TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/neutral_moebius_source_20261029/PROOF.txt
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000001 | ACTUAL NEUTRAL QUARTER SOURCE: CONSTRUCTIVE MOEBIUS STANDARD PAIR
000002 | 29 October 2026. Scoped mathematical result, not complete physical TFPT.
000003 | 
000004 | 1. Changed identifying step and outcome
000005 | 
000006 | The previously excluded step set the SAME gapped quarter-source energy equal
000007 | to a Borchers translation generator. That exclusion remains correct. Here
000008 | the parabolic generator is derived from the actual existing source currents,
000009 | with the SAME sea vacuum and the SAME neutral cyclic Hilbert space. It is
000010 | not the old energy. No new oscillator, source kernel or coupling is chosen.
000011 | 
000012 | Success criterion: actual positive translation with full scale relation,
000013 | proper local half-line algebras and their common standard vector; retain
000014 | which source/state/time have been used. Failure of any of these would stop
000015 | this route. Do not treat a finite Lie algebra as a continuum standard pair.
000016 | 
000017 | 2. Original identities reused, not newly postulated
000018 | 
000019 | Retain the one-copy quarter CAR source alpha=1/4 or3/4, negative sea Omega,
000020 | original currents J_n, relative charge Q=J0 and positive excitation energy
000021 |     H_alpha=N+Q^2/2+bQ,    b=alpha-1/2.
000022 | The original same-source relation on the finite-energy core is
000023 |     Lcal_n^alpha=L_n^0+bJ_n+(b^2/2)delta_(n,0),
000024 |     L_n^0=(1/2)sum_k :J_(n-k)J_k: .
000025 | It is already proved in original-admissible-source-closure-20261025/
000026 | QUARTER-STRESS-CURRENT-RECOVERY.txt, sections2-4, with the transported original
000027 | winding, actual CAR operators and local current algebra. The source's full
000028 | neutral Fock space is already identified through Maya/Schur/current words.
000029 | 
000030 | Set H0=closure span{J_-n1 ... J_-nk Omega:nj>0}. Here Q=0, so
000031 |     L_0^0=H_alpha|H0,   L_-1^0 Omega=L_0^0 Omega=L_1^0 Omega=0.
000032 | The original twisted stress instead has
000033 |     Lcal_-1^alpha Omega=b J_-1 Omega,  Lcal_0^alpha Omega=b^2 Omega/2.
000034 | In the declared class Lcal_n-cJ_n-d delta_n0, the requirements that the SAME
000035 | Omega be Moebius invariant force c=b and d=b^2/2. These coefficients are not
000036 | fitted. This is uniqueness only in that affine improvement class.
000037 | 
000038 | The actual old original_stress_action and boson_stress_action functions
000039 | are reused directly by the checker. The finite CAR window -8,...,7 is exact
000040 | for the tested total grades<=6: leakage outside it first occurs at grade9.
000041 | Those checks verify the first identification, not an infinite spectrum.
000042 | 
000043 | 3. Actual all-mode positive parabolic generator
000044 | 
000045 | On H0 define on the common finite-energy core
000046 |     P=L_0^0-(L_1^0+L_-1^0)/2,
000047 |     D=(L_1^0-L_-1^0)/(2i).
000048 | The Virasoro relations give [D,P]=iP. Their self-adjoint closures are those
000049 | of the existing current-net stress operators, not arbitrary extensions.
000050 | 
000051 | Put e_n=J_-n Omega/sqrt(n), n>=1. On the original one-current subspace,
000052 |     P e_n=n e_n-sqrt(n(n-1))e_(n-1)/2-sqrt(n(n+1))e_(n+1)/2.
000053 | Use the complete normalized Laguerre basis
000054 |     e_n -> L_(n-1)^(1)(x)/sqrt(n) in L2(R+,x exp(-x)dx).
000055 | The Laguerre recurrence identifies p with multiplication by x/2.
000056 | The orthogonality integral is n delta_nm before normalization. Completeness
000057 | is the usual Laguerre completeness for this exponential weight; equivalently
000058 | polynomial density follows from its exponential moments. The Jacobi core is
000059 | essentially self-adjoint, for example by the divergent Carleman sum of the
000060 | inverse offdiagonal coefficients. Hence spec(p)=[0,infinity) and ker(p)=0.
000061 | 
000062 | The Moebius generators L0,L+/-1 on the Q0 current Fock space are number-
000063 | preserving second quantizations of their one-current actions. For +/-1 no
000064 | pair-creation term occurs in Sugawara. Thus
000065 |     P=dGamma(p)>=0,   spec(P)=[0,infinity),   ker(P)=C Omega.
000066 | The one-current subspace supplies all positive spectral values; every
000067 | nonvacuum finite-particle sector has zero kernel. This is an all-mode proof,
000068 | not a limit guessed from finite eigenvalues.
000069 | 
000070 | The Laguerre picture also gives the explicit dilation generator
000071 |     D f=i[x f'(x)+(1-x/2)f(x)].
000072 | Its actual unitary group is, with r=exp(-t),
000073 |     (V_t f)(x)=r exp((1-r)x/2) f(rx).
000074 | Change of variables proves unitarity; its group law is exact and
000075 |     V_t p V_t*=exp(-t)p.
000076 | Second quantization proves the corresponding complete group law on H0.
000077 | 
000078 | A minimal witness to P!=H_alpha is
000079 |     P J_-1 Omega=J_-1 Omega-J_-2 Omega/2,
000080 |     H_alpha J_-1 Omega=J_-1 Omega.
000081 | The same neutral source supports compact rotation energy and a different
000082 | gapless parabolic generator. No contradiction with the earlier gap theorem.
000083 | 
000084 | The ACTUAL old source quarter also fits this same group. The previously
000085 | proved identity in joint-native-algebra-selection-20261016/
000086 | ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt:166-180 is
000087 |     t_v=exp(-4pi iK_alpha) exp(i pi(1+alpha/2)Q), H_alpha=8K_alpha.
000088 | On the present Q0 space this gives
000089 |     t_v=R=exp(-i pi L0^0/2),   R L_n^0 R*=i^n L_n^0.
000090 | Writing A=L0^0,B=(L1^0+L_-1^0)/2, the adjoint action sends B->-D,D->B.
000091 | The actual quarter thus transports P through
000092 |     A-B -> A+D -> A+B -> A-D -> A-B.
000093 | All four are positive unitary conjugates, corresponding to moved Cayley
000094 | cuts. On the common finite-energy core (and the closure of these sums),
000095 |     H_alpha|H0=A=(P0+P2)/2=(P1+P3)/2.
000096 | Thus the gapped rotation energy is an average of opposite gapless parabolic
000097 | energies, not either one. They do not commute: [P0,P2]=-2iD on the common
000098 | core. There is no simultaneous scalar spectral averaging argument that
000099 | would transfer their gaplessness to their operator mean.
000100 | For the local result transport M_k=R^k M R^-k,
000101 | N_k=R^k N R^-k and their modular groups together. Do not claim that all four
000102 | P_k half-sidedly preserve the same fixed M. This is compatibility with the
000103 | GIVEN original source clock on its
000104 | neutral sector. It does not identify this operation with the differently
000105 | marked native compiler J, assert fixed-cut invariance, remove charged parity
000106 | on the full CAR space, or choose a physical P1 clock.
000107 | 
000108 | 4. Geometry, standard pair and reflection
000109 | 
000110 | For the circle coordinate theta choose x=-cot(theta/2). Translation in x
000111 | has circle vector field1-cos(theta), and dilation x->exp(s)x has vector
000112 | field-sin(theta). They are implemented by exp(iaP), exp(-isD), respectively.
000113 | This is a Cayley coordinate and a derived symmetry action, not an asserted
000114 | change of the original physical time parameter.
000115 | 
000116 | The already constructed local current net is
000117 |     A_J(I)={exp(iJ(f)):f real smooth,supp(f) subset I}'' on H0.
000118 | Current essential self-adjointness, its positive-energy Moebius action,
000119 | strong additivity and vacuum properties are the standard U1 current-net
000120 | construction reviewed in Tanimoto, sections3.1-3.2. The original-source
000121 | current identification above verifies which representation is being used.
000122 | 
000123 | For a>0 set M=A_J(0,infinity), N=A_J(a,infinity). Reeh-Schlieder gives Omega
000124 | cyclic and separating for M,N and A_J(0,a); locality already makes the last
000125 | algebra a subalgebra of N' intersect M. Haag duality/strong additivity give
000126 |     N' intersect M=A_J(0,a).
000127 | Thus the inclusion is standard. The conformal Bisognano-Wichmann theorem
000128 | on this actual vacuum net gives
000129 |     Delta_M^(it)=exp(i2pi t D),
000130 |     Delta_M^(it) exp(isP) Delta_M^(-it)=exp(i exp(-2pi t)sP),
000131 |     Delta_M^(it) N Delta_M^(-it)=A_J(a exp(-2pi t),infinity) subset N
000132 |       for t<=0.
000133 | Moreover N=exp(iaP)Mexp(-iaP), and as the corresponding self-adjoint
000134 | generators (or on a suitable common invariant core),
000135 |     log Delta_N-log Delta_M=2pi aP.
000136 | Specify the explicit half-side t<=0; naming conventions for +/-hsm differ.
000137 | The modular conjugation J_M acts geometrically on interval algebras by
000138 |     J_M A_J(I) J_M=A_J(-I).
000139 | This does NOT yet identify it with the original physical P1 reflection Theta.
000140 | 
000141 | Primary references actually read:
000142 | Y. Tanimoto, arXiv:1807.11723v2, sections3.1-3.2, pp5-7:
000143 | https://arxiv.org/pdf/1807.11723
000144 | D. Guido,R. Longo,H.-W. Wiesbrock, arXiv:hep-th/9703129v2, section1,
000145 | especially Reeh-Schlieder/BW eqs3-4, standardity and Lemma1.3:
000146 | https://arxiv.org/pdf/hep-th/9703129
000147 | These are established general theorems applied to the specified source,
000148 | not new discoveries attributed to the finite checker.
000149 | 
000150 | 5. The source/state boundary that must be preserved
000151 | 
000152 | H0 is the original NEUTRAL CYCLIC space. On the full charged CAR space,
000153 | Omega is not cyclic for A_J. No full charged standard-pair claim follows.
000154 | The eightfold native composite retains eight original preimage arcs and its
000155 | background/cocycle. Its support and geometric Ward action cannot be replaced
000156 | by a new point-local field or an unshifted primary merely from this theorem.
000157 | 
000158 | The existing elliptic source is rho_beta proportional exp(-beta H_alpha),
000159 | in particular beta=4pi, not Omega. Even its Q0 conditioned oscillator state
000160 | has <J1 J_-1>_beta=1/(1-exp(-beta))>1, whereas the vacuum value is1.
000161 | On the one-current block, put q=exp(-beta); then
000162 |     [rho_beta,P]_(1,2)*Z=-sqrt(2)(q-q^2)/2 !=0.
000163 | Consequently the derived vacuum P does not leave that Gibbs state invariant.
000164 | Its BW/HSMI modular action must not be substituted for the Gibbs/HS double
000165 | or for the primitive P1 state. Other thermal modular constructions are not
000166 | excluded by this test and are not asserted here.
000167 | 
000168 | 6. Correct original source target; avoid adding a second quantization law
000169 | 
000170 | The original operational Seedop already contains
000171 |     (A_loc,tau_t,Theta,omega,[u_Sigma],D_coll).
000172 | If THIS full source is supplied, K(F,G)=omega((Theta F)G) on positive-time
000173 | words is already specified. Its OS/GNS reconstruction requires no extra
000174 | independently chosen CAR/Berezin measure. The first missing comparison is
000175 | the actual local word map iota, preserving its entire state,time,reflection
000176 | and products. This corrects the priority of the previous fermionic-action
000177 | search; deriving a bilinear is one possible constructive route, not a
000178 | universally necessary additional prerequisite for the full Seedop route.
000179 | 
000180 | The original functor Seedop -> SBord_adm goes FROM the operational source
000181 | TO the geometric Cauchy datum, not conversely. Thin P1 does not state the
000182 | kernel's argument space; the geometric primitive kernel is a spectral/Cauchy
000183 | package and explicitly omits a vacuum datum. Its physical source must not
000184 | be inferred by silently inverting that functor. These are recovered source
000185 | facts, not a new absence proof or a new GNS theorem.
000186 | 
000187 | The newly constructive neutral standard pair is available once that source
000188 | comparison is actually established. It is not a new physical selector of
000189 | omega, geometry, coupled fields or 3+1D. The next origin test remains the
000190 | actual two-face neutral word K(F,G) for bilinear F,G, and ultimately all
000191 | words under iota, with original admissible restriction and real support.
000192 | Do not replace that physical word value by a Wick value from this reference.
000193 | Common T1-T8 closure and complete automated TFPT discovery remain open.