TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/VARIABLE-BACKGROUND-PROOF.txt
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000001 | L20261102D. Original local force source on spatially variable backgrounds
000002 | 2 November 2026. Conditional application theorem, not fundamental TFPT closure.
000003 | 
000004 | 1. The actual question and the reused source
000005 | 
000006 | The shared source audit L20261102C remains controlling. The original full
000007 | S_adm/Z_rel/OS/Gamma route exists. No extra active native clock or eightfold
000008 | CAR model is a global origin requirement. This calculation addresses a
000009 | specific application assumption in the ALREADY EXISTING local Higgs-force
000010 | reader, R079/R080. That reader was proved for a constant Higgs background.
000011 | The original pointwise oneform already allows H(theta). We discharge the
000012 | constant-background restriction for that reader, without selecting a new
000013 | geometry, coupling, probability, field alphabet, physical clock or vacuum.
000014 | 
000015 | Original sources read:
000016 | LOCAL-SPACETIME-FORCE-SOURCE-PROOF.txt:1-161, especially1-17,67-114;
000017 | ALLMODE-COLLECTIVE-TIME-SOURCE-PROOF.txt:1-178;
000018 | DYNAMICAL-HIGGS-SOURCE-PROOF.txt:10-29;
000019 | the original v252 full96 constructor and the quark36 reducing reader.
000020 | The Atlas finding for R080 explicitly assumes constantHstar. No assertion
000021 | of worldwide novelty is made. The finite diagnostics below are not a proof
000022 | of the infinite-dimensional statement, which is proved next.
000023 | 
000024 | 2. Same original operator family and its exact domain
000025 | 
000026 | Let K=L2(S1,C^d), D0=-i partial_theta+alpha on periodicH1, alpha1/4 or3/4,
000027 | and gamma*=gamma,gamma^2=I. Keep the original finite internal matrices,
000028 | charges, actual oneform Ta=-i gamma dPhi/dxa, and fixed Yukawas.
000029 | For the actual quark reader d=36, not96 redundant CAR species.
000030 | Let V(theta) be the given bounded Hermitian multiplication potential.
000031 | The original smooth pointwise V_H=-i gammaPhi(H(theta)) is included.
000032 | Then h=gammaD0+V is selfadjoint on the SAME H1 by bounded perturbation.
000033 | Each admitted smooth background is bounded on this circle. Uniform claims
000034 | over several backgrounds explicitly require sup||V||<=M.
000035 | 
000036 | Use the existing conditional state rule C_h=expit(2ell h), ell>0 given,
000037 | pairing0, and the existing h-time. This is a family of the original
000038 | conditional states, not a fixed state silently kept while h changes.
000039 | No primitive P1 identification or interacting H pushforward is assumed.
000040 | 
000041 | Let F(t,theta)=sum_a f_a(t,theta)Ta be Hermitian, odd with respect to gamma,
000042 | smooth and compactly supported in time and space. In fact integrability of
000043 | F,partial_tF,partial_thetaF in L1_t L_infty_theta suffices, with the stated
000044 | domain-preserving spatial regularity and vanishing time boundary terms.
000045 | Set A_F=int e^(ith) F(t) e^(-ith) dt as a strong/weak operator integral.
000046 | It is bounded selfadjoint, with ||A_F||<=a0=int||F(t)||dt.
000047 | No operator-norm Bochner measurability is presumed without justification.
000048 | 
000049 | 3. The short identity that removes the constant-background assumption
000050 | 
000051 | Oddness kills the unbounded derivative part in the anticommutator:
000052 | 
000053 |  {h,F(t)} = B_F(t) = -i gamma partial_thetaF(t)+{V,F(t)}.
000054 | 
000055 | B_F is bounded. Write Bint=int e^(ith)B_F(t)e^(-ith)dt and
000056 | Dt=int e^(ith)partial_tF(t)e^(-ith)dt. On Domh, first as weak forms,
000057 | 
000058 |  {h,A_F}=Bint,       [h,A_F]=iDt,
000059 |                   2hA_F=Bint+iDt.                         (1)
000060 | 
000061 | The commutator identity is integration by parts in t. For u,v in Domh,
000062 | the sum of the two forms identifies <hu,A_Fv> with a bounded form in u,v.
000063 | Selfadjointness of h gives A_Fv in Domh and hA_Fv=(Bint+iDt)v/2.
000064 | Approximate arbitrary v by Domh and use closedness of h. Thus A_F maps ALL
000065 | K into Domh and (1) is a bounded operator equality, not only a formal rule.
000066 | 
000067 | Put a1=int||partial_thetaF(t)||dt and at=int||partial_tF(t)||dt.
000068 | Equation(1) gives ||hA_F||<=(a1+2||V||a0+at)/2.
000069 | Since D0A_F=gamma(hA_F-VA_F),
000070 | 
000071 |  ||A_F||_2 <= r_alpha [(1+2||V||)a0+(a1+at)/2],              (2)
000072 |  r_alpha^2=d sum_(n inZ) 1/[1+(n+alpha)^2] <infinity.
000073 | 
000074 | Here A_F=(D0+i)^-1[(D0+i)A_F], and the first factor is Hilbert-Schmidt.
000075 | This proves the WHOLE space/time insertion is Hilbert-Schmidt, directly
000076 | for variable bounded V and general tests. No momentum factorization,
000077 | constantH, uniform Perron gap, eigenvector derivative, mode sampling or
000078 | positive mass gap is required. Zero modes cause no problem because the
000079 | factor is D0+i. A moving gamma or a changed principal symbol would require
000080 | their actual connection terms and is not covered by this theorem.
000081 | 
000082 | 4. All existing ordered source responses now apply to the same family
000083 | 
000084 | For real tests Ak=A_Fk in S2, the R080 source proof is already complete.
000085 | It applies to ANY bounded covariance0<=C<=I, hence to each existing C_h:
000086 | 
000087 |  chi=det2[I+C(U-I)] exp Tr C(U-I-iL),
000088 |  U=prod_k exp(i lambda_k Ak), L=sum_k lambda_k Ak.           (3)
000089 | 
000090 | The original order, det2 AND trace correction are required. The proof gives
000091 | strong unitary and adjoint limits and selfadjoint Stone generators K(A).
000092 | It does not replace them by commuting random variables or Wick from an
000093 | averaged two-point function. Source positivity is inherited from the same
000094 | conditional CAR state. No Euclidean RP claim for the full bosonic source
000095 | is inferred from characteristic-function positivity.
000096 | 
000097 | There is also explicit uniform control of ALL response orders. For a finite
000098 | cutoff A_N, C_N its same-state compression and W_N=dGamma(A_N)-Tr C_N A_N,
000099 | the finite Klich identity yields for real s
000100 | 
000101 |  log omega(exp(s W_N))
000102 |  <=Tr C_N(exp(sA_N)-I-sA_N)
000103 |  <=(s^2/2) exp(|s| ||A_F||) ||A_F||_2^2.                   (4)
000104 | 
000105 | For the first inequality use the positive matrix
000106 | Y=I+C_N^(1/2)(exp(sA_N)-I)C_N^(1/2)>0 and logdetY<=Tr(Y-I).
000107 | The second is scalar Taylor functional calculus and0<=C_N<=I.
000108 | No commutation of C_N and A_N is needed. Both signs of s give exponential
000109 | tail bounds, all single-reader moments and uniform integrability.
000110 | 
000111 | For genuinely ORDERED mixed response coefficients, do not infer everything
000112 | from separate marginal moment bounds. Instead analytically continue (3).
000113 | For complex z_k, set U=prod exp(z_k Ak), L=sum z_k Ak,
000114 | S=sum |z_k| ||Ak||, B=sum |z_k| ||Ak||_2. Product differentiation gives
000115 | ||U-I||_2<=e^S B and ||U-I-L||_1<=e^S B^2/2.
000116 | The second bound follows by integrating the second derivative of
000117 | prod exp(t z_k Ak); every term has two S2 factors and bounded intervening
000118 | exponentials. Thus (3), with i lambda replaced by z, is locally uniformly
000119 | entire in all z. S2/S1 cutoff convergence holds uniformly on complex
000120 | compacts. Cauchy estimates give every ordered response derivative and its
000121 | cutoff limit. This asserts response moments; it makes no unnecessary claim
000122 | about an unrestricted common domain for arbitrary unbounded products.
000123 | 
000124 | Equations(2)-(4) are uniform over bounded V families with bounded displayed
000125 | test seminorms. They do not prove uniformity over an unbounded global H
000126 | ensemble whose law has not been supplied. Exact projectivity of the given
000127 | complete source, when available, separately propagates each integrable
000128 | local word and its tail integrals unchanged; no second uniform moment
000129 | axiom is then needed for those same cylinder observables.
000130 | 
000131 | The native computation may recompute h_N=P_NhP_N, C_N=expit(2ell h_N)
000132 | and A_N using that finite evolution. This is not exactly P_NA_FP_N.
000133 | Nevertheless it has the SAME limit. Put htilde_N=gammaD0+P_NVP_N on K,
000134 | and F_N=P_NFP_N. The finite anticommutator has boundary defects relative
000135 | to P_N B_F P_N, but its norm is still bounded by
000136 | ||partial_thetaF||+2||V||||F||. Thus (D0+i)A_N is uniformly bounded.
000137 | Since htilde_N converges strongly-resolvent to h and F_N converges strongly
000138 | to F, dominated strong time integration gives A_N->A_F strongly.
000139 | The fixed resolvent factor gives uniform high-frequency S2 tails:
000140 | ||(I-P_K)A_N||_2<=||(I-P_K)(D0+i)^-1||_2 const ->0.
000141 | Hermiticity gives the corresponding right tails. Finite K corners plus
000142 | both tail bounds then give A_N->A_F in S2. The finite C_N converge strongly
000143 | to C_h, with norm<=1, so C_N(U_N-I)->C_h(U-I) in S2 and the trace correction
000144 | also converges. This validates the actual native Galerkin path without
000145 | mistaking its boundary error for zero or its finite data for the proof.
000146 | For this re-evolved cutoff, (4) uses its own norms or the shared bounds
000147 | from(2); ||A_N||<=||A_F|| is only automatic for direct compressions.
000148 | 
000149 | 5. Locality and the meaning of this result
000150 | 
000151 | The R080 finite propagation proof only uses the fixed principal symbol
000152 | gamma, its norm1, and Hermiticity of the zero-order potential. Its local
000153 | current energy identity still holds with spatially variable V. Compact
000154 | tests with all pairs spacelike on the GIVEN cylinder have A_FA_G=0 both
000155 | orders. The same full-state cyclicity and ordered source proof therefore
000156 | give commuting unitary groups. This extends the original given local
000157 | force net; it does not derive3+1 spacetime from an internal index.
000158 | 
000159 | Z0, the bound reference, eta and the absolute vacuum scalar have not been
000160 | normalized away. Each conditional source here is centered relative C_h.
000161 | It is not silently centered relative a different fixed reference. If a
000162 | fixed C_ref is used and C_h-C_ref is S2 on the SAME actual circle,
000163 | Tr[(C_h-C_ref)A_F] exists and supplies the finite scalar phase of that
000164 | centering change. This does not recover absoluteZ_rel[0].
000165 | 
000166 | The missing equality to test in the FULL original source remains precise:
000167 | Z_n[J]/Z_n[0] must equal the stated conditional/joint field response with
000168 | the SAME field legs, actual state, ordering and reference. If that identity
000169 | is given for this reader, constant-background regularity is now discharged
000170 | and its entire local response family is controlled. This application
000171 | theorem does not assert that identity or replace the physical source by
000172 | this CAR reader. F1 and the overall fundamental solution are not marked
000173 | complete. Existing E8/clocks/flavor/alpha/geometry results are unchanged.
000174 | 
000175 | 6. Native diagnostics and independent review
000176 | 
000177 | check_variable_background.py imports the original v252 full96 constructor,
000178 | builds all four original oneform derivatives, verifies the reducing quark36
000179 | sector, and uses nonconstant trigonometric coefficients in those same four
000180 | directions. Both original quarter branches are checked at N2,4,8.
000181 | The weak anticommutator identity is tested on interior Fourier modes;
000182 | the finite cutoff boundary defect is not called zero. Omitting the spatial
000183 | derivative fails that identity, providing a decisive countercontrol.
000184 | Finite h diagnostics verify (1), original nonconstant momentum mixing,
000185 | Hermiticity, the exact centered ordered determinant identity and (4).
000186 | Run: OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python3 -B -OO
000187 | work/source_continuation/check_variable_background.py
000188 | Observed:59 checks pass. These diagnostics are not a continuum numerical
000189 | certificate, a mass/coupling fit, an experiment, a Lean run or ledger change.
000190 | The analytical argument carries the full admitted source family.