TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/HARDY-BMU-FIRST-NONZERO-ENTRY.txt Original file SHA-256: 57a86a6e51d78730e23d2e9763fa93f66dd1e001efdc9f3de5a383501334452b Inclusive original lines: 1-132 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 | ACTUAL SIGNED HARDY Bmu READER: FIRST EXACT NONZERO MATRIX ENTRY 000002 | Independent original-data calculation, 30 October 2026. 000003 | No new tangent, state, coefficient line, Yukawa input or Atlas mutation. 000004 | 000005 | RESULT 000006 | The original mixed product mu_01, canonically induced from the scalar 000007 | S0 L0^q cable, has a NONZERO finite reader entry. The genuine q/p frames 000008 | do not reduce this entry to a closed q-disk Cauchy contour. In the actual 000009 | G-column/induction basis it is 000010 | M_(0,2)=(4 X_*/pi) I P_gamma,+, 000011 | X_*=[2 cosh(ell_NS/2)]^-1, 000012 | I=0.1258739334853158906973132600522979652656 000013 | -0.4697679151157443151336120497675343216716 i. (1) 000014 | A closed elementary expression and strict positive lower bound prove I!=0 000015 | below. P_gamma,+ is the ACTUAL finite grading projector, retaining all 000016 | its D5/charge slots. It is not a newly selected weak Higgs carrier. 000017 | The stated common spin/root branch is inherited from the old G/section 000018 | normalization; reversing a recorded defect-frame sign conjugates the 000019 | matrix and cannot turn this entry into zero. 000020 | 000021 | 1. Reused actual source, input and induction, with no rank9 doubling 000022 | Use INDUCED-HARDY-DEFECT-BMU-FINITE-READER.txt and its actual G proof. 000023 | F_v117=pi_*L0^q, with G columns g,c*g,c^2*g ordered as in the original 000024 | report. On scalar induction these columns correspond to the q0,q1,q2 000025 | disks. Put E_r=V_h E0_r, E0_r=z^(1/2)xi_r/sqrt(2pi) in the horizontal 000026 | NS frame. At Phi=0, h0=A_NS gamma_F, p_*=1/2 on E0_r. 000027 | The already given J is Swap tensor j, j=i gamma_F F_R R_x. 000028 | Since R pulls back q_r to p_(r+1), JE_r is supported on the OUTER 000029 | p_(r+1) cable. It is not the same q disk with an unrelated constant 000030 | p frame assigned to it. Its spin coefficient is i gamma_F z^(-1/2). 000031 | The actual induced input is pi_*mu_01; equivalently it includes the 000032 | required c translates in the full pullback frame. It is not a naked 000033 | noninvariant product inserted without its coefficient descent. 000034 | 000035 | 2. Actual spin and Kummer differential, fully normalized 000036 | The already used meromorphic spin frames obey 000037 | Phi^2=(1+w^3)dw/eta, Chi=Phi/z, eta^2=1-w^6, 000038 | z=eta/(1-w^3), hence Phi Chi=dw EXACTLY. 000039 | For the existing q0 and p1 spin sections one therefore obtains 000040 | alpha_01=h0^q h1^p dw/[(w+1)(w-omega)], 000041 | (h0^q)^2=(w+1)/(w+omega), 000042 | (h1^p)^2=(w-omega)/(w-omega^2). (2) 000043 | This is the original mu_01 flat coefficient with its actual Hom legs, 000044 | not a replacement ordinary scalar differential. The normalized R lift 000045 | R*h0^q=h1^p is retained. 000046 | Multiplication by z simplifies its square exactly: 000047 | (z alpha_01)^2=-(w+omega^2)(w-1)dw^2/(1-w^3)^2. 000048 | On the shared q2/p1 boundary ray w=i t choose the continuous old branch 000049 | z alpha_01=zeta(w)dw/(1-w^3), 000050 | zeta(w)=-omega sqrt(1-w)sqrt(1+omega w), zeta(0)=-omega. (3) 000051 | Each root is continued from 1 along the ray. Replacing (3) by a direct 000052 | principal sqrt of zeta^2 introduces a false sign jump. In the original 000053 | G calibration tau=i omega^2, rho=principal sqrt(2/[3(omega-1)]), the 000054 | q2 h0 and R-induced p1 h1 values at this endpoint are both -omega, 000055 | which gives exactly the sign in (3). A different inherited rho sign is 000056 | the corresponding defect-basis sign, not a new scalar normalization. 000057 | 000058 | 3. Exact endpoint/sign test: why Cauchy does not annul the entry 000059 | The old disk boundary is Re(w^3)=0, six rays. q_s is adjacent only to 000060 | p_(s+1) and p_(s+2). The given J support p_(r+1) therefore permits 000061 | r=s or r=s+1, before the actual Kummer signs are used. 000062 | Between the eta-positive and eta-negative arcs, h0^q has signs 000063 | q0:-1, q1:-1, q2:+1; 000064 | h1^p has signs 000065 | p0:+1, p1:-1, p2:-1. 000066 | These follow from its simple zero/pole in the local eta parameter at 000067 | the corresponding marked center, not from bare Kummer values as fibers. 000068 | z itself changes sign. For (r,s)=(0,2), the q2/p1 product sign is -1, 000069 | so the two z alpha contributions ADD. In CCW z-boundary orientation 000070 | both run infinity to0 along w=i t; their sum is -2I, where 000071 | I=integral_(0 to infinity) zeta(i t) i dt/(1+i t^3). 000072 | The diagonal pairs have Kummer product +1, so their two contributions 000073 | cancel instead. No matrix trace or selected normal component was used 000074 | to infer the nonzero offdiagonal entry. The individual matrix element 000075 | sees two shared q2/p1 arcs, not the full closed boundary of the q2 disk. 000076 | 000077 | 4. Original normal Clifford, both faces and exact prefactor 000078 | Use the ALREADY evaluated reader -N c(dz)=z(I+Z). In the whole charged 000079 | chart its value on mu is 2z (alpha_01/dz) P_gamma,+. 000080 | The two graph-face factors at p_*=1/2 have c d=X_*, so their sum is2X_*. 000081 | With , the actual NS and J factors give 000082 | (i gamma_F z^-1/2)* [2z(alpha_01/dz)P_gamma,+] z^1/2 000083 | =-2i z^2(alpha_01/dz)P_gamma,+. 000084 | Using dtheta=dz/(i z), the total finite form is 000085 | -(2X_*/pi) P_gamma,+ integral_shared z alpha_01. 000086 | The -2I oriented integral from section3 proves (1), with no missing 000087 | sheet factor: the normalized scalar induction cancels the 1/sqrt(3) 000088 | of the full pullback W exactly as in the preceding reader report. 000089 | The row/column labels are actual G-column coefficients, not reinterpreted 000090 | new physical generations. All other carrier slots stay attached. 000091 | 000092 | 5. Closed elementary value and an analytic nonzero proof 000093 | Set a=3^(-1/4)exp(i pi/12), and u=zeta/(w-1). Then 000094 | w=(u^2-omega^2)/(u^2+1), zeta=omega u/(u^2+1). 000095 | The ray runs u:omega -> i. A rational primitive is 000096 | F(u)=(2/3^(3/4)) exp(-5i pi/12) 000097 | [atan(u/a)-sqrt(2)atan(u/(sqrt(2)a))]. 000098 | Along the actual path the principal endpoint values continue without an 000099 | atan cut. Indeed Im[(u/a)^2]=sqrt(3)(t^2-1)/[2(t^2+1)]; its only 000100 | imaginary-axis crossing is t=1, at u/a=i sqrt((3-sqrt(3))/2), below i. 000101 | The second atan has the same crossing divided by sqrt(2). Hence with 000102 | x=3^(1/4)cos(5pi/12), 000103 | A=atan2(2x,1-sqrt(3)), B=atan2(sqrt(2)x,1-sqrt(3)/2), 000104 | one has the EXACT value 000105 | I=(2/3^(3/4))exp(-5i pi/12)[A-sqrt(2)B]. (4) 000106 | Moreover sqrt(3)<7/4 implies 2x3pi/4 and B<5pi/12, so 000108 | A-sqrt(2)B>pi(9-5sqrt(2))/12>0. 000109 | This is a strict analytic nonzero proof. Direct continuous-root quadrature 000110 | and (4) agree within1.3e-71 at70-digit working precision. The algebraic 000111 | squared-differential remainder is exactly0. Durable calculation output: 000112 | workspace outputs/multi-model-source-search-20261029/HARDY-BMU-FIRST-ENTRY.json. 000113 | 000114 | 6. Precise source and operator scope 000115 | This is the named ORIGINAL NS/Hardy kinetic moment p_*=1/2 with its given 000116 | collar parameter ell_NS. The old quarter maps satisfy 000117 | W0 f(theta)=exp(i theta/2)f(2theta), 000118 | A_NS W0=2W0 A_(1/4), 000119 | so X_* is also the quarter moment only with ell_quarter=2ell_NS. 000120 | The primitive NS seam shift mixes the two quarter sectors; it is not 000121 | silently U_base on one quarter alone. No physical radial-time or scale 000122 | identification is inferred from this change of frame. 000123 | 000124 | The result is a nonzero finite SOURCE RESPONSE, not a derived Yukawa. 000125 | Its displayed family operator is neutral and finite-grading even; in the 000126 | given full CAR source it has the ordinary same-charge finite bilinear 000127 | channel and its adjoint. The corresponding original finite Higgs/Dirac 000128 | insertion is grading odd and uses its already specified weak/D5 carrier. 000129 | No equality to that insertion, original P1 output policy, a nonzero 000130 | unweighted trace, or time stationarity of the finite reader is asserted. 000131 | The full adjoint slots are retained; they do not annul this offdiagonal 000132 | operator entry. No extra O/quantization/state/time condition was imposed.