TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/MARKED-FIBER-K3-TRANSPORT-ADDENDUM.txt Original file SHA-256: 7b2fb212bb98bb11d71361b13057cb459aa4f60f13a96f79eb35c191b4b17ef0 Inclusive original lines: 1-196 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 | MARKED FIBER K3 TRANSPORT AND THE FULL SIGN-LINE DECK TEST 000002 | ========================================================== 000003 | 000004 | Corrected narrow addendum, 30 October 2026. This is a new file; the 000005 | registered ANTISYMMETRIC-E8-KERNEL-BRIDGE report is unchanged. 000006 | 000007 | CORRECTED VERDICT 000008 | ----------------- 000009 | 000010 | The geometric base map is ALREADY PROVED. For 000011 | 000012 | C: eta^2=1-w^6, 000013 | X_f=P1 minus mu4, 000014 | 000015 | the existing sign-line dictionary gives 000016 | 000017 | pi:C -> X_f, 000018 | z=eta/(1-w^3), 000019 | z^2=(1+w^3)/(1-w^3), 000020 | pi^* E_v117 = L_0^q direct-sum L_1^q direct-sum L_2^q. (1) 000021 | 000022 | Thus no new base identification is needed. What must remain separate is the 000023 | choice of a coordinate basis on C from a coordinate basis on X_f. The map 000024 | pi and the local-system pullback do not identify arbitrary function frames 000025 | or turn a global section operator into a pointwise End(F) matrix. 000026 | 000027 | There are now two distinct exact statements. 000028 | 000029 | 1. FINITE MARKED FIBER. The matrices A4 and KQ are coefficient actions. 000030 | The September marked intertwiner S transports any ALREADY GIVEN positive 000031 | three-dimensional coefficient operator uniquely at the selected fiber. 000032 | 000033 | 2. GLOBAL SIX-SECTION SOURCE. The bare antisymmetrizer on the six sections 000034 | mu_ab does NOT descend through the full independent Kummer deck group 000035 | V4_q x V4_p. Therefore the finite formula S K3 S^(-1) does not prove that 000036 | the naked six-section source contains a global rank-three positive 000037 | subsource. A coupled endpoint kernel such as the separately studied 000038 | B_mu construction may compensate the deck characters, but that repair is 000039 | not proved here. 000040 | 000041 | 000042 | 1. THE COEFFICIENT CHART AND ITS METRIC TRANSPORT 000043 | ------------------------------------------------- 000044 | 000045 | The literal KQ matrix comes from the checked algebraic Fuchs frame, 000046 | 000047 | G(Qy)=U G(y) KQ, 000048 | KQ=Gbase^(-1) U^(-1) Gbase 000049 | =[[0,1,0],[-1,0,0],[0,0,1]]. (2) 000050 | 000051 | KQ is therefore a transported vertical coefficient action, not a new map of 000052 | the bases. Its displayed matrix uses Gbase, not the September marked 000053 | intertwiner S alone. The latter is a second finite coefficient chart: 000054 | 000055 | S^dagger S=N, 000056 | S P S^(-1)=M, 000057 | S(-Gint)S^(-1)=U. (3) 000058 | 000059 | Its fixed-generator intertwiner space is one-dimensional, so the remaining 000060 | overall phase cancels in conjugation. For an ALREADY GIVEN three-dimensional 000061 | operator put 000062 | 000063 | a=y^2+delta^2+epsilon^2, 000064 | P^sharp_N=N^(-1) P^dagger N=P^(-1), 000065 | H3_N=a I-y delta(omega P+bar(omega) P^sharp_N), 000066 | K3_N=H3_N^(-1/2). (4) 000067 | 000068 | The exact marked-fiber transport is 000069 | 000070 | H3_v117=S H3_N S^(-1) 000071 | =a I-y delta(omega M+bar(omega) M^dagger), 000072 | 000073 | K3_v117=S K3_N S^(-1)=H3_v117^(-1/2). (5) 000074 | 000075 | This is metric rather than a bare similarity: S^dagger S=N gives 000076 | 000077 | _standard=_N. 000078 | 000079 | Consequently N-selfadjointness and positivity become ordinary Hermitian 000080 | selfadjointness and positivity in the marked v117 chart. Conversely, 000081 | 000082 | K3_N=S^(-1) K3_v117 S. (6) 000083 | 000084 | Equations (5)-(6) establish only the transport of a supplied rank-three 000085 | coefficient operator. They do not construct that operator from the six 000086 | global mixed-spin sections. 000087 | 000088 | 000089 | 2. THE DECISIVE FULL-DECK COUNTERTEST 000090 | ------------------------------------ 000091 | 000092 | The actual products are global sections 000093 | 000094 | mu_ab in V_ab=H0(C,K_C tensor L_a^q tensor L_b^p), 000095 | a != b. (7) 000096 | 000097 | The q and p Kummer covers have independent deck groups. If chi_a and psi_b 000098 | denote their three nontrivial V4 characters, then 000099 | 000100 | mu_ab has character (chi_a,psi_b) of V4_q x V4_p. (8) 000101 | 000102 | The six ordered pairs (a,b), a!=b, give six distinct characters. In 000103 | particular take 000104 | 000105 | epsilon_q=(+,-,-), epsilon_p=(+,+,+). 000106 | 000107 | Then 000108 | 000109 | mu_01 -> +mu_01, 000110 | mu_10 -> -mu_10, 000111 | 000112 | and hence 000113 | 000114 | mu_01-mu_10 -> mu_01+mu_10. (9) 000115 | 000116 | The antisymmetric three-space is therefore not invariant under the full 000117 | deck group. This is a type obstruction, not a normalization issue. 000118 | 000119 | The complete argument is equally short. Since the six deck characters are 000120 | distinct and their spaces V_ab are one-dimensional, every endomorphism 000121 | commuting with V4_q x V4_p is diagonal in the mu_ab basis. The actual R 000122 | action is transitive on all six labels. A diagonal endomorphism commuting 000123 | also with R must have all six diagonal entries equal. Thus 000124 | 000125 | Comm(V4_q x V4_p, R)=C I_6. (10) 000126 | 000127 | There is no common rank-three projector. Equivalently, the full finite 000128 | deck-plus-R representation on the six global-section lines is irreducible 000129 | over C. Restricting to the diagonal A4 subgroup makes mu_ab and mu_ba carry 000130 | the same character and permits the algebraic antisymmetrizer, but that 000131 | restriction is an additional quotient of the actual independent deck data. 000132 | It cannot be made silently. 000133 | 000134 | The raw identities 000135 | 000136 | s^*mu_ab=-mu_ba, 000137 | R^*mu_ab=rho^(-1)mu_(b+1,a+1) 000138 | 000139 | remain correct as pullbacks between the one-dimensional GLOBAL section 000140 | spaces. They do not make mu_ab(y)-mu_ba(y) a pointwise End(F) element in a 000141 | single trivially identified fiber. Applying s to one endpoint is a 000142 | correspondence between q and p line data; it is not a constant local matrix 000143 | unless the corresponding endpoint transport and deck compensation are kept. 000144 | 000145 | 000146 | 3. PRECISE SURVIVING KERNEL STATEMENT 000147 | ------------------------------------- 000148 | 000149 | The finite algebraic map 000150 | 000151 | A(mu_ab-mu_ba)/sqrt(2)=e_a wedge e_b 000152 | 000153 | is still a normalized intertwiner after the diagonal A4 coefficient data 000154 | have been identified. The bare six-cycle spectral calculation and the 000155 | matrix formulas (4)-(6) also remain correct within that finite coefficient 000156 | category. What fails is their promotion to a global three-dimensional 000157 | source on the full sixteen-sheet Kummer cover. 000158 | 000159 | Likewise the quarter action is only covariant. In the algebraic horizontal 000160 | column chart it sends 000161 | 000162 | K3_col -> KQ K3_col KQ^dagger, (11) 000163 | 000164 | with the c-orientation transported at the same time. Neither the marked S 000165 | nor the Hodge map adds KQ invariance. 000166 | 000167 | The actual Fuchs row metric is 000168 | 000169 | H(z)=G(z)^(-dagger) G(z)^(-1), 000170 | 000171 | and G(z)^dagger H(z)G(z)=I. This horizontal coefficient frame is compatible 000172 | with the already existing pi in (1). It does not remove the independent p 000173 | deck characters in (7)-(10). The remaining original question is therefore 000174 | not the existence of pi. It is whether the actual coupled B_mu endpoint 000175 | kernel, diagonal skew operation and Hodge contraction furnish a deck-neutral 000176 | global operator before the finite K3 transport is applied. 000177 | 000178 | 000179 | SOURCES CHECKED 000180 | --------------- 000181 | 000182 | * SIGN-LINE-COVER-DICTIONARY.txt, 000183 | SHA256 555f75a2d3c67347fa5f8e3c4f7f46e79d94cb4172176c84010224585dc842e7. 000184 | * MIXED-SPIN-PRODUCT-U6-SOURCE-REVIEW.txt, 000185 | SHA256 319d9b965c909058c237b523236c2d9f4521989b8154fffbd527987d1b3ba1c8. 000186 | * source-native-clock-flavor-bridge-20260922/PROOF.md, 000187 | SHA256 a3e88f4cb025994fad73d2bb0b6408fd3970f184b05d8606d5c429eb2fb2036b. 000188 | * source-native-clock-flavor-bridge-20260922/checker.py, 000189 | SHA256 3937adb4be23d48ecd017eb25bc2463bb28f9aab9ddd01cbc1680a0438089f47. 000190 | * ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt, 000191 | SHA256 3a57907902631b58e0abee61e9a424c3ac03ff7a8fa074cc4d27e3379b406cf3. 000192 | * FUCHS-FRAME-INDEPENDENT-TRACE-REVIEW.txt, 000193 | SHA256 b6014bc7ff48a743dd62768e66e307b33e552b95829afeb06355cd5381c690a1. 000194 | * ANTISYMMETRIC-E8-KERNEL-BRIDGE.txt, 000195 | SHA256 37ddfb2a11aba09a23a6326fd3c929352edca3c8bb575f19778a2e03cce2454c. 000196 |