TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/boundary_origin_20261027/uniqueness/SAME-SPIN-REFERENCE-SIMPLIFICATION.txt Original file SHA-256: 8a0586c93b371905105d33f04c1deb796c9f603c0fed09c8cc90dbb33bd64481 Inclusive original lines: 1-191 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 | SAME SPIN REFERENCE SIMPLIFICATION 000002 | The original seam shift recovers the ungraded native complex module class on one spin Hardy space 000003 | 2026-10-27 000004 | 000005 | VERDICT 000006 | 000007 | The proposed simplification is correct at the ordinary complex Fredholm line 000008 | and ungraded finite module levels. Work on the same Hardy space with 000009 | coefficient 000010 | 000011 | E=S_0 tensor P_spin. 000012 | 000013 | Compare multiplication by the original seam coordinate z with multiplication 000014 | by 1. The identity operator is invertible, while the z shift is injective 000015 | with cokernel E|D_q. Therefore 000016 | 000017 | delta(E,M_z) tensor delta(E,M_1)^(-1) 000018 | =det H^0(D_q,E|D_q)^(-1) 000019 | =N_q(S_0)^(-1) tensor B_O^(-1). (1) 000020 | 000021 | The extra line B_O^(-1) is trivial under c and beta and has Q character -1. 000022 | It changes alpha to -alpha and changes no other finite relation. The accepted 000023 | native classification already retains exactly this global sign ambiguity. 000024 | Thus (1) produces the full ungraded native complex module class without 000025 | introducing a second, unspun physical reference space. 000026 | 000027 | To obtain the literal signed representative W_q=H_Theta tensor N_q^(-1), one 000028 | must still remove the ordering line B_O or choose the unordered norm 000029 | normalization. The present theorem does not identify those two signed 000030 | representatives. It shows that this distinction is already contained in the 000031 | allowed alpha sign at the ungraded complex module level. It does not identify 000032 | their super determinant or Pfaffian sewing functors. 000033 | 000034 | 000035 | 1. THE SAME SPACE RELATIVE FREDHOLM LINE 000036 | 000037 | Let H_E be the direct sum of the three Hardy spaces of E over the components 000038 | of the inverse image of the open z disk. The source half form supplies the 000039 | Hardy norm, and a unitary flat metric on P_spin supplies the coefficient norm. 000040 | The four boundary ramification points have measure zero and do not contribute 000041 | an interior evaluation term. 000042 | 000043 | Multiplication by 1 is the identity on H_E. Multiplication by z satisfies 000044 | 000045 | ker M_z=0, 000046 | coker M_z=H_E/zH_E=H^0(D_q,E|D_q). 000047 | 000048 | With 000049 | 000050 | delta(T)=det ker(T) tensor det coker(T)^(-1), 000051 | 000052 | one has the canonical identities 000053 | 000054 | delta(E,M_1)=C, 000055 | delta(E,M_z)=B(E)^(-1), (2) 000056 | 000057 | where B(E)=det H^0(D_q,E|D_q). No comparison between two Hilbert spaces or 000058 | two coefficient bundles occurs in (2). 000059 | 000060 | 000061 | 2. THE SPIN NORMALIZATION REMOVES THE MOVING COEFFICIENT 000062 | 000063 | For three line fibers, the ordered evaluation determinant factors as 000064 | 000065 | B(S_0 tensor P_spin) 000066 | =Nm_Dq(S_0) tensor Nm_Dq(P_spin) tensor B_O. (3) 000067 | 000068 | The accepted spin relative Poincare normalization proves, equivariantly over 000069 | the full Picard family, 000070 | 000071 | Nm_Dq(P_spin)=O. (4) 000072 | 000073 | Using N_q(S_0)=Nm_Dq(S_0), equations (2)-(4) give 000074 | 000075 | delta(E,M_z) tensor delta(E,M_1)^(-1) 000076 | =N_q(S_0)^(-1) tensor B_O^(-1), (5) 000077 | 000078 | which is (1). This is familywise. It applies to the actual sign line and its 000079 | c orbit and needs no fixed coefficient fiber. 000080 | 000081 | The separate spin versus scalar cable construction proves the more rigid 000082 | identity N_q^(-1) itself because its two ordering determinants cancel. The 000083 | same spin construction proves the ungraded module class with one fewer physical 000084 | premise: its reference is the identity on the same source Hardy space. 000085 | 000086 | 000087 | 3. THE ORDERING LINE IS EXACTLY THE ALLOWED ALPHA SIGN 000088 | 000089 | The actions on the scalar ordering line are determined only by the 000090 | permutation of D_q: 000091 | 000092 | c|B_O=+1, 000093 | beta|B_O=+1, 000094 | Q|B_O=-1. (6) 000095 | 000096 | Indeed, c is a three cycle, Q fixes q_0 and exchanges q_1,q_2, and beta=Q^2 000097 | fixes all three scalar fibers. Taking the inverse does not alter these signs. 000098 | 000099 | On N_q^(-1), beta acts by +i and c acts trivially. Tensoring with B_O^(-1) 000100 | therefore keeps 000101 | 000102 | gamma^3=I, 000103 | alpha^2=iI, 000104 | alpha gamma alpha^(-1)=gamma^(-1), (7) 000105 | 000106 | but replaces alpha by -alpha. It acts trivially on the N64 Heisenberg group 000107 | and hence does not change its central character or Pauli quotient action. 000108 | The spin rigidified theta theorem proves that alpha and -alpha are precisely 000109 | the two lifts over each accepted quotient action and belong to the same 000110 | classified ungraded native complex module class. 000111 | 000112 | Consequently B_O^(-1) is not an additional central phase fit. It records the 000113 | difference between an ordered three point determinant and the unordered norm 000114 | line, already visible as the unavoidable sign of the Quarter lift. 000115 | 000116 | 000117 | 4. THE SUPER DETERMINANT PARITY DOES NOT CANCEL ON ONE CABLE 000118 | 000119 | A Fredholm determinant line has super parity equal to its index modulo two. 000120 | Here 000121 | 000122 | index(M_z)=-3=1 mod 2, 000123 | index(M_1)=0 mod 2. (8) 000124 | 000125 | Thus the same spin relative defect in (1) is an odd superline. In the 000126 | factorization (5), B_O^(-1) is also odd. Its Q character can be absorbed by 000127 | alpha -> -alpha after forgetting the grading, but its odd parity cannot be 000128 | removed by that sign change. 000129 | 000130 | The two cable construction has a different graded type. Its spin shift and 000131 | scalar shift both have index -3 and hence both determinant lines are odd. 000132 | Their relative tensor product is even. Therefore the equality 000133 | 000134 | spin cable / scalar cable =N_q^(-1) 000135 | 000136 | is an equality of even relative superlines, while (5) is only the same 000137 | ordinary complex character after forgetting one odd ordering line. The two 000138 | statements are not equal as graded sewing functors. 000139 | 000140 | The primitive base shift has index -1 and is likewise odd. Pulling it to the 000141 | three sheet cable changes the integer index from -1 to -3 but preserves its 000142 | parity. This is compatible with an odd defect; it does not by itself derive 000143 | fermionic statistics, a Pfaffian orientation or the physical source 000144 | quantization rule. 000145 | 000146 | 000147 | 5. WHAT THIS REMOVES AND WHAT IT DOES NOT 000148 | 000149 | After forgetting the super grading, the theorem removes the need to assume 000150 | that the physical reference operator lives on an unspun scalar cable. It is 000151 | enough, for the finite native complex module class, that the original source 000152 | uses the relative determinant line of its actual spin shift M_z against the 000153 | canonical identity M_1 on the same Hardy space. 000154 | 000155 | It does not yet prove that the physical P1 source action evaluates this 000156 | relative Fredholm line. The remaining physical statement is now narrower: 000157 | 000158 | the original seam determinant or boundary amplitude must use 000159 | delta(E,M_z) relative to delta(E,M_1), 000160 | 000161 | with the Hardy polarization induced by the actual Calderon and reflection 000162 | data. No separate scalar coefficient bundle, state or Hilbert space is 000163 | needed for this statement. 000164 | 000165 | Nor does (5) select a fixed signed alpha representative. A later observable 000166 | that distinguishes alpha from -alpha would still require the unordered norm 000167 | normalization or an equivalent orientation datum. All finite N64, gamma, 000168 | beta and projective Quarter relations already agree after forgetting the 000169 | grading. 000170 | 000171 | For a physical graded or Pfaffian sewing statement, the scalar cable or an 000172 | equivalent odd reference remains material because it cancels the Fredholm 000173 | parity. The source must decide whether its amplitude is the odd one cable 000174 | defect or the even two cable relative defect. This choice cannot be dismissed 000175 | as alpha gauge and is not fixed by the finite complex module classification. 000176 | 000177 | 000178 | SOURCES CHECKED 000179 | 000180 | * research-20261027-boundary-origin/source/ 000181 | ORIGINAL-BOUNDARY-NORM-ORIGIN.txt 000182 | 000183 | * research-20261026-real-boundary/ 000184 | MARKED-Q-DIVISOR-CENTRAL-TWIST.txt 000185 | 000186 | * research-20261026-real-boundary/ 000187 | SPIN-RIGIDIFIED-THETA-MODULE.txt 000188 | 000189 | * research-20261027-boundary-origin/originals/ 000190 | INVERSE-Q-POINT-NORM-ORIGIN.txt 000191 |