TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/LOCAL-SPACETIME-FORCE-SOURCE-PROOF.txt
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000001 | R080. Original local Higgs derivative: whole space/time CAR source and locality.
000002 | 1 November 2026. Scoped mathematical source construction, not physical F1 closure.
000003 | 
000004 | 1. GIVEN OBJECTS AND FIRST LOCAL INSERTION
000005 | Reuse R078/R079 actual real v252/254 Phi(H), fixed Yukawas/MR, grading and
000006 | nonredundant reducing particle-quark36 CAR reader. All other original master
000007 | sectors and vertices remain. On the already given circle use
000008 | h=gamma(-i partial_theta+alpha)+V*, V*=-i gamma Phi(H*), alpha=1/4 or3/4,
000009 | periodic H1 domain, constant original H*, {gamma,V*}=0, gamma²=I.
000010 | Cp=expit(2ell hp), ell>0 as given, defines the same pairing-zero quasifree
000011 | state C=direct_sum Cp and same original time flow. No physical clock selection.
000012 | Ta=-i gamma dPhi/dxa is Hermitian and odd. The local POINTWISE extension of
000013 | this original real-linear oneform gives delta h(t)=sum_a M_Fa(t,theta) Ta.
000014 | This is the derivative of the GIVEN H-coupled consumer, not a primitive
000015 | identification of its force with H or derivation of an H probability law.
000016 | Bounded real variations preserve the one-particle H1 domain.
000017 | Define the actual original interaction-picture linear insertion
000018 | A_F=sum_a int exp(ith) M_Fa(t) Ta exp(-ith)dt.
000019 | For real tests it is bounded/selfadjoint. For F=f(t)g(theta)e_a it has exactly
000020 | (A_F)pi,qj=g_(p-q) fhat(Epi-Eqj) (Up*TaUq)ij,
000021 | p,q in Z+alpha, hp Up=Up diag(Epi), fhat(w)=int f(t)exp(iwt)dt.
000022 | No free spectral weights, mixing matrices or interactions have been added.
000023 | Changed premise from R079: genuine spatial multiplication couples p to q;
000024 | the old product over independent momentum determinants cannot be reused.
000025 | 
000026 | 2. WHOLE ONE-PARTICLE INSERTION IS HILBERT-SCHMIDT
000027 | First let real f be Schwartz and real bounded g on S1, hence ghat in l2.
000028 | The constant coupled energies satisfy |Epi|>=|p| and
000029 | ||hp|-|p|I|<=||V*||. Spectral sign projectors obey
000030 | Pp^epsilon=(I+epsilon sign(p)gamma)/2+O(1/(1+|p|)).
000031 | All constants are uniform outside finitely many low modes; internal dimension
000032 | is finite and original alpha avoids p=0. Split the squared matrix sum:
000033 | (i) Opposite ENERGY signs. |Epi-Eqj|>=|p|+|q|. At fixed k=p-q, rapid fhat
000034 | decay makes the p sum uniformly bounded. Multiplication by |ghat_k|² and
000035 | summation over k is finite. Alternatively a rapid joint double-sum bound works.
000036 | (ii) Equal ENERGY signs and equal MOMENTUM signs. Odd Ta kills the leading
000037 | equal gamma projectors, giving ||Pp^epsilon Ta Pq^epsilon||
000038 |  <=const[(1+|p|)^-1+(1+|q|)^-1]. At fixed k, summing the square over p is
000039 | uniformly bounded by the convergent inverse-square series; then sum |ghat_k|².
000040 | (iii) Equal ENERGY signs and opposite MOMENTUM signs. One must keep temporal
000041 | filtering here too. |Epi-Eqj|>=||p|-|q||-2||V*||. For p>0,q<0 and k=p-q,
000042 | |p|-|q|=2p-k. A Schwartz bound gives a uniform lattice-strip sum
000043 | sum_(0<p<k) |fhat(Epi-E_(p-k)j)|²
000044 |  <=const sum_p (1+|2p-k|)^(-r)<=const, r>1.
000045 | Then again sum_k |ghat_k|²<infinity. The other quadrant is identical.
000046 | Thus A_F belongs to S2, with ||A_F||2<=const_f ||g||L2.
000047 | In particular a bounded rectangular SPATIAL window is allowed. A proposed
000048 | H^(1/2) necessity wrongly discards the fhat strip. No such exclusion is used.
000049 | For general smooth compactly supported F(t,theta), its joint transform is
000050 | rapid in time frequency and spatial mode. The same three sums give an S2
000051 | seminorm bound, continuous in the test topology. Finite sums and their smooth
000052 | limits therefore cover arbitrary original compact space/time tests.
000053 | Non-S2 by itself is not a general nonimplementability theorem. The argument
000054 | here uses explicit S2 continuity, not an incorrectly substituted mixed-C
000055 | Shale projector criterion. Weighted variance is
000056 | Tr C A(1-C)A=||(1-C)^(1/2) A C^(1/2)||2² <=||A||2².
000057 | 
000058 | S1 cannot generally replace S2. At the actual radial H*=(0,1), V*=T3,
000059 | the original singular channels have h_p=[[p,s],[s,-p]], T3=s sigma_x.
000060 | Diagonal values in the energy basis are +/-s²/sqrt(p²+s²).
000061 | If ghat0 fhat0!=0 and an original singular value s!=0, absolute diagonals
000062 | have a harmonic tail. The trace-class criterion fails. Tr CA_N can grow
000063 | logarithmically although the connected source converges. No factor1/2 here.
000064 | 
000065 | 3. COMPLETE CENTERED SOURCE: REQUIRED det2 AND TRACE CORRECTION
000066 | Let A1,...,Ar be these Hermitian S2 insertions, L=sum_k lambda_k Ak and
000067 | U=prod_k exp(i lambda_k Ak) in the ORIGINAL order. Then U-I is S2,
000068 | R=U-I-iL is S1. Indeed exp(iA)-I-iA is S1 by the quadratic Taylor remainder;
000069 | all remaining product terms have at least two S2 factors. These mappings
000070 | are continuous S2->S2 and (A1,...,Ar)->R in S1 on bounded sets.
000071 | For the original finite Fourier cutoff P_N, A_kN=P_N Ak P_N and C_N=P_NCP_N,
000072 | the centered finite CAR identity is
000073 | chi_N=exp(-i Tr C L_N) det[I+C(U_N-I)].
000074 | Using det2(I+X)=det[(I+X)exp(-X)] gives the exact limiting source
000075 |  chi(A1,...,Ar;lambda)=det2[I+C(U-I)] exp(Tr C[U-I-iL]).             (1)
000076 | S2 continuity of det2 and S1 continuity of R prove cutoff convergence.
000077 | Both factors are REQUIRED by the original centering. det2 alone changes
000078 | even connected two-point responses. An uncentered determinant or separate
000079 | infinite linear trace is not substituted. No free counterterm is chosen.
000080 | The finite identity is Klich's trace formula. det2 definition/continuity are
000081 | standard trace-ideal results, checked in Britz et al arXiv2007.12834 pp1-2,
000082 | equations1.3-1.5. No ordinary determinant multiplicativity for det2 is assumed.
000083 | 
000084 | 4. GENUINE SELFADJOINT REALIZATION AND ALL ORDERED WORDS
000085 | Write W_N(A)=dGamma(A_N)-Tr C A_N. Variance gives
000086 | ||(W_N(A)-W_M(A))Omega||²<=||A_N-A_M||2² ->0.
000087 | Put V_N(lambda)=exp(i lambda W_N(A)). Formula(1) for the TWO entries
000088 | (-lambda A_N,+lambda A_M) tends1 as N,M->infinity. This follows from S1
000089 | continuity of R at(A,A), not from a false positive square expansion.
000090 | Correct second-order remainder is A_N A_M-(A_N²+A_M²)/2
000091 |  =-(A_M-A_N)²/2+[A_N,A_M]/2. It need not be selfadjoint or positive.
000092 | Thus ||(V_N-V_M)Omega||²=2-2Re omega(V_N*V_M)->0, uniformly for compact
000093 | lambda intervals. V_N implements the CAR automorphism exp(i lambda A_N).
000094 | Since A_N->A in S2 and hence operator norm, those automorphisms converge in
000095 | norm on each finite CAR polynomial B. The identity
000096 | V_N B Omega=alpha_N(B)V_N Omega proves strong convergence on dense CAR Omega
000097 | and hence everywhere, including adjoints. Limits V_A(lambda) are inverse
000098 | unitaries and a strongly continuous group. Stone yields selfadjoint K(A).
000099 | Using the integrated finite group equation identifies K(A) with the L2 Wick
000100 | sum on CAR-polynomial Omega; commutators with B converge by bounded A_N->A.
000101 | No operator-norm convergence of V_N or essential-selfadjointness core claim.
000102 | Strong multiplication and (1) give every finite ordered unitary word.
000103 | Gram positivity passes from the same original finite CAR state. Higher
000104 | responses are retained, without Gaussian closure. The state is the given C,
000105 | not a newly selected physical initial state, an H measure or Euclidean RP law.
000106 | 
000107 | 5. EXACT LOCALITY IN THE GIVEN CYLINDER
000108 | The principal matrix gamma has norm1. For the original first-order equation
000109 | partial_t psi=ih psi, rho=psi*psi and j=-psi*gamma psi obey
000110 | partial_t rho+partial_theta j=0 and |j|<=rho. The bounded Hermitian
000111 | zero-order alpha gamma+V* cancels in this identity. Standard moving-region
000112 | energy estimates imply support speed<=1 on S1, including winding routes.
000113 | Equivalently, squared evolution has normally hyperbolic principal part.
000114 | Bär/Ginoux/Pfäffle arXiv0806.1036 Theorem3.2.11 confirms this support theorem
000115 | on the stipulated globally hyperbolic cylinder; the geometry is not derived.
000116 | 
000117 | Let real compact smooth tests F,G have EVERY pair of points satisfying
000118 | dist_S1(theta,phi)>|t-s|. Finite propagation then gives exactly
000119 | M_F(t) Ta exp(i(s-t)h) M_G(s) Tb=0, so A_F A_G=A_G A_F=0.
000120 | The four unitary word V_AF(-lambda)V_AG(-mu)V_AF(lambda)V_AG(mu)
000121 | implements identity on CAR. Its expectation from(1) is1, since its
000122 | one-particle U=I and L=0. Therefore it fixes Omega. It commutes with every
000123 | CAR B and hence fixes dense B Omega, so it is I on the entire GNS space.
000124 | This uses cyclicity, NOT an irreducible-Fock assumption for a mixed state.
000125 | The unitary groups commute and their Stone generators commute strongly.
000126 | No residual Schwinger central term survives. Finite Fourier cutoffs need not
000127 | be local; exact locality is in the S2 source limit, proved analytically.
000128 | Define A_force(O) as the von Neumann algebra generated by V_AF(lambda)
000129 | with suppF subsetO. It is isotonic and local on THIS given1+1 cylinder.
000130 | Gaussian time tests are not compact and cannot certify strict separation.
000131 | Timelike-related tests may also commute in special cases; no converse is
000132 | claimed. Nonzero spacelike correlations, if present, do not violate locality.
000133 | Constant H*, circle and finite ell do not imply global boost invariance,
000134 | physical OS reflection, physical3+1 or universal spin2.
000135 | 
000136 | 6. NATIVE CHECK AND NEXT ORIGINAL BRIDGE
000137 | The unchanged v252 full96 builder supplies all four Ta with original
000138 | seed252 Yukawas/MR, before the checked reducing quark36 readout. Native
000139 | spatial multipliers and original times give at N16
000140 | chi(A,B)=.9497002610252191+.011194758682075152i,
000141 | chi(B,A)=.9484380914431697-.008548765590089118i.
000142 | Wick variance of A=.07723107284175024. Det2 with its trace correction agrees
000143 | with the exact finite centered determinant. A separate16-state Fock marginal
000144 | on TWO different original momenta checks the whole mixed-mode identity.
000145 | For compact spacelike supports the actual original singular-channel control
000146 | has product HS defect decreasing from.0403447549957 atN16 to.0000500173424
000147 | atN128. This is a finite cutoff error, not a claimed exact numerical zero or
000148 | physical signal. The analytical support proof supplies zero in the limit.
000149 | Bounded rectangular spatial probes also converge;28 unique controls pass
000150 | under -B -OO. Test widths and source amplitudes are probes, not fitted physics.
000151 | 
000152 | This completes a supported local original circle FORCE source, extending
000153 | R079's spatially integrated component. The physical primitive map from the
000154 | whole E8/Naht/Sadm/Wk source to these supported operations and the common H(x)
000155 | weights is still to be executed. It is not legitimate to impose K(F)=H(F).
000156 | The old geometric fixed-point/measure passage is tested separately in
000157 | ORIGINAL-GEOMETRIC-MEASURE-AUDIT.txt; its literal global kernel map fails
000158 | before a finite fixed measure. No free compactification/weight is inserted.
000159 | Reuse the explicit conditional circle net; do not demand its reconstruction
000160 | again as missing. Preserve Ref/Eta/H0/Z0, other sectors/vertices,P/Q,charged
000161 | words,memory,State/Time/RP,E8/clocks,Yukawa/flavor/alpha. F1 remains in progress.