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Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/NEUTRAL-QUADRATIC-AND-MIXED-RESPONSE.md
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000001 | # Neutral quadratic source fields and their first mixed response
000002 | 
000003 | 2026-10-16. Continuation of UR.NATIVE.JOINT_OPERATOR.01, scoped/PARTIAL.
000004 | No physical gate, ledger, paper or load-bearing verification status changes.
000005 | The existing E8_1 vacuum, actual native code, and the already constructed
000006 | original neutral source fields are premises. No fitted coupling is added.
000007 | Use R_neut for the weight-two source field; R5 is the marked logical clock.
000008 | 
000009 | ## 1. The actual source candidate, not a new Hamiltonian
000010 | 
000011 | Original `neutral_source_response._native_neutral_branch` starts with the
000012 | 64 chiral source roots and omits one actual SU4 family weight, retaining48.
000013 | Its symmetrized normal products satisfy the original lattice identity
000014 | 
000015 |     B_alpha = (:E_alpha E_alpha^dagger: + :E_alpha^dagger E_alpha:)/2
000016 |             = (1/2) :(alpha.h)^2: .
000017 | 
000018 | The two ordered regular terms have opposite alpha(-2) contributions;
000019 | these cancel in the symmetric product. Thus
000020 | 
000021 |     B48 = 12 T_E8 + R_neut,
000022 |     R_neut = T_AR,  T_A=(1/2):h^T A h:,
000023 |     AR = 4 P_A2 - 8 P_U1,
000024 |     tr AR=0, tr AR^2=96, ||R_neut Omega||^2=48.
000025 | 
000026 | These coefficients are the ORIGINAL charge moments, not adjusted to the
000027 | logical clock. The source selection of a family omission is still an
000028 | input: the original reader accepts `omitted_family`. The existing required
000029 | map to a physically selected three-family space is not discharged here.
000030 | For each of the four choices the same trace/norm holds; their AR matrices
000031 | sum to zero. The full64 field is B64=16 T_E8. This identity is a control,
000032 | not an imposed physical averaging law. Its 5+2+1 eigenspaces live in the
000033 | eight-current source; they are not the logical five-code's 3+2 split.
000034 | 
000035 | ## 2. Classify all local neutral weight-two zero modes at once
000036 | 
000037 | In the GIVEN lattice VOA, the full lattice-charge-zero weight-two space is
000038 | 
000039 |     h(-2)Omega + Sym^2(h(-1))Omega, dimensions 8+36.
000040 | 
000041 | This is lattice charge neutrality, not an assumed Standard Model singlet
000042 | condition. For an energy-preserving zero mode use o(v)=v_(wt(v)-1):
000043 | 
000044 |     o(h(-2)Omega)=-h(0),
000045 |     o(T_A)=sum_(n>0) h(-n)^T A h(n) + (1/2)h(0)^T A h(0).
000046 | 
000047 | For a Hermitian field A is real symmetric in a real orthonormal Cartan
000048 | basis. On the neutral oscillator code h(0) vanishes. All code states use
000049 | four mode-one oscillators, so only n=1 acts and o(T_A)=dGamma(A).
000050 | These identities concern the local field zero mode. The earlier individual
000051 | mode roundtrip K48=sum J_(lambda,-1)^dagger J_(lambda,-1) is NOT o(B48).
000052 | 
000053 | Let Pi be the actual five-code projector, f_j its normalized basis, and
000054 | J the actual native real complex structure. For any real symmetric A set
000055 | 
000056 |     Ac=(A-JAJ)/2,   Aa=(A+JAJ)/2.
000057 | 
000058 | Then the ACTUAL code has compression and total leakage
000059 | 
000060 |     Pi dGamma(A) Pi = (tr A)/2 I5,
000061 | 
000062 |     L(A)=sum_j ||(I-Pi)dGamma(A)f_j||^2
000063 |         =4||Ac-(tr A)I8/8||_HS^2 + (5/2)||Aa||_HS^2.
000064 | 
000065 | Derivation: split the one-particle space H+ plus H-. Write B=A|H+ and
000066 | C=P_H- A|H+. Actual one-register erasure gives Pi dGamma(B)Pi=tr B I5.
000067 | The original two-register marginal is (I+Swap)/4; expanding the squared
000068 | four-register sum gives code leakage 8||B-tr B I4/4||^2. The term C creates
000069 | one H- particle and has orthogonal leakage 5||C||^2 by the original
000070 | one-body number matrix 5I4. Reality of A gives
000071 | 
000072 |     ||Ac-(tr A)I8/8||^2=2||B-tr B I4/4||^2,
000073 |     ||Aa||^2=2||C||^2.
000074 | 
000075 | The real leakage Gram relative to the MATRIX Hilbert-Schmidt metric has
000076 | eigenvalues 0 (multiplicity1), 4 (multiplicity15), 5/2 (multiplicity20).
000077 | Hence its kernel is exactly A=aI8. The complex sesquilinear Gram also has
000078 | an imaginary antisymmetric part; it need not vanish. The displayed norm
000079 | formula uses REAL coefficients. Both parts are retained in the certificate.
000080 | 
000081 | The checker constructs all five ORIGINAL support polynomials in the eight
000082 | real oscillator coordinates, computes their Fock coefficients exactly,
000083 | checks all36 logical matrix actions and the full leakage Gram, and
000084 | compares it with this independently derived formula. It does not infer
000085 | the result solely from a numerical matrix rank or substitute a new code.
000086 | 
000087 | ## 3. What the actual R_neut and B48 do on this code
000088 | 
000089 | For every common real orthogonal Cartan chart,
000090 | 
000091 |     Pi o(R_neut) Pi=0,
000092 |     240 <= L(AR) <= 384,
000093 |     Pi o(B48) Pi=48 I5,  L(A48)=L(AR),
000094 |     o(B64)|Code=64 I5,  L(A64)=0.
000095 | 
000096 | The inequalities are universal bounds, not a claimed sharp set of values
000097 | as source alignment varies. They use tr AR^2=96 and coefficients 5/2,4.
000098 | They therefore need no fitted Cartan identification. Concrete individual
000099 | matrix elements in different source charts still require the actual map.
000100 | 
000101 | In particular R_neut is NOT zero as a source field: its vacuum norm and
000102 | correlators remain as previously proved. Its first action on code leaks
000103 | out instead of giving a nontrivial INTERNAL logical Hamiltonian.
000104 | Pi here describes a mathematical restriction; repeated physical projection
000105 | or a Zeno/measurement rule is neither assumed nor derived.
000106 | 
000107 | There is also a global statement. Arbitrary time-dependent sequences
000108 | generated SOLELY by these Hermitian neutral weight-two zero modes preserve
000109 | each oscillator mode's occupation. On the initially mode-one-only code
000110 | their evolution is Sym^4(U8) for some one-particle unitary U8. If the final
000111 | operation preserves the code, it preserves its summed one-body matrix
000112 | 5 P_H+. Thus U8 preserves H+; its restriction is a code-preserving U4.
000113 | The complete original stabilizer theorem then bounds its logical action
000114 | by phase*rho5(G31), which excludes marked R5. Intermediate exit from H+
000115 | and code is allowed in this argument. No physical time selection follows.