | Then the complete local pushforward is
000121 | 
000122 |   K_full(X,Y)_(epsilon,eta)
000123 |     =h_epsilon(q) h_eta(r) s_E(epsilon q,eta r)
000124 |       (dX dY)^(1/2),                                    (7a)
000125 | 
000126 | for epsilon,eta in {+1,-1}, with the square-root branches fixed by (7).
000127 | Its universal singular part is the rank-two matrix
000128 | 
000129 |   K_sheet(X,Y)=c/[2 sqrt(qr)]
000130 |     [[1/(q-r),  i/(q+r)],
000131 |      [-i/(q+r), 1/(q-r)]] (dX dY)^(1/2)
000132 |     +regular.                                            (8)
000133 | 
000134 | The constant unitary which diagonalizes
000135 | 
000136 |   [[0,i],[-i,0]]
000137 | 
000138 | turns (8) into
000139 | 
000140 |   K_spin(X,Y)=c/(X-Y)
000141 |     diag((X/Y)^(1/4),(Y/X)^(1/4)) (dX dY)^(1/2)
000142 |     +regular.                                            (9)
000143 | 
000144 | Indeed,
000145 | 
000146 |   1/(q-r)+1/(q+r)=2q/(X-Y),
000147 |   1/(q-r)-1/(q+r)=2r/(X-Y).
000148 | 
000149 | Equation (9) is the smallest decisive kernel identity. Away from X=0 its
000150 | diagonal singularity is the ordinary identity Cauchy kernel. Moving one
000151 | argument around the branch point produces the two phases +i and -i. The
000152 | regular part of the actual s_E in (7a) changes global correlations but cannot
000153 | change these local exponents.
000154 | 
000155 | 3. Comparison with the old quarter circle
000156 | ==========================================
000157 | 
000158 | On a punctured disk, after passing to the central spin lift of the branch
000159 | meridian, the two diagonal factors in (9) are the standard regular-singular
000160 | quarter systems
000161 | 
000162 |   d+(1/4)dX/X,
000163 |   d-(1/4)dX/X.                                           (10)
000164 | 
000165 | Their scalar Cauchy kernels have the same local form
000166 | 
000167 |   K_a(X,Y)=c (X/Y)^a (dX dY)^(1/2)/(X-Y),
000168 | 
000169 | with a=+1/4 and a=-1/4. This is precisely the pair displayed in (9), up to
000170 | the order of the two diagonal entries.
000171 | 
000172 | This is the exact positive local match to the old B_1/4 and B_3/4 conjugacy
000173 | class in the ramified spin category. It is stronger than matching the numbers
000174 | +i,-i: the Cauchy principal part, the half-density normalization and the
000175 | central-lift monodromy are all carried together. It is not an isomorphism of
000176 | the ordinary flat bundles, because M_coeff has order two and M_spin has order
000177 | four.
000178 | 
000179 | The global domains are nevertheless different.
000180 | 
000181 |   * The old quarter pair lives on one smooth circle. Its +i,-i are the
000182 |     holonomies of that circle generator.
000183 | 
000184 |   * The elliptic spin trace lives on P1 minus the four branch marks. Each
000185 |     branch meridian has the local pair (-i,+i). The product of the four local
000186 |     monodromies is compatible with the trivial total loop on P1.
000187 | 
000188 |   * The original marked equator |x|=1 passes through all four branch points.
000189 |     Its full inverse image is the connected V4/E8 graph of local valence four,
000190 |     not a smooth circle. At a vertex, smoothness in q imposes the canonical