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000001 | # Native charge conjugation is an all-time charged-response discriminator
000002 | 
000003 | 15 October 2026. `UR.NATIVE.CHARGED_CONJUGATION.01`. **PARTIAL** for the
000004 | physical common-source question. The following source-specific symmetry and
000005 | response statements are exact under the stated original Hamiltonian/state.
000006 | They do not select that Hamiltonian, a chemical marking, or a P1/P2 source.
000007 | 
000008 | ## 1. Start with the original full field net
000009 | 
000010 | [OMITTED: non-scientific conversation or execution metadata]
000011 | 
000012 |     H_charge = tensor_x l2(Z8),
000013 |     T_(x,p)|{n}>=epsilon(p,n_x)|...,n_x+p,...>,
000014 |     W_(xy,p)=T_(y,p) T_(x,p)^*,
000015 |     epsilon(p,n)=(-1)^beta(p,n),
000016 |     beta(p,n)=sum_i p_i n_i G_ii/2+sum_(i>j) p_i n_j G_ij mod2.
000017 | 
000018 | This uses the complete original even unimodular G, not a parity truncation.
000019 | All integer charges remain. The onsite operators and diagonal projectors
000020 | generate B(l2(E8)) at each site. The source has global U(1)^8 charge
000021 | Q_tot=sum_x n_x. Only the finite frame group is gauged; fixing the later
000022 | sector C0={sum_x n_x=0} does not turn all charged onsite fields into
000023 | endomorphisms of that sector.
000024 | 
000025 | For the later scalar-coupled source in
000026 | `native-source-reconstruction-20261014/PROOF.md`, retain its given finite
000027 | spatial lattice, scalar space, couplings and domain:
000028 | 
000029 |     H = K_scalar+V(n,phi)+H_hop,
000030 |     V = D(n)+phi^T[-Delta_a+m²I+2u diag(e(n_x))]phi/2,
000031 |     e(n)=n^T G n/2,
000032 |     D = sum_x e(n_x)/N+nu sum_edges e(n_x-n_y),
000033 |     H_hop=J sum_(edges,p)(2I-W_p-W_p^*).
000034 | 
000035 | The original coercive diagonal/scalar Hamiltonian, bounded hopping and
000036 | their common selfadjoint finite-volume realization are reused. The source
000037 | premises include a,m>0, u,nu,J>=0 and the specified eight arithmetic hopping
000038 | directions. They are not derived afresh here.
000039 | 
000040 | ## 2. The actual cocycle permits charge conjugation
000041 | 
000042 | Define the linear unitary, not antiunitary,
000043 | 
000044 |     C|{n}>=|{-n}>,
000045 | 
000046 | acting as identity on the scalar Hilbert space. C²=I. Bilinearity gives
000047 | epsilon(-p,-n)=epsilon(p,n) and epsilon(p,-n)=epsilon(-p,n), hence
000048 | 
000049 |     C T_p C=T_-p.
000050 | 
000051 | One sign must be retained. The original adjoint law is
000052 | 
000053 |     T_p^*=epsilon(p,-p) T_-p=(-1)^e(p) T_-p.
000054 | 
000055 | In the original arithmetic basis the eight e(e_i) are 2,1,1,1,1,1,1,1.
000056 | Thus T_-p and T_p^* differ by a minus sign for seven basis directions;
000057 | they cannot be identified literally. This sign cancels in the adjoint-paired
000058 | responses below. On a bond the two onsite signs cancel as well:
000059 | 
000060 |     C W_p C=W_-p=W_p^*.
000061 | 
000062 | Quadratic evenness gives D(-n)=D(n) and V(-n,phi)=V(n,phi). Scalars are
000063 | unchanged; every hopping term is individually C invariant. The exact
000064 | Hamiltonian therefore satisfies C H C=H. C preserves the diagonal/scalar
000065 | form domain and the original core, while the hopping is bounded; this is
000066 | an identity of the existing selfadjoint operator, not only a finite matrix
000067 | or formal Taylor coefficient. It implies C exp(-itH) C=exp(-itH) and the
000068 | same for bounded positive-time heat propagators.
000069 | 
000070 | Also C Q_tot C=-Q_tot, so C preserves C0. For any beta>0 for which the
000071 | original trace-class theorem applies, both the full Gibbs density and the
000072 | normalized sector density
000073 | 
000074 |     rho_beta=exp(-beta H)/Z,
000075 |     rho_beta,0=P0 exp(-beta H)P0/Z0
000076 | 
000077 | are C invariant. The second expression is a density on the full Hilbert
000078 | space supported in C0; the full generator is still needed for charged
000079 | intermediate propagations.
000080 | 
000081 | ## 3. Equality of complete adjoint-paired charged responses
000082 | 
000083 | For any C-invariant density rho, define the bounded Euclidean return response
000084 | 
000085 |     G_(rho,p)(t)=Tr[rho T_p^* exp(-tH) T_p], t>=0.
000086 | 
000087 | The original H is nonnegative. Unitary conjugation inside the trace gives
000088 | 
000089 |     G_(rho,p)(t)=G_(rho,-p)(t)
000090 | 
000091 | for every t, not just at t=0 or to one perturbative order. The same argument
000092 | applies to the real-time ordered response
000093 | `Tr[rho T_p^* exp(itH) T_p exp(-itH)]`. C is linear, so no complex
000094 | conjugation of the amplitude or reversal of t is introduced.
000095 | 
000096 | More generally any specified bounded multitime word has the same expectation
000097 | as its C-conjugate word. Thus the full source symmetry, rather than a
000098 | single matching moment, constrains every sign-reversed marked history.
000099 | For Euclidean KMS correlation functions the same result holds in their
000100 | standard thermal analyticity domain; no unbounded exp(+tH) is introduced
000101 | in the definition used above.
000102 | 
000103 | This result requires a C-invariant starting state. A non-C-invariant
000104 | prepared state, a C-breaking boundary condition/field, or another Hamiltonian
000105 | changes the premise. It is not a theorem that every possible native vacuum
000106 | or infinite-volume phase must be C invariant.
000107 | 
000108 | ## 4. Decisive comparison with one fixed chiral quarter twist
000109 | 
000110 | In the separately stipulated chiral E8 lattice-Fock representation, the
000111 | fixed primary vacuum and its charge-shift preparations yield
000112 | 
000113 |     C_delta(v,t)=exp[-t(|v|²/2+delta.v)],
000114 |     C_delta(v,t)/C_delta(-v,t)=exp[-2t delta.v].
000115 | 
000116 | Here delta_k=(k+alpha)/8-1/2, with alpha=1/4 or 3/4 in the original quarter
000117 | cases. For the *same marked* opposite-charge pair, the native response at
000118 | the invariant start is symmetric whereas this fixed-twist response is
000119 | asymmetric whenever delta.v is nonzero.
000120 | 
000121 | The checker uses the actual original native embedding basis B, not a root
000122 | count or a dimension match. Its eight columns span R8. Therefore a nonzero
000123 | delta cannot annihilate every B e_i. For example B e_0=2e_0 has delta pairing
000124 | -15/16 at alpha=1/4 and -13/16 at alpha=3/4. At least this charged pair
000125 | already discriminates the fixed-twist target from the symmetric native
000126 | response data.
000127 | 
000128 | Consequently an exact identification preserving **all** these marked
000129 | responses, their opposite-charge assignment, the specified starting states
000130 | and a common time law is excluded. This is not an exclusion of every
000131 | unmarked partial readout, state-changing preparation, different source,
000132 | or covariant family of twists. A symmetric mixture of +/-delta can remove
000133 | this particular asymmetry, but is not thereby a native source dictionary
000134 | or a physically selected state; no such mixture is inserted as a solution.
000135 | 
000136 | ## 5. C0 and uniform chemical marking: keep the charged corners
000137 | 
000138 | A uniform linear marking has the form eta.Q_tot, with eta in the *native
000139 | integer-coordinate dual*. It vanishes exactly on C0. Because [H,Q_tot]=0,
000140 | the formal comparison H_eta=H+eta.Q_tot restricts to the same H_0 on C0.
000141 | It also obeys C H_eta C=H_-eta. The coercive original charge energy controls
000142 | this linear marking, so its finite-volume selfadjoint comparison is
000143 | well defined. Eta is additional input here, not selected by P1/P2.
000144 | 
000145 | Balanced W transports remain within C0, commute with Q_tot and have the
000146 | same sector-internal dynamics under H_eta. They still move local charges
000147 | and have the original nontrivial cocycle and dynamics. C0 never implies
000148 | that local charge information or all neutral responses are absent.
000149 | 
000150 | By contrast T_(x,p) maps C0 to C_p. One cannot use it as an endomorphism
000151 | inside H_0 or silently delete the exterior propagation. With the full
000152 | charge-sector generator and a normalized C-invariant start rho_0 supported
000153 | in C0, exact sector conservation gives
000154 | 
000155 |     G_(rho_0,p)^eta(t)
000156 |       =Tr[rho_0 T_p^* exp(-tH_eta) T_p]
000157 |       =exp[-t eta.p] G_(rho_0,p)^0(t).
000158 | 
000159 | The undeformed pair is equal by section 3. Since the heat operator is
000160 | strictly positive on nonzero vectors, these normalized returns are positive,
000161 | and their ratio is exactly
000162 | 
000163 |     G_(rho_0,p)^eta(t)/G_(rho_0,-p)^eta(t)=exp[-2t eta.p].
000164 | 
000165 | Thus the uniform marking is invisible to the dynamics confined to C0 but
000166 | visible in onsite insertion/removal responses with charged intermediate
000167 | sectors. For a proposed geometric label map v=Bp, the coefficients would
000168 | have to satisfy eta=B^T delta. This is a conditional coordinate relation,
000169 | not a derivation or selection of eta. It matches only this odd charged
000170 | response factor, not the full even energy, oscillator response, state,
000171 | locality or interacting field algebra.
000172 | 
000173 | This charged-corner distinction is the positive reusable interface. An
000174 | automated search must distinguish sector-internal W data from full onsite
000175 | T insertion data and must not claim a chiral twist has been selected merely
000176 | because a new linear chemical coefficient can be stipulated.
000177 | 
000178 | ## 6. Exact check and scope