TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/FULL-SOURCE-REGULATOR-PROOF.txt
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000104 | derivative. This does not say every gravitational observable vanishes after
000105 | normalization. It identifies a scalar response that must be transported from
000106 | the SAME original full source, with its already fixed reference and scale.
000107 | The principle was present in finding:source-reference-amplitude-retained;
000108 | the pair flow in section2 now carries it through the actual v42 regulator.
000109 | 
000110 | 4. Whole quadratic regulator class with a safe reflected composition
000111 | 
000112 | The old finding:positive-spectral-reweighting-does-not-establish-original-source-rp
000113 | already excludes generic preservation by positive spectral weights alone.
000114 | That old result is reused. The changed premise here is a REFLECTION-SPLIT
000115 | QUADRATIC IR regulator, allowing an exact complete criterion rather than a
000116 | profile scan. This does not repair a non-RP original S_adm by assumption.
000117 | 
000118 | Class: x_-,x_+ in R^n, reflection swaps the two halves, with no fixed-plane
000119 | fields in this statement. Let the finite regulator be real symmetric and PSD,
000120 |   R=[[A,C],[C,A]],  A=A^T, C=C^T,  R>=0.
000121 | The original unregulated rho is any finite reflection-positive positive
000122 | measure on these variables. Define w_R=exp(-x.R.x/2).
000123 | 
000124 | THEOREM: w_R preserves RP for EVERY such original measure iff C<=0.
000125 | This is a universal preservation criterion, stronger than necessity for a
000126 | particular fixed source. A fixed original source may preserve RP for additional
000127 | regulators; its actual full reflected form decides those cases.
000128 | 
000129 | Sufficiency: put B=-C>=0. The reflected kernel is
000130 |   w_R(u,v)=exp(-u.A.u/2) exp(u.B.v) exp(-v.A.v/2).
000131 | The middle kernel has the feature expansion
000132 |   exp(u.B.v)=sum_(multiindex alpha) (B^1/2 u)^alpha
000133 |                           (B^1/2 v)^alpha/alpha!.
000134 | Thus w_R=sum_alpha Theta(f_alpha) f_alpha is positive definite. Multiplying
000135 | any RP source by it preserves the reflected form. There is no unchecked limit:
000136 | R>=0 implies A-B>=0, and the squared feature norm is
000137 |   sum |f_alpha(x)|^2=exp(-x.(A-B).x)<=1.
000138 | Finite feature sums therefore have kernels bounded by1. For bounded tests and
000139 | finite rho, dominated convergence gives the RP form. Integrable unbounded tests
000140 | follow by suitable truncation. Normalization is positive and changes no sign.
000141 | 
000142 | Necessity: choose the original RP product Gaussian with covariance I_(2n).
000143 | After reweighting its covariance is (I+R)^-1. Let S=I+A. Its reflected cross
000144 | covariance is
000145 |   G_cross=[(S+C)^-1-(S-C)^-1]/2.
000146 | Both S+C and S-C are positive definite since R>=0. RP of linear positive-half
000147 | observables requires G_cross>=0. Inversion reverses Loewner order in BOTH
000148 | directions, so this is equivalent to S+C<=S-C, i.e. C<=0. If C is not negative
000149 | semidefinite a linear reflected observable already gives a counterexample.
000150 | No commutation of A and C or simultaneous diagonalization is assumed.
000151 | 
000152 | The cone R>=0,C<=0 is closed under sums and nonnegative multiples. Sequential
000153 | regulator weights therefore compose in the same cone. For a finite original
000154 | measure, R_k>=0 and R_k->0 give w_Rk->1, dominated by1; original bounded
000155 | correlators and Z[0] are recovered. Time-local factors are the C=0 case.
000156 | This supplies a constructive permissible scheme in the declared finite class,
000157 | not a new physical coupling, a universal necessary P1 axiom, or an arbitrary
000158 | continuum regulator. Actual sector commutation and source domain remain
000159 | additional checks. A regulator built from an unrelated operator is not derived.
000160 | 
000161 | Exact test: original covariance [[1,1/2],[1/2,1]], R=[[1,1],[1,1]], and P_adm=I
000162 | on the already retained test block. R>=0, reflection symmetry and structural
000163 | commutation all hold, yet regulated covariance is [[7,-1],[-1,7]]/16.
000164 | The linear reflected form becomes -1/16. This is a test of the weak implication,
000165 | not an original TFPT countermodel. R=[[2,-1],[-1,2]] instead has negative cross
000166 | and preserves RP. The analytic theorem covers all sizes and parameter values.
000167 | Automatic candidate checking in this class reduces to exact semidefiniteness
000168 | of R and -C, reflection typing and the independent original P_adm commutator.
000169 | 
000170 | Primary context: Jaffe/Janssens, Reflection Positive Doubles, arXiv:1607.07126,
000171 | TheoremIV.10 and VII.3/VII.4 give reflected cone/cross-coupling criteria under
000172 | their stated hypotheses. The finite unbounded-coordinate proof above is
000173 | provided explicitly; it is a specialization of known RP mathematics.
000174 | Original v42 uses the Wetterich flow; Wetterich, arXiv:1710.05815, eqs12-19,
000175 | confirms the full generating functional and its regulated Legendre relation.