TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_recursion_20261028/NATIVE-F20-JOINT-CLOCK-TYPECHECK.txt Original file SHA-256: 636378eff2bbc119d856bb416fad3bc8ce5adac790cef2ef46c6e225cec07371 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 | NATIVE F20 FOUR VERSUS THE ORIGINAL FAMILY CLOCK 000002 | 000003 | 2026-10-28 | source and exact representation check | no Atlas change 000004 | 000005 | 000006 | RESULT 000007 | 000008 | The newly identified irreducible F20 four does not by itself carry the original 000009 | native family clock. In the selected native Pauli chart, the actual v634 000010 | matrix sigma satisfies 000011 | 000012 | sigma^3=I, 000013 | tr(sigma)=1, 000014 | char(sigma)=(lambda-1)^2(lambda^2+lambda+1), 000015 | 000016 | but exact matrix multiplication gives 000017 | 000018 | rank [C5,sigma] = 4, 000019 | rank (G sigma G^-1-sigma) = 4, 000020 | rank (G sigma G^-1-sigma^-1) = 4. (1) 000021 | 000022 | Thus this selected C5,G pair and the frozen sigma satisfy none of the three 000023 | candidate joint relations. 000024 | 000025 | There is also a basis independent obstruction if one requires the original 000026 | cyclotomic direct product relation on an irreducible F20 four. Since C5 has 000027 | four distinct eigenvalues, [C5,sigma]=0 makes sigma diagonal in the C5 000028 | eigenbasis. G cycles the four eigenlines. If 000029 | 000030 | G sigma G^-1=sigma, 000031 | 000032 | then all four diagonal entries are equal. An order-three sigma is scalar and 000033 | has trace 4, 4 omega, or 4 omega^2. If instead 000034 | 000035 | G sigma G^-1=sigma^-1, 000036 | 000037 | the entries alternate a,a^-1,a,a^-1. Their trace is 4 for a=1 and -2 for 000038 | a=omega or omega^2. Neither case has trace 1. This proves a joint-clock gap 000039 | for the irreducible F20 four under either stated extra relation. It is not a 000040 | no-go for the full TFPT construction. 000041 | 000042 | 000043 | 1. WHICH RELATION THE ORIGINAL SOURCES ACTUALLY REQUIRE 000044 | 000045 | The plus relation belongs to the original cyclotomic compositum. 000046 | 000047 | v315 states that the cyclic clock factors as the direct product 000048 | 000049 | Z/30 = Z/5 x Z/6, 000050 | 000051 | and that Q(zeta30) is the compositum of the carrier field Q(zeta5) and the 000052 | family field Q(zeta3). It explicitly distinguishes this arithmetic bridge 000053 | from dynamical coupling. See verification/v315_coxeter_coupling.py:1-30, 000054 | 58-89 and 106-114, SHA256 000055 | 000056 | 490969af0827cf8829f236e3999710bb526396dc9eb2ee0672d8bdd5c6af7b0c. 000057 | 000058 | v419 identifies the carrier Galois generator with the Frobenius 000059 | zeta5 -> zeta5^2 and proves 000060 | 000061 | G C5 G^-1=C5^2, 000062 | G^4=I. 000063 | 000064 | It also retains the primewise splitting 000065 | 000066 | Gal(Q(zeta30)/Q)=(Z/5)^x x (Z/3)^x. 000067 | 000068 | See verification/v419_seam_galois_carrier.py:1-33 and 67-103, SHA256 000069 | 000070 | cc6152ad7ea68c67725020e386e49eec32810ab25bb483c76bae99964399c2fd. 000071 | 000072 | Consequently, the carrier Frobenius element with CRT residues 000073 | 000074 | 7 mod 30 = 2 mod 5 and 1 mod 3 000075 | 000076 | fixes the family unit zeta3. On the actual cyclotomic module the required 000077 | relations are therefore 000078 | 000079 | [C5,sigma_family]=0, 000080 | G sigma_family G^-1=sigma_family. (2) 000081 | 000082 | Equation (2) is required if the F20 four is claimed to be the complete 000083 | original cyclotomic carrier plus family module. The source does not require 000084 | every native four-dimensional realization to obey (2). In particular, v417 000085 | contains a separate three-dimensional operator realization in which the seam 000086 | deck and family rotation do not commute. See 000087 | verification/v417_eisenstein_cp_operator.py:34-40 and 126-140, SHA256 000088 | 000089 | c5f1fe26b9e6fd30bf2ad45354de6819a184c4a86b491f19ab21cb53b21c2b7e. 000090 | 000091 | That separate realization prevents promotion of (2) to a universal TFPT 000092 | commutation axiom. 000093 | 000094 | The minus relation belongs to another operator. The marked Theta source 000095 | requires 000096 | 000097 | alpha sigma alpha^-1=sigma^-1, 000098 | alpha^2=iI. (3) 000099 | 000100 | PAULI-SPIN-CENTRAL-LIFT.txt:23-62, 101-127 and 161-193 constructs an exact 000101 | native candidate 000102 | 000103 | alpha_native=zeta8 (XI) U, 000104 | 000105 | and verifies (3), alpha^4=-I and alpha^8=I. Its SHA256 is 000106 | 000107 | e5ef50c899eba80dca03e1b7023edf816df3b45a669534fccf2fef215b0ae8e6. 000108 | 000109 | Nothing in that source identifies alpha_native with the F20 generator G. 000110 | Therefore the minus case above is a valid obstruction to a proposed 000111 | identification of those two operations, but it is not an original relation 000112 | imposed on the v419 Frobenius. 000113 | 000114 | 000115 | 2. THE TWO QUARTER LIFTS ARE DIFFERENT 000116 | 000117 | The equality of order eight and fourth power -I had hidden a type mismatch. 000118 | For the marked Theta lift, 000119 | 000120 | alpha_native^2=iI, 000121 | 000122 | so alpha_native has projective order two. For the new symplectic normalization 000123 | 000124 | Ghat=zeta8^-1 G, 000125 | 000126 | the exact square is 000127 | 000128 | [ 0 1 0 0] 000129 | [-1 0 0 0] 000130 | Ghat^2 = [ 0 0 0 i], 000131 | [ 0 0 i 0] 000132 | 000133 | which is nonscalar and has trace zero. Ghat has projective order four. A 000134 | central phase does not change scalar versus nonscalar square, so no conjugacy 000135 | up to central phase can identify these lifts. The existing 000136 | check_distinct_quarter_lifts.py proves the same invariant independently. 000137 | 000138 | This correction preserves both positive constructions. The F20 four and its 000139 | Spin5 dictionary remain exact. The marked Theta alpha remains an exact 000140 | C3 semidirect C8 lift. They are different group operations until a new 000141 | generator-level dictionary proves otherwise. 000142 | 000143 | 000144 | 3. THE ORIGINAL RANK-EIGHT JOINT MODULE 000145 | 000146 | The original arithmetic carrier and family clocks already coexist on 000147 | Q(zeta30), which has rational rank eight. In the basis 000148 | 000149 | 1,zeta30,...,zeta30^7 000150 | 000151 | modulo Phi30, define 000152 | 000153 | C5 = multiplication by zeta30^6, 000154 | sigma_30 = multiplication by zeta30^10, 000155 | G_7 = the automorphism zeta30 -> zeta30^7. 000156 | 000157 | Exact reduction modulo Phi30 gives 000158 | 000159 | C5^5=sigma_30^3=G_7^4=I, 000160 | [C5,sigma_30]=0, 000161 | G_7 C5 G_7^-1=C5^2, 000162 | G_7 sigma_30 G_7^-1=sigma_30. (4) 000163 | 000164 | This is the positive original joint-clock carrier. It is not the native 000165 | complex four: after complexification sigma_30 has four omega and four omega^2 000166 | eigenvalues, hence trace -4 rather than 1. The rank-eight arithmetic module 000167 | therefore avoids the four-dimensional F20 obstruction by changing the 000168 | representation and its scalar field, not by embedding the frozen v634 sigma 000169 | unchanged. 000170 | 000171 | The first remaining bridge is now precise. One must either construct an 000172 | explicit typed functor from the original Q(zeta30) arithmetic module to the 000173 | native/Theta family module that explains the change of character, or keep the 000174 | F20 carrier dictionary and the Theta family clock as separate compatible 000175 | operators on a larger object. Equal quarter order cannot supply this bridge. 000176 | 000177 | 000178 | 4. ACTUAL NATIVE SOURCE 000179 | 000180 | v634 defines the E8 complex structure J and the coordinate-pair permutation 000181 | sigma in one root representation, proves sigma^3=I and [J,sigma]=0, and later 000182 | proves 000183 | 000184 | c=J sigma, 000185 | sigma=c^4, 000186 | J=c^9. 000187 | 000188 | See verification/v634_st31_structure.py:90-152 and 721-749, SHA256 000189 | 000190 | 5ff53ed031ca03103187ac22edf723aabce88bee3df1085c52486ecb0044d37c. 000191 | 000192 | On its frozen complex four, this is the matrix used in (1): 000193 | 000194 | [0 0 1 0] 000195 | [1 0 0 0] 000196 | sigma = [0 1 0 0]. 000197 | [0 0 0 1] 000198 | 000199 | 000200 | VERIFICATION