TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/LOCAL-SOURCE-MODULAR-TIME-PROOF.txt
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000001 | UR.UNIFIED.LOCAL-SOURCE-MODULAR-TIME.01
000002 | 31 October2026. R060. Given original free source, not a new state choice.
000003 | 
000004 | 1. Decision and retained original data
000005 | The actual elliptic annulus source already has occupation covariance
000006 |  C_alpha e_r=c_alpha(r)e_r, c_alpha(r)=(1+exp(beta(r+alpha)))^-1,
000007 |  alpha=1/4 or3/4, beta=4pi, r in Z, e_r(theta)=exp(ir theta).
000008 | Use L2(S1,dtheta/(2pi)) in a periodic trivialization. Multiplication by
000009 | exp(i alpha theta) on a local chart restores the original twisted section;
000010 | it does not change the neutral current or its support. This is the given
000011 | free CAR quantization, not a derivation of the primitive P1 measure.
000012 | The complete current functional, including its charge sectors, is
000013 |  omega_alpha(W(f))=Theta_alpha(f0)/Theta_alpha(0) exp(-V_beta(f)/2),
000014 |  V_beta(f)=sum_(n>=1) w_n |f_n|^2, w_n=n coth(beta n/2),
000015 |  f_n=(2pi)^-1 int f(theta) exp(-in theta)dtheta,
000016 |  Theta_alpha(t)=sum_q exp[-beta(q^2/2+(alpha-1/2)q)] exp(itq).
000017 | Original Weyl words retain their central BCH phase. All Fourier modes stay.
000018 | The neutral vacuum Mobius standard pair is ALREADY constructively proved.
000019 | Its existence and the known Hardy/elliptic state distinction are reused.
000020 | New question: does its interval-preserving geometric current flow preserve
000021 | this actual elliptic state on a genuinely supported local operation?
000022 | Success is an exact state-invariance decision plus an actual interval CAR
000023 | modular producer. Abort identifying the two times at the first mismatch.
000024 | 
000025 | 2. One exact supported probe, without Fourier truncation
000026 | Fix I=(-pi/2,pi/2) in the original source circle. The vacuum Mobius interval
000027 | vector field is v(theta)=cos(theta); an overall nonzero time rescaling is
000028 | irrelevant to the following nonvanishing test. Let gamma_t be its flow.
000029 | Its current transport is f_t=f composed gamma_-t, so dot(f)=-cos(theta)f'.
000030 | This is the weight1 current smearing law, not a scalar-field weight guess.
000031 | Choose any nonzero even nonnegative C-infinity function b supported in
000032 | [-delta/2,delta/2], 0<delta<pi/8. Normalize b_0=1 if desired.
000033 | Set chi=b*tilde(b) with normalized circle convolution and tilde(b)(x)=b(-x).
000034 | Then chi is real even smooth, supported in [-delta,delta], and
000035 |  chi_n=|b_n|^2>=0, chi_1 chi_2>0.
000036 | The strict last assertion follows because cos(nx)>0 on supp(b), n=1,2.
000037 | Set theta0=pi/4 and f(theta)=chi'(theta-theta0). Then supp(f) compact in I,
000038 |  f0=0, f_n=in chi_n exp(-in theta0).
000039 | This is an actual local current Weyl test function in the given source.
000040 | No finite Fourier polynomial is claimed to have proper support.
000041 | 
000042 | 3. Entire finite-beta class, all-mode derivative
000043 | From dot(f)_n=-(i/2)[(n-1)f_(n-1)+(n+1)f_(n+1)], summation gives
000044 |  dot(V_beta)=sum_(n>=1) d_n Im(conj(f_n)f_(n+1)),
000045 |  d_n=(n+1)w_n-n w_(n+1)
000046 |     =n(n+1)[coth(beta n/2)-coth(beta(n+1)/2)]>0.
000047 | Smoothness justifies differentiating and summing. For our supported probe,
000048 |  Im(conj(f_n)f_(n+1))=-n(n+1) sin(theta0) chi_n chi_(n+1).
000049 | Consequently
000050 |  dot(V_beta)=-sin(theta0) sum_(n>=1) [n(n+1)]^2
000051 |    [coth(beta n/2)-coth(beta(n+1)/2)] chi_n chi_(n+1)<0.       (A)
000052 | Every summand has the same sign; the first is strictly negative. This
000053 | decides every finite beta>0 for this same supported-profile class, including
000054 | the actual fixed beta4pi. It does not change or fit beta. At beta infinity
000055 | all d_n vanish, agreeing with the preserved vacuum interval modular flow.
000056 | For b0=1, |b_n|<=1. Put q=exp(-beta), N=M+1. An explicit absolute tail
000057 | bound after n=M is
000058 |  2 [N(N+1)]^2 q^N /[(1-q)(1-q((N+2)/N)^2)],                 (B)
000059 | whenever the final denominator is positive. It follows by bounding each
000060 | coth difference by 2q^n/(1-q) and a geometric bound on successive terms.
000061 | The proof of strict sign is analytic and does not use numerical cutoffs.
000062 | 
000063 | 4. Charge factors, bounded observables and the two-operation word
000064 | For s real nonzero, W(sf_t) is bounded and lies in the same fixed interval
000065 | algebra. At t=0, (f_t)_0=0. Theta_alpha(0)>0 and
000066 |  Theta'_alpha(0)/Theta_alpha(0)=i omega_alpha(Q).
000067 | Therefore the charge factor has zero FIRST derivative of its log modulus,
000068 | even though f_t may acquire nonzero mean. A c-number affine current shift
000069 | or Weyl central phase likewise changes only the phase. Thus
000070 |  d_t log|omega_alpha(W(sf_t))|_(t=0)=-s^2 dot(V_beta)/2 !=0. (C)
000071 | The same witness can be expressed as a genuine supported two-operation
000072 | word: fix s1,s2 nonzero with s1+s2!=0 and take
000073 |  W(s1 f_t) W(s2 f_t)=W((s1+s2)f_t).
000074 | Its central phase is exactly zero because c(f_t,f_t)=0; no intermediate
000075 | compression is made. Equation(C) holds with s=s1+s2. These are two
000076 | operations on the same supported probe, not a test of disjoint regions.
000077 | One bounded observable already suffices to disprove state invariance.
000078 | Every modular automorphism preserves its defining state. Hence the vacuum
000079 | interval Mobius current transport cannot be the restricted elliptic-state
000080 | modular flow under this unchanged current/support dictionary. Any nonzero
000081 | constant rescaling or reversed time leaves the obstruction nonzero.
000082 | This is a local-flow witness, beyond the already known global rho/P
000083 | commutator and local Hardy/thermal state difference. It is not a claim
000084 | that distinct states can never share a flow or that no geometry exists.
000085 | 
000086 | 4b. The entire stated geometric-flow class at the ACTUAL quarter source
000087 | The preceding supported witness also motivates a stronger class decision.
000088 | Consider ANY real C5 vector field v on I, continuous at its endpoints, whose
000089 | one-parameter point diffeomorphisms preserve I and act on this SAME current
000090 | by f->f composed gamma_-t, allowing arbitrary scalar current shifts.
000091 | State invariance forces invariance of the CONNECTED two-current kernel.
000092 | For d=x-y !=0 sufficiently small this analytic kernel is
000093 |  K_alpha(d)=Var_alpha(Q)-1/[4 sin^2(d/2)]
000094 |        +2 sum_(n>=1) n q^n/(1-q^n) cos(nd)
000095 |    =-1/d^2+a0+a2 d^2+O(d^4),
000096 |  a0=Var_alpha(Q)-1/12+2 S1, a2=-1/240-S3,
000097 |  Sp=sum_(n>=1) n^p q^n/(1-q^n), q=exp(-4pi).
000098 | The original nonzero mean is removed by CONNECTED covariance. Scalar
000099 | affine shifts cannot change it. The necessary invariance equation is
000100 |  (v(x)-v(y))K'(d)+(v'(x)+v'(y))K(d)=0.                  (E)
000101 | Expand at x=z+d/2,y=z-d/2. The constant and d^2 coefficients give
000102 |  -v'''/6+2a0 v'=0,
000103 |  -v^(5)/240+(a0/4)v'''+4a2 v'=0.
000104 | Thus v'''=12a0 v', v^(5)=144a0^2 v', and
000105 |  (4/5)(5a2+3a0^2)v'=0.                               (F)
000106 | The coefficient is strictly negative for BOTH original quarters. Indeed
000107 |  Var_alpha(Q)=sum_r c_alpha(r)(1-c_alpha(r))>0
000108 |  <=[exp(-pi)+exp(-3pi)]/(1-exp(-4pi)),
000109 |  0<S1<=q/(1-q)^3, S3>0.
000110 | With x=exp(-pi)<1/20, monotonicity and an EXACT rational comparison give
000111 |  Var_alpha(Q)+2S1 <=(x+x^3)/(1-x^4)+2x^4/(1-x^4)^3
000112 |  <[(1/20)+(1/20)^3]/[1-(1/20)^4]
000113 |        +2(1/20)^4/[1-(1/20)^4]^3 <1/12.
000114 | Therefore -1/12<a0<0 and a2<-1/240, hence
000115 |  5a2+3a0^2 < -1/48+1/48=0.
000116 | Equation(F) forces v'=0 throughout I. Interval preservation fixes both
000117 | endpoints, so a constant v must be zero. The entire stated C5 geometric
000118 | point-transport class has no nontrivial state-preserving local flow here.
000119 | This actual-quarter class proof does not claim the arbitrary-beta class,
000120 | field-mixing/dressed currents, all geometries or full CAR kernel nonlocality.
000121 | It does exclude a modular interpretation within this explicit current class.
000122 | 
000123 | 5. Positive producer from the actual interval occupation covariance
000124 | Let iota_I:L2(I)->L2(S1) extend by zero and define
000125 |  C_alpha,I=iota_I^* C_alpha iota_I.
000126 | This is the actual spatial restriction of the independently given source,
000127 | not the five-code P of R059 and not a freely chosen covariance. For u!=0,
000128 |  <u,C_alpha,I u>=sum_r c_alpha(r)|<e_r,iota_Iu>|^2>0,
000129 |  <u,(1-C_alpha,I)u>=sum_r(1-c_alpha(r))|<e_r,iota_Iu>|^2>0.
000130 | Thus both endpoint kernels vanish, with no claimed uniform spectral gap.
000131 | Indeed u_m=exp(im theta)u for a fixed normalized supported u has C_I
000132 | expectation tending to0 as m->+infinity and to1 as m->-infinity by dominated
000133 | convergence of the original Fourier occupations. Thus both endpoints are
000134 | in the spectrum, although not eigenvalues; k_I is unbounded on both sides.
000135 | Spectral calculus defines the selfadjoint generally unbounded operator
000136 |  k_alpha,I=log(1-C_alpha,I)-log(C_alpha,I),
000137 |  Dom(k)={u:int_(0,1)|log((1-lambda)/lambda)|^2 d<u,E_lambda u><infinity}.
000138 | Equivalently, on spectral subspaces away from endpoints and then on its
000139 | spectral-calculus domain,
000140 |  k=int_0^infinity [(C_I+s)^-1-(1-C_I+s)^-1] ds.            (D)
000141 | The original local state is the quasifree CAR state with covariance C_I;
000142 | its complete CAR words are determined by its actual Wick rule. In its
000143 | standard GNS representation the canonical modular group is the quasifree
000144 | group with this generator. With annihilation a(u) antilinear in u and
000145 | sigma_t(A)=rho^(it) A rho^(-it) in a finite CAR restriction, the convention
000146 | is sigma_t(a(u))=a(exp(-it k)u). The finite identity fixes the sign; the
000147 | quasifree modular theorem supplies the infinite GNS statement.
000148 | This does NOT posit a traceclass density matrix for the full local algebra.
000149 | The logarithm is taken AFTER spatial restriction. It is not the compression
000150 | of beta(-i partial_theta+alpha), nor a physical time calibration.
000151 | The CAR local modular group is not asserted to preserve the smaller current
000152 | subnet; the positive CAR construction and the current-flow exclusion have
000153 | these distinct, explicit scopes.
000154 | 
000155 | 6. Direct original-source evaluation, with the infinite sea retained
000156 | The checker also evaluates an actual finite CAR subalgebra of the interval,
000157 | not an alternative finite lattice source. Test functions
000158 |  u_j(theta)=sqrt(2) exp(2ij theta) 1_I(theta), j=0,1,2,
000159 | are orthonormal in the stated L2 measure. CAR smearing is bounded for these
000160 | L2 functions; they need not be smooth current profiles. Write C=P_sea+D,
000161 |  P_sea=1_(r<0), delta_c(r)=c_alpha(r)-1_(r<0).
000162 | For j,k>=0 the entire Hardy-sea matrix element is exactly
000163 |  <u_j,P_sea u_k>=(-1)^(j+k)/(2pi^2) S(j+1/2,k+1/2),
000164 |  S(a,a)=trigamma(a), S(a,b)=[digamma(b)-digamma(a)]/(b-a).
000165 | This follows by summing ALL negative odd modes; negative even overlaps
000166 | vanish for these j. Add the exponentially convergent exact delta_c series
000167 | with overlaps <e_r,u_j>=sqrt(2)sin((2j-r)pi/2)/(pi(2j-r)),
000168 | including its removable value1/sqrt(2) at r=2j.
000169 | The omitted correction for |r|>R has operator-norm bound
000170 |  [exp(-beta(R+1+alpha))+exp(-beta(R+1-alpha))]/(1-exp(-beta)).
000171 | The original sea is never cut. The resulting3mode covariance defines the
000172 | actual restriction to CAR(span{u0,u1,u2}). Its exact finite modular density,
000173 | two-point, four-point and two-number-unitary word are compared independently
000174 | with the8dimensional CAR Fock matrices. This is a finite subalgebra check;
000175 | its log matrix is not claimed to approximate the full interval k_I.
000176 | 
000177 | 7. What is retained and what is decided
000178 | R059's whole-word locality and all Q intermediate terms remain required.
000179 | R060 excludes the whole stated geometric current-time class at the actual
000180 | quarter source and supplies its actual local free-CAR modular producer.
000181 | It does not derive the original P1 insertion/measure, admissible carrier,
000182 | geometric reflection, full E8 interacting source, physical3+1D geometry,
000183 | spin2 pole, Einstein dynamics or factor6. Original charged/neutral source
000184 | maps, E8 hull, clocks, Yukawa/Flavor/Alpha and all prior matter results are
000185 | retained. No Gaussianization of the Higgs-integrated R054 source is allowed.
000186 | No inverse reconstruction from an independently fed DY is used.
000187 | 
000188 | Primary comparison: Blanco/Perez-Nadal, arXiv1905.05210, SecII Eqs3-6,
000189 |  https://arxiv.org/pdf/1905.05210 . Their G=<psi psi*> equals1-C in the
000190 | occupation convention here, so -log(G^-1-1)=log((1-C)/C). Their explicit
000191 | NS/R torus nonlocal kernels are NOT transferred to quarter holonomy.
000192 | Araki, PublRIMS6(1970/71)385-442, Sec6 Theorem3 and separating GNS,
000193 |  https://ems.press/content/serial-article-files/41673 .
000194 | Verdict:EXACT_SCOPED_LOCAL_TIME_WITNESS_AND_GIVEN_FREE_CAR_READOUT;
000195 | GEOMETRIC_CURRENT_POINT_FLOW_CLASS_EXCLUDED_AT_ACTUAL_QUARTERS;
000196 | PHYSICAL_ORIGIN_OPEN.
000197 | Experiment contract only. No new Lean, paper, empirical or ledger claim.