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000001 | ORIGINAL INSERTIONS, MIXED CURRENTS AND THE PRODUCT TIME TEST
000002 | 29 October 2026. Continuation of tfpt-unified-source-derivation-20261029.
000003 | Verdict: ESTABLISHED_SCOPED; ORIGINAL_PHYSICAL_P1_IMAGE_UNPROVED.
000004 | 
000005 | 1. Target, reused sources and precise changed premise
000006 | 
000007 | The target is the actual original source image, including its time and state.
000008 | DERIVATION.txt in the parent contract already proves a sufficient all-word
000009 | recognition theorem. We do not repeat it or assume its hypotheses in P1.
000010 | The old full-vector cocycle exclusion is retained in its actual scope:
000011 | collar-current-source-bridge-20261021/DIRAC-LIFT-ADMISSIBILITY.md, sections1-3.
000012 | It explicitly leaves mixed vector/axial currents outside that exclusion.
000013 | Here we use exactly that different generator inside the already supplied
000014 | full complex CAR source, then test the existing finite product coefficient.
000015 | No new independent chiral determinant, free state or fitted coupling is added.
000016 | This conditional CAR source itself was not derived from thin P1.
000017 | 
000018 | Original finite operators are v252_full_finite_triple.py, lines86-168;
000019 | v254_inner_fluctuations.py, lines47-60 and85-99. The insertion audit
000020 | seam-collar-multifield-20261016/P1-INSERTION-TANGENT.txt already separates
000021 | primitive action variation, finite one-forms and the actual source jet.
000022 | The explicit qM witness and full left/right statement are already in
000023 | seam-joint-source-origin-audit-20261016/HIGGS_VARIATION_PROOF.txt:135-182,
000024 | finding:finite-sm-higgs-variation-retains-seed. We reuse that result;
000025 | it is not counted as a newly derived P1 source.
000026 | We also retain the previously proved positive-compression carrier exclusion;
000027 | it is neither a new result nor a no-go for the entire original construction.
000028 | 
000029 | 2. Follow an actual finite one-form before naming it a current
000030 | 
000031 | On a particle doublet, put a_F=pi(0,q,0), b_F=pi(1,0,0).
000032 | In left/right internal order the actual representation and Dirac block give
000033 |   a_F=[[q,0],[0,0]], b_F=[[0,0],[0,I]],
000034 |   D_F=[[0,M],[M*,0]], M=diag(Y_up,Y_down).
000035 | Consequently a_F b_F=0 and
000036 |   a_F[D_F,b_F]=[[0,qM],[0,0]].
000037 | Its selfadjoint completion is V_q=[[0,qM],[M*q*,0]].
000038 | The analogous lepton block uses Y_nu,Y_e. The actual full representation
000039 | also has antiparticles and Majorana terms; the checker uses its full96 block.
000040 | Its selected one-form has only the particle upper-right part, as above.
000041 | The Yukawa and Majorana matrices remain supplied inputs of that constructor.
000042 | 
000043 | For D=D_M tensorI+gamma5 tensorD_F, a=f tensora_F, b=g tensorb_F,
000044 |   a[D,b]=f[D_M,g] tensora_F b_F
000045 |           +fg gamma5 tensora_F[D_F,b_F].
000046 | For this particular witness the first term is ZERO. Therefore this insertion
000047 | is an actual internal Higgs/Yukawa direction, not already a conserved current.
000048 | Central coordinate one-forms instead recover Clifford multiplication c(dg).
000049 | Neither one-particle insertion is automatically a many-body P1 field image.
000050 | 
000051 | 3. An explicit right current in the already given complete CAR source
000052 | 
000053 | Retain the original two-component product collar, its two boundary faces,
000054 | and alpha=1/4 or3/4. These components are not two particles or two faces.
000055 |   A=-i partial_theta+alpha, Gamma=diag(1,-1),
000056 |   Jn=[[0,-1],[1,0]], S=Jn Gamma=[[0,1],[1,0]],
000057 |   B0=Gamma A, D0=Jn(partial_r+B0).
000058 | The old pure graph on BOTH faces is
000059 |   G(B)=[[a,c],[c,1-a]],
000060 |   a=(1+exp(-2ell B))^-1, c=(2cosh(ell B))^-1.
000061 | On its second face n(B0)=diag(n(A),1-n(A)),
000062 |   n(A)=(1+exp(2ell A))^-1.
000063 | This is a thermal face of the existing pure joint state, not a newly selected
000064 | Hardy vacuum. Normal ordering and charged CAR remain the old source premise.
000065 | 
000066 | Use smooth real functions on that SAME face and even quadratic currents
000067 |   J_V(f)=:Psi* fI Psi:, J_A(f)=:Psi* fGamma Psi:,
000068 |   J_R(f)=(J_V(f)+J_A(f))/2=:Psi* fP_R Psi:,
000069 |   J_L(f)=(J_V(f)-J_A(f))/2, P_R=(I+Gamma)/2.
000070 | These are existing smeared quadratic generators, under the old implementability
000071 | conditions. Restricting to their algebra does not replace the full determinant.
000072 | The old infinite telescoping identity gives opposite component cocycles c,-c.
000073 | For component coefficients x=(x_R,x_L),y=(y_R,y_L), bilinearity gives
000074 |   c(x,y)=(x_R y_R-x_L y_L)c.
000075 | In particular c_VV=c_AA=0, c_VA=c_AV=2c, c_RR=c.
000076 | For the original f=sin(4theta),g=1-cos(4theta), c=-2i;
000077 | the mixed value is -4i and the right value -2i. Cross entries use different
000078 | profiles; there is no conflict with antisymmetry under exchanging BOTH inputs.
000079 | The normal-pairing exclusion does not apply to Gamma: [Gamma,Jn]!=0.
000080 | This is standard vector/axial current algebra applied to the retained source,
000081 | not a new general anomaly theorem or a four-dimensional particle identification.
000082 | 
000083 | On trigonometric H1 vectors let X_n e_k=e_(k-n). Then [A,X_n]=-n X_n and
000084 |   [B0,IX_n]=-n Gamma X_n,
000085 |   [B0,Gamma X_n]=-n IX_n,
000086 |   [B0,P_R X_n]=-n P_R X_n.
000087 | These are exact one-particle identities and determine the corresponding CAR
000088 | automorphism action. With the original eightfold modes replace n by8n.
000089 | Calling this actual P1 OS time would still require the original time dictionary.
000090 | Its thermal face does NOT imply a positive vacuum generator H Omega=0.
000091 | The parent recognition theorem must use the actual KMS alternative here,
000092 | retaining the joint zero-mode law, delta and original support dictionary.
000093 | 
000094 | 4. Keep the existing finite product term in that time test
000095 | 
000096 | In this explicit two-dimensional collar take a constant finite selfadjoint Phi.
000097 | The even product expression and its actual tangential factor are
000098 |   D_Phi=D0+Gamma tensorPhi=Jn(partial_r+B_Phi),
000099 |   B_Phi=Gamma A-S tensorPhi.
000100 | This is a bounded perturbation of B0 on its original H1 domain. A acts
000101 | trivially on the finite space; no varying normal profile is presumed.
000102 |   [S,P_R]=Jn,
000103 |   [B_Phi,P_R X_n]+nP_R X_n=-Jn tensorPhi X_n.
000104 | Thus for nonzero Phi the bare geometrically right current algebra exists
000105 | at equal time but is not invariant under this same source time evolution.
000106 | This already follows from the bounded n=0 projection commutator; no
000107 | differentiation of a putative massive Fock Hamiltonian on a massless vacuum
000108 | or comparison of a finite cutoff CCR is being used.
000109 | 
000110 | For an internal orthoprojection E, the pure-right channel obeys
000111 |   [B_Phi,P_R tensorE]=[[0,E Phi],[-Phi E,0]].
000112 | It reduces B_Phi iff Phi E=0. Selfadjointness gives E Phi=0 as well.
000113 | This proves a necessary and sufficient kernel criterion in THIS product class.
000114 | It does not show whether P1 selects any such channel.
000115 | 
000116 | For a general even-dimensional product the correctly typed formula is
000117 |   D_M=Jn(partial_r+B_t),
000118 |   B_prod=B_t tensorI-Jn gamma5 tensorD_F.
000119 | Here {Jn,gamma5}=0, [B_t,gamma5]=0, {B_t,Jn}=0.
000120 | Jn gamma5 is a selfadjoint involution anticommuting with B_t and gamma5.
000121 | The displayed two-component Gamma is the chirality of the chosen old collar.
000122 | We do not identify that fiber with a complete four-dimensional spinor module.
000123 | 
000124 | 5. Classification of ALL constant time-reducing projections in this class
000125 | 
000126 | Let K be any finite complex space and Phi=Phi*. A constant orthoprojection E
000127 | on C2 tensorK reduces B_Phi(p)=p Gamma-S tensorPhi for all Fourier momenta p
000128 | iff it commutes separately with Gamma and S tensorPhi. Any two distinct
000129 | momenta suffice for this coefficient comparison. Thus exactly
000130 |   E=diag(E_R,E_L), E_R Phi=Phi E_L, E_L Phi=Phi E_R.
000131 | The diagonal entries are orthoprojections. The equations imply commutation
000132 | with Phi^2 and so a splitting by its squared-mass eigenspaces.
000133 | For K_mu=ker(Phi^2-mu^2), mu>0, set U_mu=Phi/mu on K_mu.
000134 | U_mu is a selfadjoint unitary. Every solution there is
000135 |   E_R=Q_mu, E_L=U_mu Q_mu U_mu
000136 | with an arbitrary orthoprojection Q_mu. On K0=kerPhi the two projections
000137 | E_R^0,E_L^0 are independent. Conversely every such choice satisfies all the
000138 | equations. This proves completeness, including repeated and signed masses.
000139 | The linear commutant has complex dimension sum_(mu>0)dim(K_mu)^2+2dim(K0)^2.
000140 | In particular the massive right and left ranks are equal.
000141 | 
000142 | The corresponding scalar current cocycle is zero on the massive part.
000143 | Proof uses the existing full-graph Lagrange lemma rather than a regulator.
000144 | On K0-perp put F=signPhi and N'=Jn tensorF. Then N' is unitary,
000145 |   N' B_Phi N'*=-B_Phi, [E,N']=0.
000146 | For the full two-face graph N_face=diag(-N',N') satisfies
000147 |   N_face G(B_Phi) N_face*=1-G(B_Phi).
000148 | A one-face generator fE commutes with N_face. If the off-diagonal blocks
000149 | are Hilbert Schmidt, the old trace-class cocycle lemma gives c(fE,gE)=0.
000150 | This is its usual circle smoothness class, not an unrestricted higher
000151 | dimensional trace formula. N' is an algebraic pairing, not an OS reflection.
000152 | On K0 the original independent massless right and left components remain.
000153 | Consequently the full channel's central coefficient, in units of c, is
000154 |   kappa(E)=rank(E_R^0)-rank(E_L^0).
000155 | An invertible Phi therefore leaves no nonzero scalar chiral level in ANY
000156 | constant reducing channel in this class. Momentum-dependent nonlocal
000157 | projections, variable profiles and actual normal index modes are not classified.
000158 | 
000159 | 6. Retain the product grading and the true reflection question
000160 | 
000161 | For the existing finite triple {Phi,gamma_F}=0. Therefore
000162 |   [B_Phi,Gamma tensorgamma_F]=0.
000163 | The positive TOTAL graded sector is a genuine reducing channel. It contains
000164 | R tensorker(gamma_F-I) AND L tensorker(gamma_F+I).
000165 | Its central coefficient is tr_(kerPhi)(gamma_F); the massive part is balanced.
000166 | The complete actual v252 complex96 space has48 positive and48 negative
000167 | internal grades. Since Phi pairs the opposite internal grades on every
000168 | nonzero squared-mass eigenspace, tr_(kerPhi)(gamma_F)=tr_K(gamma_F)=0
000169 | also for singular Phi. Hence the full TOTAL graded channel has zero scalar
000170 | level for every such finite odd coefficient. An individually selected kernel
000171 | channel can still have a nonzero level, as section5 proves. This equality
000172 | is not a physical particle count or a Pfaffian statement.
000173 | Its invertible constructor control below has zero kernel;
000174 | it demonstrates the cancellation, not a derivation of physical Yukawa ranks.
000175 | For arbitrary invertible input Y_nu,Y_e,Y_u,Y_d, the actual full_D is invertible
000176 | even with Majorana MR: left neutrino equations first force right neutrinos
000177 | to zero, also in the antiparticle block; right equations then force the left
000178 | ones to zero. Charged blocks are directly invertible. This does not fix inputs.
000179 | 
000180 | There is also an explicit candidate for the geometric product reflection.
000181 | On the old collar r->-r with fiber S preserves D0. For constant Gamma Phi
000182 | the fiber S alone flips that term. If {gamma_F,Phi}=0, the combined fiber
000183 |   R_prod=S tensorgamma_F
000184 | preserves D_Phi together with r reversal. This follows by conjugating both
000185 | normal and tangential symbols and the potential. It is not proof of its
000186 | domain, reference, admissibility or OS positivity. In the original relative
000187 | split an R-odd term can itself be intended, so failure of the bare S symmetry
000188 | is not by itself inadmissibility of that term.
000189 | 
000190 | 7. Reuse the EXISTING normal-index result without a new operator choice
000191 | 
000192 | The normal index line was already computed in original-admissible-source-
000193 | closure-20261025/NORMAL-INDEX-TRANSPORT.txt, sections1-3. Its source-product
000194 | identification was explicitly conditional. It is not missing work to repeat:
000195 |   B_normal=B_a tensor gamma_N+I tensor D_N,
000196 |   ker D_N^+=C, ker D_N^-=0 for the stated spin twist by L2=O(1),
000197 |   B_normal V_chi=V_chi B_a on the whole domain.
000198 | The old horizontal holonomy test and moving-kernel Kato criterion remain
000199 | necessary in a genuine family. The present constant finite Phi test does not
000200 | replace the normal elliptic D_N by the supplied finite Yukawa matrix D_F.