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Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/RELATIONAL-CHANNEL.txt
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000001 | RELATIONAL CHANNEL ON CONJUGATE(H5) TENSOR H5
000002 | ==============================================
000003 | 
000004 | Scope and additional contract
000005 | -----------------------------
000006 | 
000007 | The doily quartic theorem supplies the original native quartic representation
000008 | H5, its exact positive form, its six ovoid effects, and all sixty marked G31
000009 | actions.  This file adds one explicit kinematic contract:
000010 | 
000011 |     the state space contains conjugate(H5) tensor H5,
000012 | 
000013 | with diagonal action conjugate(rho_5(g)) tensor rho_5(g).
000014 | 
000015 | The presence or dynamical population of this tensor sector is not derived from
000016 | the compiler.  No coupling strength, transition rate, P2 carrier, or D5 carrier
000017 | identification is asserted.  Everything below follows exactly once this
000018 | additional sector contract is accepted.
000019 | 
000020 | Isometric use of the existing bridge
000021 | ------------------------------------
000022 | 
000023 | In the centered ovoid basis the metric is H=I5+J5.  In the native unnormalised
000024 | quartic word basis the primitive positive form is
000025 | 
000026 |     G = diag(1,3,3,3,6).
000027 | 
000028 | For the existing primitive bridge T, the checker verifies
000029 | 
000030 |     T^T (G/12) T = H.
000031 | 
000032 | Thus T is an exact isometry after the displayed fixed normalization.  Changing
000033 | the common scale of T and G does not change any ray, projector, probability,
000034 | or channel below.
000035 | 
000036 | Invariant vector and singlet Choi construction
000037 | -----------------------------------------------
000038 | 
000039 | Let e_1,...,e_5 be a G orthonormal basis and define
000040 | 
000041 |     |Omega> = (1/sqrt(5)) sum_i conjugate(e_i) tensor e_i.
000042 | 
000043 | In the native word basis the same ray is vec(G^-1).  The checker solves the
000044 | full simultaneous fixed space for all sixty vector actions and obtains
000045 | dimension one.  Therefore |Omega> is the unique invariant vector ray in the
000046 | contracted relational sector.  This vector statement does not make its density
000047 | the unique invariant density operator.
000048 | 
000049 | Its projector has rank one, trace one, and both reduced density matrices are
000050 | 
000051 |     rho_left = rho_right = I5/5.
000052 | 
000053 | If the state is |Omega><Omega|, the unnormalised operator
000054 | 5 |Omega><Omega| has input partial trace I5, is positive, and has rank one.
000055 | Under the canonical conjugate input identification it is exactly the Choi
000056 | operator of the identity channel on the native H5.  This is an operational
000057 | construction inside the declared relational sector.  It does not identify H5
000058 | with another five dimensional carrier.
000059 | 
000060 | Six ovoid effects and their exact joint law
000061 | -------------------------------------------
000062 | 
000063 | The six primitive native ovoid rays q_a obey
000064 | 
000065 |     q_a^T G q_a = 10,
000066 |     q_a^T G q_b = -2 for a != b.
000067 | 
000068 | The effects
000069 | 
000070 |     E_a = q_a q_a^T G / 12
000071 | 
000072 | are positive rank one operators for G and sum to I5.  All sixty native quartic
000073 | actions permute them exactly.
000074 | 
000075 | Conditional on the state |Omega><Omega|, measuring the conjugate effect on the
000076 | first half and the native effect on the second half gives
000077 | 
000078 |     p(a,b) = <Omega| conjugate(E_a) tensor E_b |Omega>
000079 |            = Tr(E_a^dagger E_b)/5.
000080 | 
000081 | The complete exact table has
000082 | 
000083 |     p(a,a) = 5/36,
000084 |     p(a,b) = 1/180 for a != b,
000085 |     p(a) = p(b) = 1/6.
000086 | 
000087 | Consequently
000088 | 
000089 |     p(second outcome = a | first outcome = a) = 5/6.
000090 | 
000091 | The joint table sums to one.  It is invariant under all sixty marked actions.
000092 | 
000093 | Conjugation convention
000094 | ----------------------
000095 | 
000096 | The first tensor factor is the conjugate representation.  The identity used
000097 | throughout is
000098 | 
000099 |     <Omega| conjugate(A) tensor B |Omega> = Tr(A^dagger B)/5.
000100 | 
000101 | There is no transpose substitution.  The checker contains a complex control
000102 | with A=i E_00 and B=E_00.  The direct tensor expectation and adjoint formula
000103 | both give -i/5, while Tr(A^T B)/5 gives +i/5.  The wrong convention therefore
000104 | fails explicitly.
000105 | 
000106 | Invariant relational operator and exact spectrum
000107 | --------------------------------------------------
000108 | 
000109 | Besides the singlet projector, the effects give
000110 | 
000111 |     R = sum_a conjugate(E_a) tensor E_a.
000112 | 
000113 | R is Hermitian in the real isometric chart, nonzero, non-scalar, and invariant
000114 | under all sixty actions.  The checker proves the annihilating polynomial
000115 | 
000116 |     R (R - (5/6) I25) (R - (2/3) I25) = 0
000117 | 
000118 | and the three exact nullities.  Hence
000119 | 
000120 |     spec(R) = (5/6)^1, (2/3)^5, 0^19.
000121 | 
000122 | The one dimensional largest eigenspace is precisely the Omega ray.  Therefore
000123 | Rayleigh Ritz gives a sharp conditional selection statement: if one adds the
000124 | rule that the realized normalized state maximizes Tr(rho R), the unique
000125 | maximizer is |Omega><Omega| and the maximum is 5/6.  Choosing that variational
000126 | rule, or choosing the Hamiltonian -R and a ground-state rule, is an additional
000127 | state postulate.  Neither follows from G31 covariance or from the POVM.
000128 | 
000129 | Symmetry alone gives a direct counterexample.  The density I25/25 is invariant
000130 | under all sixty actions and has the same two local marginals I5/5.  For that
000131 | state
000132 | 
000133 |     Tr((I25/25) R) = 1/6,
000134 |     p(a,b) = 1/36 for every a,b.
000135 | 
000136 | Thus invariant densities need not have the Omega correlations.  The correlated
000137 | law 5/36 on the diagonal and 1/180 off the diagonal belongs specifically to
000138 | the Omega state.
000139 | 
000140 | Normalized Lueders channel
000141 | --------------------------
000142 | 
000143 | Each existing effect E_a has rank one and sole nonzero eigenvalue 5/6.  Its
000144 | positive square root is therefore
000145 | 
000146 |     K_a = sqrt(E_a) = sqrt(6/5) E_a.
000147 | 
000148 | The checker verifies
000149 | 
000150 |     sum_a K_a^dagger K_a = I5,
000151 |     sum_a K_a K_a^dagger = I5.
000152 | 
000153 | Consequently the square-root instrument gives a trace-preserving, unital
000154 | channel.  With column vectorization in the real isometric chart its Liouville
000155 | matrix is
000156 | 
000157 |     L = sum_a conjugate(K_a) tensor K_a = (6/5) R,
000158 | 
000159 | so its exact spectrum is
000160 | 
000161 |     spec(L) = 1^1, (4/5)^5, 0^19.
000162 | 
000163 | The fixed operator space is one dimensional and contains I5.  Hence the unique
000164 | normalized stationary density is I5/5.  Here stationary means fixed by this
000165 | specified channel.  The POVM determines the channel only after adopting the
000166 | Lueders square-root instrument; it does not supply a physical time evolution,
000167 | a coupling strength, or a rate.
000168 | 
000169 | Liouville and Choi index roles
000170 | ------------------------------
000171 | 
000172 | R is an operator on conjugate(H5) tensor H5.  After multiplication by 6/5 its
000173 | indices give the Liouville matrix L of the Lueders channel.  The Choi matrix is
000174 | constructed separately as
000175 | 
000176 |     J_L = sum_a vec(K_a) vec(K_a)^dagger,
000177 |     vec(K)[input*5 + output] = K[output,input].
000178 | 
000179 | The checker verifies all 625 entries of the reshuffle
000180 | 
000181 |     J_L[(i,a),(j,b)] = L[(b,a),(j,i)].
000182 | 
000183 | Thus R must not be called a Choi matrix merely from its tensor notation.  A
000184 | special algebraic coincidence occurs here: every K_a is a real symmetric rank
000185 | one outer product, so the reshuffle fixes the summed array and J_L and L have
000186 | the same 25 by 25 entries.  Their index meanings remain different.  The Choi
000187 | construction independently has rank six, trace five, is positive, and has both
000188 | partial traces I5.
000189 | 
000190 | What is and is not selected
000191 | ---------------------------
000192 | 
000193 | Within the additional tensor-sector contract, symmetry fixes exactly one
000194 | vector ray.  It does not choose that ray as the state because invariant mixed
000195 | densities also exist.  The identity-channel Choi construction and the displayed
000196 | correlated law follow after specifying the Omega state.  Maximizing the already
000197 | given R selects Omega uniquely, but the maximizing rule remains extra.  The
000198 | Lueders channel follows after specifying the square-root instrument.  None of
000199 | these statements selects a local pure state or one preferred ovoid, and none
000200 | supplies physical dynamics, preparation probability, or interaction strength.