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Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/HARDY-BMU-ALL-SIX-INPUT-READER.txt
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000001 | THE COMPLETE SIX-INPUT HARDY RESPONSE IN ONE ACTUAL G/HOM FRAME
000002 | 30 October 2026. New continuation of the existing source calculation.
000003 | No original paper, source state, weight, Atlas entry or pinned report changed.
000004 | 
000005 | RESULT
000006 | All six existing offdiagonal induced mu_ab inputs now have their WHOLE
000007 | three-by-three finite response evaluated. There are54 tested entries,
000008 | 12 nonzero continuous-root periods and42 exact support/parity zeros.
000009 | In the SAME original horizontal G coefficient frame, put
000010 |   alpha=(4Xstar/pi) I02, beta=(4Xstar/pi) I21,
000011 |   F=alpha E02+beta E21,
000012 |   N=-i(alpha E12+beta E20),
000013 |   omega=exp(2pi i/3), Xstar=[2cosh(ell_NS/2)]^-1.
000014 | Then the full table is
000015 |   mu01 -> F,             mu12 -> omega^2 F,   mu20 -> omega F,
000016 |   mu10 -> N,             mu21 -> omega^2 N,   mu02 -> omega N.       (1)
000017 | Every matrix in(1) retains its original P_gamma,+ normal/internal factor.
000018 | These are six given input responses, not a freely selected linear
000019 | combination, physical families or the whole physical Yukawa source.
000020 | 
000021 | 1. REUSED SOURCE AND THE ACTUAL TWO-END FRAME MAP
000022 | 
000023 | Retain E=V_h E0, B=Gq J*, J=Swap tensor i gamma_F F_R R_x and the
000024 | given Phi=0 NS source h0=A_NS gamma_F. The full source and its state,
000025 | both graph faces, spin, metric and signed continuation are unchanged.
000026 | The NS n=0 energy is1/2. It is not silently identified with quarter1/4
000027 | or an independently physical transfer time. The earlier report
000028 | HARDY-BMU-FULL-MU01-READER.txt already proves the complete mu01 result.
000029 | 
000030 | Here is the additional common-frame proof, instead of merely cycling
000031 | row/column labels. In ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt:81-84,118-124,
000032 | the actual column j contains h_(-j) and its V4 character is epsilon_(-j).
000033 | For the scalar cable on q-line a, the fundamental frame is therefore
000034 |   G_a=G P^(-a), P=[[0,0,1],[1,0,0],[0,1,0]].                     (2)
000035 | Its column j contains h_(a-j), and column0 is the requested chi_a.
000036 | Thus this is the actual induced representation, not dimension matching.
000037 | Equation(2) is a new direct consequence of the existing G columns, not
000038 | a formula claimed to be printed in the old original paper.
000039 | 
000040 | P is real, orthogonal and detP=1, so every G_a has the same coefficient
000041 | metric G^(-*)G^-1. The original fixed interior NS domain and its kinetic
000042 | Hardy projection are preserved by this constant column permutation.
000043 | The canonical spin has c*sqrt(dz)=sqrt(dz); c fixes z and the normal
000044 | and internal symbols. Since this GIVEN Phi=0 h0 is family blind, the
000045 | same P also intertwines V_h and the whole G graph. Accordingly
000046 |   E0^(a)=E0 P^(-a), E^(a)=E P^(-a).                            (3)
000047 | This does not assert P covariance of a fixed arbitrary family-coupled Phi.
000048 | 
000049 | The actual normalized R lift gives R*h_a^q=h_(a+1)^p. Hence the p-line b
000050 | in the output is J applied to the q-line b-1. A scalar cable matrix has
000051 | different source and target induction frames:
000052 |   Mscalar_ab=(J E^(b-1))* M_mu_ab E^(a)
000053 |              =P^(b-1) Mcommon_ab P^(-a).
000054 | Therefore the correct COMMON ORIGINAL G frame is
000055 |   Mcommon_ab=P^(1-b) Mscalar_ab P^a.                           (4)
000056 | Both ends have been transported. Replacing(4) by a guessed Ad(P) on one
000057 | fixed family matrix loses the original Hom type.
000058 | 
000059 | 2. ALL54 ENTRIES, SIGNED SQUARE ROOTS AND ORIENTATION
000060 | 
000061 | For the already existing input write
000062 |   alpha_ab=h_a^q h_b^p dw/[(w+omega^a)(w-omega^b)],
000063 |   (z alpha_ab)^2=-(w+omega^(a+2))(w-omega^(b+2))dw^2/(1-w^3)^2. (5)
000064 | The direct exact rational calculation verifies(5) for all six inputs.
000065 | Its actual lift obeys c*alpha_ab=omega^-1 alpha_(a-1,b-1), including
000066 | the differential factor. No naked coefficient root is substituted.
000067 | 
000068 | The old G calibration at z=+1 from the original interior branch gives
000069 |   h=(h0,h1,h2)=(-omega,-omega,+omega).
000070 | On q_s the h_a value is h_(a-s); on p_t the h_b value is h_(b-t),
000071 | by the same normalized R map. For output r, t=r+1 mod3. Thus the fixed
000072 | endpoint square root in(5), written zeta_ab(w)dw/(1-w^3), starts at
000073 |   zeta_ab(0)=-h_(a-s)h_(b-t)/omega^(a+b).                     (6)
000074 | Each factor of
000075 |   zeta_ab(w)=zeta_ab(0)
000076 |        sqrt(1+w/omega^(a+2))sqrt(1-w/omega^(b+2))
000077 | continues from1 along its actual ray. Principal-rooting the whole
000078 | squared expression would introduce erroneous sign jumps.
000079 | 
000080 | Only r=s or r=s+1 have overlapping q_s/p_(r+1) cable supports. The
000081 | Kummer eta sign is negative on q_s for s=a,a+1 and on p_t for t=b,b+1.
000082 | A product+ cancels the two eta arcs, whereas a product- adds them.
000083 | For r=s the ray angle is5pi/6+2pi s/3, with CCW z boundary running
000084 | t:0->infinity. Its finite coefficient is minus the ray integral.
000085 | For r=s+1 the ray angle is7pi/6+2pi s/3, with CCW direction
000086 | t:infinity->0. Its coefficient is plus the ray integral. The shared
000087 | normal/NS prefactor is4Xstar/pi, as in the established mu01 calculation.
000088 | 
000089 | For visibility, BEFORE the two-end change(4), the entire scalar table is
000090 |   mu01: alpha E02+beta E21,
000091 |   mu12: omega^2(alpha E10+beta E02),
000092 |   mu20: omega(alpha E21+beta E10),
000093 |   mu10: -i diag(alpha,beta,0),
000094 |   mu21: -i omega^2 diag(0,alpha,beta),
000095 |   mu02: -i omega diag(beta,0,alpha).                            (7)
000096 | All unspecified entries are exact zeros by the above support/eta rule.
000097 | Equations(4) and(7) give(1). The42 zeros are structural, not a numerical
000098 | threshold. Boundary intersections have no extra point mass in the
000099 | already given integrable L1 form reader.
000100 | 
000101 | 3. EXACT REVERSE PERIODS, NOT A NEW PHASE FIT
000102 | 
000103 | Keep the two previously proved periods
000104 |   C=2*3^(-3/4) exp(-5pi i/12),
000105 |   I02=C L, I21=-C[L+pi(sqrt2-1)], L>0.
000106 | Their exact atan2/continued-primitive proof and values are retained in
000107 | HARDY-BMU-FULL-MU01-READER.txt:69-102. In particular
000108 |   I21/I02=-3.347597664025180155360831848819267811936281725626... .
000109 | 
000110 | For the reverse mu10 rays use the actual integral substitution
000111 |   w_new=omega^2 conjugate(w_old).
000112 | The endpoint signs(6) give
000113 |   zeta_new=-omega^2 conjugate(zeta_old),
000114 | so the differential and denominator give the reverse ray integral
000115 |   Jnew=-omega conjugate(Iold).
000116 | These diagonal entries have the opposite CCW ray orientation in the
000117 | finite form, so their coefficients are omega conjugate(Iold).
000118 | Since omega conjugate(C)/C=-i exactly and L is real, the two entries
000119 | are precisely -i I02 and -i I21. This proves the reverse half of(7).
000120 | The substitution evaluates the fixed integrals; it does not declare a
000121 | new anti-reflection symmetry or a new state for the Hardy source.
000122 | 
000123 | 4. WHAT WHOLE COEFFICIENT PRODUCTS NOW DO, AND WHAT IS NOT CLAIMED
000124 | 
000125 | Both alpha and beta are nonzero. In ordinary coefficient matrices,
000126 |   F^2/(alpha beta)=E01, -N^2/(alpha beta)=E10,
000127 |   E01 E10=E00, E10 E01=E11, E22=I-E00-E11,
000128 |   E00 F E22/alpha=E02, E22 F E11/beta=E21,
000129 |   i E11 N E22/alpha=E12, i E22 N E00/beta=E20.                  (8)
000130 | Consequently the unital coefficient algebra generated by the two
000131 | ACTUAL full responses is all M3(C); its common commutant is scalar.
000132 | This is an exact small algebra calculation, with no parameter scan.
000133 | It rules out a common untouched family coefficient ray of this alphabet.
000134 | It is not a proof that physical insertions multiply as these matrices:
000135 | second quantization is not multiplicative, and an original whole source
000136 | consumer still has to specify how such response words are used.
000137 | 
000138 | For the original real bilinear/Hodge reader, define
000139 |   h(skew_B M)=(M12-M21,M20-M02,M01-M10)/2.
000140 | Then h(F)=(-beta,-alpha,0)/2 and h(N)=(-i alpha,-i beta,0)/2.
000141 | They are independent since alpha^2 != beta^2. The already established
000142 | weak WEDGE/Clifford/KO consumer applies linearly to both; no missing-X
000143 | claim is revived. Its symmetric family coefficients S=M+M^T have
000144 | singular values(r,r,0), r=sqrt(|alpha|^2+|beta|^2), for each of the
000145 | six phase-related inputs. The existing complete odd96 block has rank64.
000146 | This remains a calculated contribution, not a whole mass hierarchy.
000147 | 
000148 | The holomorphic normal symbol gives P_gamma,+; its ordinary adjoint
000149 | and actual J_F-conjugate are fixed, respectively M* P_gamma,+ and
000150 | bar(M) P_gamma,-. Both graph faces and eta arcs were already included.
000151 | No additional C6 branch is physically chosen to vanish here.
000152 | 
000153 | 5. SOURCE ANCHORS AND REPLAY
000154 | 
000155 | All report anchors below are under this same aps_source_review folder:
000156 |   ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt:38-42,81-84,118-138;
000157 |   FUCHS-FRAME-INDEPENDENT-TRACE-REVIEW.txt:37-45,50-88;
000158 |   CANONICAL-BRANCHED-SPIN-PULLBACK.txt:23-49;
000159 |   GEOMETRIC-R-SPIN-COLLAR-REVIEW.txt:16-29,114-132;
000160 |   INDUCED-HARDY-DEFECT-BMU-FINITE-READER.txt:18-40,45-69,87-102;
000161 |   HARDY-BMU-FIRST-NONZERO-ENTRY.txt:21-88;
000162 |   HARDY-BMU-FULL-MU01-READER.txt (unchanged, complete nine-entry proof);
000163 |   ORIGINAL-PRIMITIVE-WEDGE-CONSUMER-HARDY-REVIEW.txt;
000164 |   HARDY-BMU-WEAK-CLIFFORD-ODD-LIFT-REVIEW.txt;
000165 |   HARDY-BMU-KO-AND-FOUR-SECTOR-COMPLETION.txt.
000166 | 
000167 | Durable replay in the workspace outputs/multi-model-source-search-20261029:
000168 |   check_all_six_hardy_reader.py,
000169 |   HARDY-BMU-ALL-SIX-INPUT-READER.json.
000170 | The replay records all54 signed entries, the12 separately integrated
000171 | periods, both endpoint changes, exact squared differentials and(8).
000172 | It does not call or modify an original suite, ledger or Atlas.
000173 | Root independently confirmed the G_a and J endpoint maps, both actual
000174 | reverse periods, all matrix units and the scalar common commutant.