TFPT scientific source excerpt Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/GAUSSIAN-REFLECTION-PHASE-RADIAL.txt Original file SHA-256: e51580087db4b7765f931f872916b64a2d2fa21d5417292048a7362455577d1e Inclusive original lines: 1-190 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | EXACT GAUSSIAN REFLECTION TESTS OF THE DERIVED NUMERATOR KERNELS 000002 | 3 November 2026. Analytical proof, no new physical parameter or profile. 000003 | 000004 | Verdicts and scope 000005 | ------------------ 000006 | 1. The standalone Gaussian phase covariance with inverse K(x)=x F(x), 000007 | F(x)=int_0^1 exp[-a(1-a)x]da, fails scalar time reflection positivity. 000008 | There is an analytic negative one-pulse reflection test. No spectral 000009 | representation assumption or finite numerical witness is needed. 000010 | 2. The actual radial numerator coefficient proposed by root is correct. 000011 | For the stable single-heat class mu>1/2 it is positive for all momenta, 000012 | yet its standalone Gaussian covariance also fails time reflection 000013 | positivity. The negative remainder survives subtraction of the entire 000014 | allowed polynomial contact. The radial coordinate is gauge invariant 000015 | near the nonzero original Higgs background. 000016 | 3. Positive Laplace mixtures inherit these obstructions under the precise 000017 | inverse-moment/coefficient-integrability assumptions below. No claim is 000018 | made for profiles outside those assumptions, original unsquared f, 000019 | signed relative weights, or a fully coupled primitive source. 000020 | 000021 | The phase at fixed connection is gauge variant. Its Gaussian failure is 000022 | not a physical TFPT exclusion: physical RP is required on the actual 000023 | admissible algebra. The radial norm is gauge invariant, but the inverse 000024 | of its isolated quadratic block is not the physical marginal covariance 000025 | of a joint system with nonzero mixed blocks. Nor must an arbitrary 000026 | Gaussian expansion about a background inherit RP of an interacting 000027 | measure. The calculations exclude the declared standalone Gaussian 000028 | numerator candidates; reference, eta, full matrix Majorana terms, 000029 | geometry/scalaron mixing, selector and actual interacting integration 000030 | remain to be included before a whole-source statement. 000031 | 000032 | 1. Reflection form with all contacts removed exactly 000033 | --------------------------------------------------- 000034 | Use dimensionless four-momentum q=p/chi and fix spatial |k|. For any 000035 | scalar covariance C(q0^2+|k|^2), a polynomial P(q0^2+|k|^2) Fourier 000036 | transforms to derivatives of delta(time). Its time-reflected quadratic 000037 | form is zero for tests with support strictly inside positive time, 000038 | because reflected separation is t+s>0. Thus these contact polynomials 000039 | can be removed for such tests WITHOUT an arbitrary physical subtraction. 000040 | 000041 | Suppose R_k(q0)=C(q0^2+|k|^2)-P(q0^2+|k|^2) is integrable. Define 000042 | g_k(tau)=int_R dq0/(2pi) exp(i q0 tau) R_k(q0). 000043 | If R_k<0 for all real q0, then g_k is continuous and g_k(0)<0. 000044 | Consequently g_k(2epsilon)<0 for every sufficiently small epsilon>0. 000045 | The single positive-time pulse delta(t-epsilon) has reflection form 000046 | g_k(2epsilon)<0. Smooth compactly supported approximants, with support 000047 | remaining strictly positive, preserve this strict negativity. A plane 000048 | spatial wave is replaced by a smooth Fourier packet in a sufficiently 000049 | small neighborhood of k, again preserving the strict sign. Thus this 000050 | is a legitimate negative scalar RP test, not just a formal plane wave 000051 | or a source touching the reflection plane. 000052 | 000053 | This elementary test supplies the needed counterexample whenever the 000054 | subtracted covariance is eventually negative as a function of its 000055 | nonnegative argument: choose |k|^2 beyond that threshold. 000056 | 000057 | 2. Exact one-pulse obstruction for K=xF 000058 | -------------------------------------- 000059 | The exact form factor satisfies 000060 | 4x F'(x)+(x+2)F(x)=2, (1) 000061 | with F(0)=1 and F'(x)<0. This follows by integration by parts after 000062 | writing F=exp(-x/4) int_(-1/2)^(1/2) exp(x u^2)du. 000063 | For K=xF, (1) is 000064 | K'=1/2-K/4+K/(2x). (2) 000065 | Jensen gives 000066 | K(4)=4 int_0^1 exp[-4a(1-a)]da 000067 | >=4 exp(-2/3)>2. 000068 | The last inequality is exact: log2>2/3 follows, for example, from the 000069 | strictly positive remaining terms of the atanh series for log2. 000070 | At any point x>0 where K=2, (2) gives K'=1/x>0. Therefore K cannot 000071 | cross downward through2 after x=4. Hence 000072 | K(x)>2 and C(x)=1/K(x)<1/2 for EVERY x>=4. (3) 000073 | 000074 | The exact large-x endpoint expansion is 000075 | F(x)=2/x+4/x^2+24/x^3+240/x^4+O(x^-5), 000076 | K(x)=2+4/x+24/x^2+240/x^3+O(x^-4), 000077 | C(x)=1/2-1/x-4/x^2-40/x^3+O(x^-4). (4) 000078 | It follows either by endpoint Laplace asymptotics or from 000079 | F(x)=2 int_0^(1/4) exp(-x u)(1-4u)^(-1/2)du. 000080 | 000081 | Choose the FIXED dimensionless spatial momentum |k|=2, not a physical 000082 | parameter choice. Then R_k(q0)=C(q0^2+4)-1/2 is strictly negative for 000083 | all q0 by (3), and integrable by (4). Section1 gives a negative ONE-PULSE 000084 | reflection test at sufficiently small positive epsilon. The removed1/2 000085 | is only the actual contact part of this covariance; it makes no 000086 | contribution to a strictly positive-time reflection form. This proves 000087 | Gaussian RP failure directly, including the contact issue. 000088 | 000089 | A positive Stieltjes spectral representation plus a bounded contact 000090 | would also be contradicted by approach to1/2 from below, as root noted. 000091 | The direct proof here avoids extra assumptions about subtraction or 000092 | growth of a proposed spectral measure. 000093 | 000094 | For the entire ordinary charged single-heat phase trace, all original 000095 | mass/color/conjugate multiplicities multiply this same K by a positive 000096 | constant: m_i^2 exp(-m_i^2/chi^2) times multiplicities6,6,2. 000097 | Thus the same failure applies to that isolated whole charged phase 000098 | Gaussian candidate whenever its coefficient is nonzero. It does not 000099 | dispose of the original Majorana/mixed/reference/source requirements. 000100 | 000101 | 3. Positive Laplace phase mixtures: exact moment conditions 000102 | ---------------------------------------------------------- 000103 | Let w be the ACTUAL nonzero positive profile weight, including any actual 000104 | positive charged coefficients and exp(-s m_i^2/chi^2) factors. Let 000105 | K_w(x)=x int_(s>0) F(sx) dw(s), 000106 | M_j=int s^(-j)dw(s). 000107 | Assume 0=0 and tends2. Also 000113 | h(y)=y^2F(y)-2y is bounded and tends4. Apply dominated convergence with 000114 | the actual weights s^-1dw and s^-2dw respectively. No third inverse 000115 | moment is needed for the stated first correction. Because w is positive 000116 | and nonzero, M_2>0. The remainder C_w-C_w(infinity) is therefore 000117 | negative for all sufficiently large x and is O(1/x). 000118 | Choosing spatial |k|^2 beyond this threshold gives the same integrable 000119 | negative remainder and the same negative one-pulse RP form. These 000120 | assumptions exclude a nonzero atom at s=0 and profiles with divergent 000121 | required inverse moments; no conclusion for those classes is supplied 000122 | by this asymptotic argument. 000123 | 000124 | 4. Original radial coefficient: independent sign/factor calculation 000125 | ------------------------------------------------------------------ 000126 | Use the ACTUAL charged singular pair at the nonzero original background, 000127 | Phi_h=(m+h(x))sigma1, 000128 | D_h=i gamma.partial+gamma5 Phi_h, 000129 | P_h=-partial^2+m^2+V1+V2, 000130 | V1=2mh I2+i gamma^mu gamma5 sigma1 partial_mu h, 000131 | V2=h^2 I2. 000132 | The full spin/internal trace of the scalar linear vertex square is 000133 | 32m^2; the derivative linear vertex gives8p^2; their mixed spin trace 000134 | vanishes. The first quadratic contact has trace8. The original full 000135 | Duhamel calculation therefore gives the h^2 coefficient 000136 | 000137 | S_h[2]= exp(-t m^2)/(4pi^2) int d4p/(2pi)^4 |h(p)|^2 000138 | [(p^2+4m^2)F(tp^2)-2/t]. (6) 000139 | 000140 | The sign and coefficient -2/t are fixed by the original h^2 contact; 000141 | discarding them would change the source. No extra1/2 is present in this 000142 | quadratic-coefficient convention; the Hessian is twice (6). 000143 | For a common original Higgs radius fluctuation, h_i=y_i h per charged 000144 | singular value. Sum (6) with the ORIGINAL squared Yukawa factors and 000145 | color/particle-conjugate multiplicities. No independent radial mass or 000146 | Yukawa value is selected by this calculation. 000147 | 000148 | 5. Exact stable radial Gaussian and its RP failure 000149 | -------------------------------------------------- 000150 | Set x=tp^2 and mu=tm^2, retaining the positive overall factors in (6). 000151 | The dimensionless radial inverse covariance block is 000152 | R_mu(x)=(x+4mu)F(x)-2. 000153 | Equation (1) gives the EXACT positivity identity 000154 | R_mu(x)=-4x F'(x)+(4mu-2)F(x). (7) 000155 | Therefore, for every actual mu>1/2, R_mu(x)>0 for all x>=0. This proves 000156 | stability of the declared standalone Gaussian class without a guessed 000157 | stabilizing term. At mu<1/2 its zero-momentum coefficient is negative; 000158 | mu=1/2 has a zero mode and requires its separate original IR handling. 000159 | 000160 | The entire large-momentum expansion is 000161 | R_mu(x)=A/x+B/x^2+C/x^3+O(x^-4), 000162 | A=4+8mu, B=24+16mu, C=240+96mu. 000163 | Consequently 000164 | 1/R_mu(x)=x/A-B/A^2+(B^2-A C)/(A^3 x)+O(x^-2), (8) 000165 | B^2-A C=-384-1536mu-512mu^2<0 for mu>=0. 000166 | The leading two terms are the WHOLE polynomial contact 000167 | P_mu(x)=x/A-B/A^2. Its reflection form vanishes for positive-time 000168 | tests. The remaining covariance is strictly negative for sufficiently 000169 | large x and O(1/x). Section1 therefore gives a negative one-pulse RP 000170 | test after fixing sufficiently large spatial momentum. This proves 000171 | that the POSITIVE stable radial Gaussian class mu>1/2 fails RP. 000172 | 000173 | This is stronger than the phase-coordinate test in its observable type: 000174 | the local radial perturbation of |H| at a nonzero H background is gauge 000175 | invariant. It remains a test of the isolated squared-heat numerator 000176 | Gaussian, not of the whole physical admissible source or its interacting 000177 | radial marginal. The latter is not furnished by an inverse isolated 000178 | block if the original joint source has nonzero mixed blocks. 000179 | 000180 | 6. Positive Laplace/charged mixtures of radial numerators 000181 | -------------------------------------------------------- 000182 | The obstruction survives positive mixtures whenever the required exact 000183 | asymptotic coefficients are finite. Use x=p^2/chi^2 and mu=m^2/chi^2. 000184 | For each actual proper time s>0 the dimensionless radial contribution is 000185 | exp(-s mu)[(x+4mu)F(sx)-2/s]. 000186 | After its actual positive weights, including squared Yukawas and charged 000187 | multiplicities, its asymptotic coefficients are 000188 | A_s=4/s^2+8mu/s, 000189 | B_s=24/s^3+16mu/s^2, 000190 | C_s=240/s^4+96mu/s^3. (9)