TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/boundary_origin_20261027/uniqueness/NORM-POWER-UNIQUENESS.txt Original file SHA-256: e10ed195c257cce245ef5502f6e7cb5414783ae631089e3af03d18bc5ca8fb05 Inclusive original lines: 1-200 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 | THE FULL FINITE CLOCK RELATIONS FIX THE NORM EXPONENT ONLY MODULO FOUR 000002 | Independent uniqueness and equivalence test for the marked q line 000003 | 2026-10-27 000004 | 000005 | VERDICT 000006 | 000007 | The accepted spin-rigidified theta-module theorem remains valid, but its 000008 | finite relations do not select the literal exponent k=1. For 000009 | 000010 | W_(q,k)=H^0(Jac(C),O(2 Theta)) tensor N_q(S_0)^(-k), 000011 | 000012 | the complete native central relation alpha^2=+i I holds exactly when 000013 | 000014 | k=1 modulo 4. (1) 000015 | 000016 | Gamma and the N64 action do not refine this congruence. With the actual 000017 | marked coordinate eta, the eighth power N_q(S_0)^8 has a canonical c and Q 000018 | invariant trivialization, so k is intrinsically periodic modulo eight. The 000019 | two surviving residues are k=1 and k=5. They have identical N64, gamma and 000020 | beta actions; their alpha operators differ by the unresolved global sign 000021 | alpha -> -alpha. 000022 | 000023 | Consequently there is one class after quotienting by the spin-lift sign that 000024 | the source itself leaves unselected, but there is no theorem that the source 000025 | uses literally one inverse norm rather than five, minus three, or another 000026 | representative of the same class. If alpha is treated as a fixed signed 000027 | operator, k=1 and k=5 are distinct marked tuples. A physical or functorial 000028 | one-copy statement therefore still requires the missing boundary, sewing or 000029 | determinant insertion rule. Minimal positive exponent is a convention or an 000030 | extra simplicity rule, not a consequence of the finite relations. 000031 | 000032 | 000033 | 1. ALL CENTRAL PHASES ARE EXACTLY PERIODIC 000034 | 000035 | Write 000036 | 000037 | H_Theta=H^0(Jac(C),O(2 Theta)), 000038 | N=N_q(S_0). 000039 | 000040 | The accepted marked-divisor calculation gives 000041 | 000042 | beta|H_Theta=+I, 000043 | beta|N=-i, 000044 | gamma|N=+1. (2) 000045 | 000046 | Since beta=alpha^2, the inverse kth power contributes 000047 | 000048 | beta|N^(-k)=(-i)^(-k)=i^k. (3) 000049 | 000050 | The N64 Heisenberg group acts only on H_Theta, so tensoring by N^(-k) does 000051 | not change that action. Gamma is also unchanged because its normalized 000052 | action on N is one. The conjugation relation 000053 | 000054 | alpha gamma alpha^(-1)=gamma^(-1) (4) 000055 | 000056 | therefore holds for every k. 000057 | 000058 | The native q-oriented central relation is beta=+i I. Equation (3) gives 000059 | 000060 | i^k=i iff k=1 modulo 4. (5) 000061 | 000062 | For these and only these exponents, alpha^4=(i^k)^2=-I and alpha has order 000063 | eight. Thus the full displayed finite relations, rather than the central 000064 | phase alone, give the same condition (5). There is no further restriction 000065 | from gamma or N64. 000066 | 000067 | 000068 | 2. ADDING FOUR COPIES IS PRECISELY THE SPIN-LIFT SIGN 000069 | 000070 | Let a be the scalar by which alpha acts on N. Compatibility with (2) gives 000071 | 000072 | a^2=-i. (6) 000073 | 000074 | Hence a is an eighth root of unity of exact order eight. The two possible 000075 | values differ by a sign, reflecting the two projective spin lifts alpha and 000076 | -alpha. 000077 | 000078 | On W_(q,k), the line contributes a^(-k) to alpha. Increasing k by four 000079 | changes this contribution by 000080 | 000081 | a^(-4)=-1, (7) 000082 | 000083 | while (3), gamma and N64 remain unchanged. Therefore 000084 | 000085 | W_(q,k+4) has alpha_(k+4)=-alpha_k (8) 000086 | 000087 | with every other finite operator unchanged. 000088 | 000089 | This sign is mathematically visible if the full source tuple contains a 000090 | fixed signed alpha and the N64 and gamma markings are held pointwise. By 000091 | irreducibility, an intertwiner commuting with all of N64 is scalar, so it 000092 | cannot conjugate alpha to -alpha. Equivalently, a Pauli conjugation that 000093 | also fixes gamma would need a Pauli label fixed by the fixed-free cubic 000094 | action, hence must be central and cannot change alpha. 000095 | 000096 | The source spin construction, however, determines alpha only up to this 000097 | global sign. The accepted native census likewise retains alpha and -alpha 000098 | as the unavoidable sign pair. Under that stated equivalence, k=1 and k=5 000099 | represent the same finite source-to-native class. This is an equivalence of 000100 | marked constructions with the unresolved spin sign, not a conjugacy of two 000101 | pointwise fixed operator tuples. 000102 | 000103 | 000104 | 3. THE MARKED CURVE CANONICALLY TRIVIALIZES N^8 000105 | 000106 | The period eight is not only a character-table coincidence. On 000107 | 000108 | C: eta^2=1-w^6 000109 | 000110 | the function eta is a local parameter at every q_a. Define the nonzero 000111 | cotangent product 000112 | 000113 | Omega=product_(a=0)^2 d eta|q_a in Norm_Dq(K_C). (9) 000114 | 000115 | The cubic c leaves eta unchanged and only cycles the three factors, so it 000116 | fixes Omega. At every q_a one has w^3=-1. For 000117 | 000118 | Q:(w,eta) -> (1/w,-i eta/w^3), 000119 | 000120 | the pullback of d eta at a branch point is i d eta. Hence 000121 | 000122 | Q Omega=i^3 Omega=-i Omega, 000123 | Q Omega^4=Omega^4. (10) 000124 | 000125 | The marked equation fixes eta up to its harmless sign. Replacing eta by 000126 | -eta changes Omega by -1 and leaves Omega^4 unchanged. 000127 | 000128 | The spin isomorphism S_0^2 isomorphic K_C gives 000129 | 000130 | N^8=(Norm_Dq S_0)^8 isomorphic (Norm_Dq K_C)^4. (11) 000131 | 000132 | Together, (9)-(11) give a c and Q invariant nonzero vector and therefore an 000133 | equivariant trivialization of N^8. This construction is functorial in the 000134 | spin object. Rescaling the chosen square-root isomorphism rescales both the 000135 | map in (11) and the induced identification of N^8 by the same factor; the 000136 | resulting trivialization is unchanged under isomorphism of spin objects. 000137 | 000138 | No smaller positive power is equivariantly trivial. Equation (6) says that 000139 | alpha has exact order eight on N. In particular, N^4 carries the nontrivial 000140 | alpha sign -1. The identity O(D_q)=S_0^3 supplies useful line isomorphisms, 000141 | but it cannot trivialize this nontrivial character. Its scalar ambiguity 000142 | also does not disappear when applied directly to the constant norm line: 000143 | the cancellation by degree zero that made the spin-relative Poincare family 000144 | canonical is absent here. 000145 | 000146 | Thus 000147 | 000148 | W_(q,k+8) is canonically equivalent to W_(q,k) (12) 000149 | 000150 | for the marked source geometry, while k and k+4 differ exactly by the spin 000151 | sign. 000152 | 000153 | 000154 | 4. WHAT IS UNIQUE AND WHAT IS STILL OPEN 000155 | 000156 | The result has three distinct levels. 000157 | 000158 | * With only the native central character fixed, k=1 modulo 4. 000159 | 000160 | * With the full signed alpha operator fixed, there are two residues modulo 000161 | eight, k=1 and k=5, and they are distinct marked tuples. 000162 | 000163 | * With the actual source equivalence alpha equivalent to -alpha, these two 000164 | residues form one finite-module class. The canonical N^8 000165 | trivialization removes all further representatives. 000166 | 000167 | This proves uniqueness of the finite N64, gamma, beta and projective-alpha 000168 | package in the already stated equivalence class. It does not prove a unique 000169 | tensor exponent in the boundary functor. Tensor powers N^(-1) and N^(-5) 000170 | are different constructions before the spin-sign equivalence is imposed, 000171 | and the finite module forgets which construction produced its operators. 000172 | 000173 | The smallest additional statement that would select literal k=1 is an 000174 | original source rule saying that the boundary or sewing operation contributes 000175 | one inverse evaluation norm. An exact sequence, anomaly grading or local 000176 | gluing formula could supply such a rule. None of the finite theta, N64 or 000177 | clock relations supplies it by itself. No physical distinction between the 000178 | remaining representatives is asserted without that rule. 000179 | 000180 | 000181 | EXACT CHECK 000182 | 000183 | python3 research-20261027-boundary-origin/uniqueness/check_norm_power_uniqueness.py 000184 | python3 -OO research-20261027-boundary-origin/uniqueness/check_norm_power_uniqueness.py 000185 | 000186 | The checker works only with zeta_8 exponents modulo eight. It verifies both 000187 | spin-lift signs, the full k table, the k+4 sign change, the k+8 periodicity, 000188 | and the Q and eta-sign invariance of Omega^4. 000189 | 000190 | 000191 | SOURCE ANCHORS 000192 | 000193 | * research-20261026-real-boundary/MARKED-Q-DIVISOR-CENTRAL-TWIST.txt, 000194 | SHA-256 da1f75831a46a0c0662f931486b77e6e2085126df452b009d1a90351d413d904 000195 | 000196 | * research-20261026-real-boundary/SPIN-RIGIDIFIED-THETA-MODULE.txt, 000197 | SHA-256 4bd17ffbb2be4b31c4eb7a4efe9cd8a1191c9065655889ce189f3b1f6c2499d2 000198 | 000199 | * research-20261026-real-boundary/SOURCE-SELECTED-THETA-SPIN.txt 000200 |