TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/HARDY-BMU-FULL-MU01-READER.txt Original file SHA-256: 887016f28979a20ba0bfd5c473ed704d253be63ce05656fa9bb9817cb631f17a Inclusive original lines: 1-136 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 WHOLE NINE-ENTRY READER OF THE SAME ORIGINAL INDUCED mu01 000002 | Exact source continuation, 30 October 2026. No source/Atlas modifications. 000003 | 000004 | RESULT 000005 | For the SAME previously evaluated induced mu01 insertion, the complete 000006 | family matrix has TWO nonzero entries, not only the previously recorded 000007 | M02. With zero-based indices, original E=V_h E0 and all finite slots, 000008 | M=(4Xstar/pi) [I02 E02+I21 E21] tensor P_gamma,+, (1) 000009 | Xstar=[2cosh(ell_NS/2)]^-1, 000010 | I21=-I02+pi(1-sqrt2)C, 000011 | C=2*3^(-3/4) exp(-5pi i/12). 000012 | All seven other entries vanish exactly by support or the signed eta-pair 000013 | cancellation. The second period is nonzero and is fixed by the actual 000014 | G/R endpoint calibration, with no new phase, state or tangent chosen. 000015 | 000016 | Reused inputs: INDUCED-HARDY-DEFECT-BMU-FINITE-READER.txt, the complete 000017 | G/weighted spin trace, HARDY-BMU-FIRST-NONZERO-ENTRY.txt and the actual 000018 | two-face source B=Gq J*, J=Swap tensor i gamma_F F_R R_x. The old nonzero 000019 | result is retained unchanged. Targeted Atlas search found that result and 000020 | the KO completion, not the entire nine-entry evaluation. This is not a 000021 | claim of historical absence outside the checked records. 000022 | 000023 | 1. ACTUAL G/R ENDPOINT CALIBRATION OF THE NEW ENTRY 000024 | 000025 | On the selected original lift approaching z=+1 from the interior z disk, 000026 | w=-r with r decreasing from1 to0. The G calibration has h0~rho z, 000027 | rho=principal sqrt(2/[3(omega-1)]), h1(0)=tau=i omega^2 and h0h1h2=1. 000028 | The h0 square has argument -5pi/6 near w=-1 and tends to -2pi/3 at w=0; 000029 | the h1 square has argument -pi/3 initially and tends to -2pi/3. No cut 000030 | is crossed. Hence at the endpoint 000031 | (h0,h1,h2)=(-omega,-omega,+omega). 000032 | The actual G columns are (g,c*g,c^2*g), using h_(-s) on sheet q_s. 000033 | Thus the scalar h0^q values on (q0,q1,q2) are 000034 | (-omega,+omega,-omega). (2) 000035 | The EXISTING normalized R map takes q_r support to p_(r+1) and transports 000036 | h0^q into h1^p. Its values on (p0,p1,p2) are consequently 000037 | (-omega,-omega,+omega). (3) 000038 | This is transported calibration, not separate principal roots in each 000039 | outer disk. In particular the q1/p0 endpoint has product -omega^2, 000040 | whereas q2/p1 has +omega^2. Their continuous z alpha branches start at 000041 | +omega and -omega respectively. 000042 | 000043 | 2. ALL NINE ENTRIES FROM THE EXISTING SUPPORT AND ETA RULES 000044 | 000045 | E_s has q_s support; JE_r has p_(r+1) support. The q_s disk meets only 000046 | p_(s+1) and p_(s+2), so only r=s or r=s+1 are supported. The scalar 000047 | eta-sheet signs of h0^q and h1^p are respectively 000048 | q0,q1,q2: (-1,-1,+1), p0,p1,p2: (+1,-1,-1). 000049 | They follow from the simple zero/pole orders in the local eta parameter. 000050 | The additional z factor changes sign. A Kummer product sign +1 therefore 000051 | gives cancellation between the two eta arcs; sign -1 gives addition. 000052 | Here is the entire matrix, including the support check: 000053 | 000054 | entry q / Jp shared support Kummer eta sign result 000055 | 00 q0/p1 yes + 0 000056 | 01 q1/p1 no 0 000057 | 02 q2/p1 yes - I02 000058 | 10 q0/p2 yes + 0 000059 | 11 q1/p2 yes + 0 000060 | 12 q2/p2 no 0 000061 | 20 q0/p0 no 0 000062 | 21 q1/p0 yes - I21 000063 | 22 q2/p0 yes + 0. 000064 | 000065 | The entries in the last column multiply the common4Xstar/pi and actual 000066 | P_gamma,+ in(1). Disjoint supports meet at most at integrable endpoints; 000067 | no omitted point mass is present in the already given L1 form reader. 000068 | 000069 | 3. ORIENTATION AND THE EXACT SECOND PERIOD 000070 | 000071 | The retained flat differential is 000072 | alpha01=h0^q h1^p dw/[(w+1)(w-omega)], 000073 | (z alpha01)^2=-(w+omega^2)(w-1)dw^2/(1-w^3)^2. 000074 | For M02 its continuous square root is 000075 | zeta02(w)=-omega sqrt(1-w)sqrt(1+omega w), w=i t. 000076 | For M21 the ACTUAL endpoint signs(2)-(3) give 000077 | zeta21(w)=+omega sqrt(1-w)sqrt(1+omega w), 000078 | w=exp(-i pi/6)t. Both individual roots continue from1 on their ray. 000079 | Define Irs=integral_(0,infinity) zeta_rs(w)dw/(1-w^3). 000080 | For BOTH rays w^3=-i t^3. On z=e^(i theta), w^3=i tan(theta), so on 000081 | each of the two shared CCW z arcs t runs infinity to0. Both supported 000082 | eta contributions therefore give -2Irs, with the same orientation and 000083 | the same original normal/NS/J prefactor as the old M02 proof. 000084 | 000085 | Use u=zeta/(w-1), a=3^(-1/4)exp(i pi/12). The same rational primitive is 000086 | F(u)=C[atan(u/a)-sqrt2 atan(u/(sqrt2 a))]. 000087 | The old path is u:omega->i. The new path is u:-omega->-i. For the new ray, 000088 | Im[(u/a)^2]=sqrt3(t^2-1)/[2(t^2-sqrt3 t+1)], 000089 | (u/a)^2 at t=1 is -3-sqrt3. 000090 | The branch u/a crosses the negative imaginary axis from right to left 000091 | at -i sqrt(3+sqrt3). The second atan argument crosses at 000092 | -i sqrt((3+sqrt3)/2). Both crossings are below -i. Each continued atan 000093 | equals its principal value PLUS pi after the crossing. Oddness of the 000094 | principal endpoint primitive gives -I02; the two continuation increments 000095 | give exactly pi(1-sqrt2)C. This proves (1), including its sign. 000096 | 000097 | Writing L=A-sqrt2 B>0 with the original atan2 constants of the first 000098 | report, J=L+pi(sqrt2-1)>0, one has I02=CL, I21=-CJ. Numerically, 000099 | I21=-0.4213752856971043792014444689166786680971744633140 000100 | +1.5725939752754447880487080558066697224505612853704 i, 000101 | I21/I02=-3.3475976640251801553608318488192678119362817256263. 000102 | Thus the common phase is exact, not inferred by numerical recognition. 000103 | 000104 | 4. WHOLE OPERATOR AND EXISTING CLIFFORD/KO CONSUMER 000105 | 000106 | The full M has rank2, M^2=(4Xstar/pi)^2 I02 I21 E01, and M^3=0. 000107 | Its singular values are (4Xstar/pi)(|I21|,|I02|,0). This is the complete 000108 | neutral reader, not a Hermitian matrix of three masses. On the actual 000109 | family-times32 space B=M tensor P_gamma,+ has rank32. 000110 | Both graph faces and both eta arcs have already been included. The 000111 | holomorphic original normal symbol annihilates gamma-minus; this is not 000112 | a selected zero physical branch. The ordinary adjoint has coefficient 000113 | M^dagger P_gamma,+. The actual finite J_F image has coefficient 000114 | bar(M)P_gamma,-. They are fixed by the whole source and require no new 000115 | independent period or state. C0 and J_F retain their different roles. 000116 | 000117 | The established original weak WEDGE consumer and its already checked KO 000118 | completion apply to the WHOLE matrix by linearity, with S=M+M^T. Since 000119 | M=C times a real matrix, S is normal and has singular values(r,r,0), 000120 | r=(4Xstar/pi)sqrt(|I02|^2+|I21|^2). The existing completed odd finite 000121 | block is rank64, with the same carrier support, charge, conjugation and 000122 | four-sector rules already proved for a general two-edge coefficient. 000123 | No new hierarchy/CKM scan or fresh96-dimensional construction is needed. 000124 | The old statement that no original weak consumer was found was already 000125 | corrected by ORIGINAL-PRIMITIVE-WEDGE-CONSUMER-HARDY-REVIEW.txt; it is not 000126 | revived here. This whole response is still a contribution, not the whole 000127 | physical Y or an independently selected source action. 000128 | 000129 | REPLAY 000130 | Workspace outputs/multi-model-source-search-20261029/ 000131 | HARDY-BMU-FULL-MU01-READER.json, 000132 | SHA256 8d95dfb8fbe78b8ea92c99db7fcc5794773f184d27e67f4ad3551436a069017a. 000133 | The exact squared differential, cut geometry and all nine support/parity 000134 | entries were checked together. Both continuous-root periods agree with 000135 | their closed forms at75-digit precision. Root independently confirmed 000136 | both periods and their exact primitive/cut identity.