TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/old_source_embedding/REVIEW-CLAUDE-P2.txt
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000004 | 
000005 | VERDICT
000006 | The positive grading is NOT merely dimension 16. With the real unimodular
000007 | SM gauge lift of the actual v252 bimodule it is explicitly equivalent, per
000008 | generation, to the ORIGINAL P2 exterior-even module under the whole group
000009 | S(U(3)xU(2)), with the correct hypercharges. An explicit unitary wedge
000010 | dictionary is given below. The raw left algebra representation alone is
000011 | not this physical gauge action. Nor does v252 implement a complete original
000012 | Spin(10) action or derive its status as a physical source symmetry.
000013 | 
000014 | A strongest conditional scalar selector also exists: SM invariance plus an
000015 | EXPLICITLY GIVEN cyclic family permutation selects nu_R^c tensor (1,1,1)
000016 | up to phase. This is not generally preserved by the actual Dirac coefficient.
000017 | Its exact preservation condition is MR s=0 AND conjugate(Ynu) s=0.
000018 | 
000019 | 1. Actual original inputs and precise code sources
000020 | 
000021 | verification/v252_full_finite_triple.py:
000022 |   63-83: three generations, explicit nu/e/u/d, L/R, color labels;
000023 |   86-99: S_LR=Y_name and S_RL=Y_name^*;
000024 |   102-116: D_F=[[S,T],[T^*,conjugate(S)]], T_nuR,nuR=MR;
000025 |   119-121: particle L has gamma_F=+1, particle R=-1;
000026 |            antiparticle signs are opposite;
000027 |   124-132: actual J_F=Kswap times complex conjugation;
000028 |   139-168: actual pi(lambda,q,m) on both blocks;
000029 |   258-259: opposite-action matrix J_F b J_F^-1=Kswap conjugate(b) Kswap.
000030 | The functions grading, rep, Kswap, Japply, dirac_block and full_D were
000031 | discovered/read through the code graph before reading their exact file lines.
000032 | 
000033 | The ORIGINAL P2 comparison is not a guessed charge table:
000034 | documents/archive--paper-latex--tfpt-v29/sections/
000035 | 001-section-overview-and-claim-map.tex:
000036 |   166-174: Y=diag(-1/3,-1/3,-1/3,+1/2,+1/2);
000037 |   187-197: S(U(3)xU(2)) and its Z6 global SM form;
000038 |   236-282: S+=Lambda_even(E3+E2), including the six actual SM multiplets.
000039 | The same primitive integer normalization X=6Y is explicitly retained in
000040 | documents/archive--paper-latex--tfpt-42/sections/
000041 | 027-subsection-primitive-completion-from-the-boundary-kernel.tex:156-175.
000042 | verification/v245_spectral_matter_content.py:44-52 and
000043 | verification/v248_ps_algebra.py:42-56 record the corresponding Y and
000044 | Y=T3R+(B-L)/2 tables. They do not replace the matrix check below.
000045 | 
000046 | 2. Actual real gauge action, including its essential opposite action
000047 | 
000048 | Use the real bimodule gauge lift
000049 |   U(u)=pi(u) J_F pi(u) J_F^-1,
000050 | with u=(lambda,q,m), |lambda|=1, q in SU(2), m in U(3).
000051 | This formula specifies the gauge action being compared. v252 supplies pi
000052 | and J but does not itself contain a separately implemented gauge_lift or Y
000053 | function; its numerical axiom checks alone are not this identification.
000054 | The following is a direct calculation with those supplied matrices.
000055 | 
000056 | For one generation det(pi(u))=(lambda det(m))^4; for three it is the twelfth
000057 | power. Thus the connected unimodular branch is lambda det(m)=1. No charge
000058 | fit is used: normalize its U(1) using the original P2 Y above and put
000059 |   lambda=exp(i t/2), m=exp(-i t/6) conjugate(U3), q in SU(2).
000060 | The physical color matrix U3 is the conjugate of the raw determinant-one
000061 | part of m because the particle color action comes from J_F pi(u) J_F^-1.
000062 | 
000063 | Multiplying the actual blocks gives
000064 |   particle Q_L: exp(+i t/6) (U3 tensor q),
000065 |   particle L_L: exp(-i t/2) q,
000066 |   antiparticle u_R^c: exp(-2i t/3) conjugate(U3),
000067 |   antiparticle d_R^c: exp(+i t/3) conjugate(U3),
000068 |   antiparticle e_R^c: exp(+i t),
000069 |   antiparticle nu_R^c: 1.
000070 | These are exactly the gamma_F=+1 slots. The dimensions are 6,2,3,3,1,1,
000071 | and the complete hypercharge multiset is
000072 |   (1/6)^6, (-1/2)^2, (-2/3)^3, (1/3)^3, (1)^1, (0)^1.
000073 | This verifies SU(3), SU(2) and hypercharge together, not just the multiset.
000074 | For comparison, raw pi(u) alone acts trivially on the lepton doublet when
000075 | q=I; its missing factor conjugate(lambda) comes precisely from the opposite
000076 | action. Using only that left representation would fail the Y=-1/2 test.
000077 | 
000078 | Equivalently this can be stated globally without a covering-parameter
000079 | ambiguity. Take (x,z) in S(U(3)xU(2)). Choose lambda^2=det(z), and set
000080 |   q=lambda^-1 z, m=lambda^-1 conjugate(x).
000081 | Then q is in SU(2), lambda det(m)=1, and changing lambda to -lambda changes
000082 | pi(u) by -1 on the full module, so U(u) is unchanged. The gauge action on
000083 | the six gamma-positive pieces is respectively
000084 |   x tensor z; det(x) z; det(x) conjugate(x);
000085 |   conjugate(x); det(z); 1.
000086 | These are the exterior-power actions in the next section. This establishes
000087 | the global SM-group dictionary as well as its Lie algebra charges.
000088 | 
000089 | 3. Explicit unitary exterior dictionary, not a dimension match
000090 | 
000091 | Choose orthonormal color basis c1,c2,c3 of E3 and weak basis w1,w2 of E2.
000092 | Let vol_c=c1 wedge c2 wedge c3, vol_w=w1 wedge w2. For color c define
000093 |   eta_c=(1/2) sum_jk epsilon_cjk c_j wedge c_k.
000094 | Each eta_c has norm one; eta_2=c3 wedge c1 includes its actual sign.
000095 | At any one fixed generation define W on the original v252 slots:
000096 |   nu_R^c                 -> 1;
000097 |   (u_L,c,d_L,c)          -> (c_c wedge w1,c_c wedge w2);
000098 |   u_R,c^c                -> eta_c;
000099 |   e_R^c                  -> vol_w;
000100 |   d_R,c^c                -> eta_c wedge vol_w;
000101 |   (nu_L,e_L)             -> (vol_c wedge w1,vol_c wedge w2).
000102 | 
000103 | The images are the sixteen distinct orthonormal even wedge basis vectors,
000104 | with their fixed signs. W is therefore unitary. The transformation laws
000105 | in section2 prove
000106 |   W U(u)|_(gamma=+) W^-1 = Lambda_even(x directsum z).
000107 | In particular the particle color action, antiparticle conjugate color,
000108 | weak doublet action and all hypercharges really intertwine.
000109 | 
000110 | For all three generations this is W tensor I_family. Since J_F flips the
000111 | grading and commutes with the real gauge lift, gamma_F=-1 is the conjugate
000112 | SM module. Thus the two internal grades can genuinely be read as the
000113 | P2 16 and conjugate 16 at this representation level. This does not by itself
000114 | identify internal chirality with physical spacetime chirality or field count.
000115 | 
000116 | The exterior construction does carry its usual full Spin(10) half-spinor
000117 | action. W can TRANSPORT that action onto the positive v252 space. However
000118 | the v252 builder implements the SM left/opposite action and a restricted
000119 | PS-type left action, not explicit matrices for all Spin(10) generators.
000120 | SM equivalence alone does not prove that the transported extra generators