TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-MIXED-TIME-RESPONSE.txt
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000001 | HIST20261103R. Exact mixed time response of the recovered historical action
000002 | Anonymized source calculation. 3 November 2026.
000003 | 
000004 | This continues HISTORICAL-AXIAL-ACTION.txt on that SAME supplied action.
000005 | It is not a new selected oscillator or a replacement for the full TFPT source.
000006 | The historical detector discussion already supplies a static magnetic field.
000007 | Here the actual wave response is evaluated on a uniform fixed Minkowski
000008 | background, constant a, zero E, k perpendicular to B0, and a transverse
000009 | photon potential parallel to B0. For other propagation angles the actual
000010 | Gauss reduction must retain its longitudinal electric response; this two
000011 | component reduction is not claimed there. Boundary cavity modes and damping
000012 | are not inserted into the microscopic action. No physical vacuum is selected.
000013 | 
000014 | 1. ACTUAL SECOND VARIATION AND GAUSS-REDUCED EQUATIONS
000015 | Use phi=f_eff*a, g=g0/f_eff, G=g B0 in a fixed oriented transverse axis.
000016 | Write A for that actual photon-potential component, not the scalar phi.
000017 | With E=-dot A the second variation of the original interaction g phi E.B0
000018 | is -G phi dot A. The complete two-component quadratic Lagrangian at fixed k is
000019 |  L2= (dot phi^2-k^2 phi^2)/2+(dot A^2-k^2 A^2)/2-G phi dot A.
000020 | It follows directly from the written original nonlinear action. Its equations
000021 |  ddot phi+k^2 phi+G dot A=j_phi,
000022 |  ddot A+k^2 A-G dot phi=j_A
000023 | retain both reaction directions. Treating a as a prescribed halo drive would
000024 | drop the first equation's backreaction and is a different response question.
000025 | Gauss law is satisfied in this transverse k.B0=0 reduction. It is not deleted
000026 | in the general theory. The canonical momentum of A is dot A-G phi, so
000027 |  H2=(p_phi^2+k^2 phi^2+(p_A+G phi)^2+k^2 A^2)/2 >=0.
000028 | This is the quadratic classical energy of the original action's reduction.
000029 | 
000030 | 2. FULL FREQUENCY SYMBOL AND EXACT TWO BRANCHES
000031 | For Fourier convention exp(-i omega t), the original response symbol is
000032 |  D(z,k)=[[k^2-z^2,-i G z],[i G z,k^2-z^2]], z=omega+i0 for retarded data.
000033 | The determinant is (z^2-k^2)^2-G^2 z^2, not a separately assigned mass.
000034 | For k>=0 define
000035 |  Omega_plus=sqrt(k^2+G^2/4)+abs(G)/2,
000036 |  Omega_minus=sqrt(k^2+G^2/4)-abs(G)/2.
000037 | They are nonnegative and solve the complete coupled determinant exactly.
000038 | Equivalently Omega_+-^2=k^2+G^2/2 +- abs(G)*sqrt(k^2+G^2/4).
000039 | Their product is k^2. At G=0 both recover the original massless k branch.
000040 | At k=0 one branch is zero and the other abs(G). This background splitting
000041 | is not a vacuum axion mass or a TFPT prediction of 64 micro-eV.
000042 | 
000043 | 3. ACTUAL RETARDED TIME ANSWER
000044 | Let Delta=Omega_plus^2-Omega_minus^2. For G!=0 and k>=0 the impulse-response
000045 | matrix is theta(t)*[[d(t),o(t)],[-o(t),d(t)]], where
000046 |  d(t)=[(Omega_plus^2-k^2) sin(Omega_plus*t)/Omega_plus
000047 |        +(k^2-Omega_minus^2) sin(Omega_minus*t)/Omega_minus]/Delta,
000048 |  o(t)=G[cos(Omega_plus*t)-cos(Omega_minus*t)]/Delta.
000049 | The zero-frequency sin(Omega*t)/Omega is interpreted by its exact limit t.
000050 | The off-diagonal signs follow from the oriented G and the actual interaction
000051 | -G phi dot A; changing a field orientation changes both consistently.
000052 | This is the exact mixed TIME kernel of these linearized original equations,
000053 | not merely a Lorentzian detector frequency envelope. It obeys
000054 |  d(0)=o(0)=0, dot d(0)=1, dot o(0)=0,
000055 |  o(t)=-G t^2/2+O(t^4),
000056 | and the full matrix equations for t>0, with unit delta-source jump at t=0.
000057 | At G=0 its diagonal is theta(t)sin(kt)/k and its off-diagonal is zero.
000058 | 
000059 | The same result is checked independently in Laplace space:
000060 |  D_L(s)=[[s^2+k^2,G s],[-G s,s^2+k^2]],
000061 |  R_L(s)=1/[(s^2+k^2)^2+G^2 s^2]
000062 |           *[[s^2+k^2,-G s],[G s,s^2+k^2]].
000063 | The exact matrix product is I. The residues above invert this rational
000064 | matrix, and its large-s behavior gives the source jumps and -G t^2/2 term.
000065 | 
000066 | A causal classical response does not determine the full quantum symmetric
000067 | correlator/state. It therefore supplies no missing OS vacuum by itself.
000068 | The common full interacting physical TFPT spectrum is not this two-branch
000069 | linear response in an externally supplied magnetic background.
000070 | 
000071 | 4. CORRECTIONS TO EXTERNAL REVIEW CLAIMS
000072 | The often used first-order propagation matrix with off-diagonal gB0/2 is
000073 | a high-frequency propagation approximation. Its sin^2(gB0 L/2) is not an
000074 | exact all-frequency conversion probability of the second-order equations.
000075 | Do not replace the exact branches and boundary/flux normalization by that
000076 | formula without its approximation and actual boundary problem.
000077 | 
000078 | The scalar phase-space map phi=f_eff*a, p_phi=p_a/f_eff has Jacobian ONE.
000079 | A configuration Jacobian alone must not become an extra full-source factor.
000080 | For positive f_eff the sign of lambda does not change g0/f_eff. Any signed
000081 | alternative field chart must transport its field orientation consistently.
000082 | The original Gaussian elimination prefactor is a different operation and
000083 | retains the actual metric, measure, contour, reference and regulator.
000084 | 
000085 | The positive energy above is H_aEM or H2 of this historical action. Naming
000086 | it H_tr,pos or a piece of the QFT005 subtraction without the actual whole
000087 | field/state/time and reference identification is not justified. Likewise,
000088 | putting S or a into A_b or Phi_b of the96 chart requires an actual original
000089 | operator embedding, not a list of plausible bosonic background variables.
000090 | For F=dA, Maxwell variation uses the exterior/codifferential (LC divergence)
000091 | and does not gain an arbitrary full-RC connection term by notation alone.
000092 | 
000093 | P25-L:218 also requires an exact coefficient check: its stated anomalous
000094 | term e^2 E theta F Ftilde/(16 pi^2), phase a=e^2 E theta/(4pi), and
000095 | -g0 a F Ftilde/4 with g0=-1/(2pi) have a factor-two mismatch if theta and
000096 | Ftilde keep the SAME definitions throughout. This is an algebraic check of
000097 | that written matching, not a universal claim about anomaly conventions.
000098 | Read actual theta normalization and full fermion map before transporting
000099 | the numerical topological coefficient into g_phys or a physical prediction.
000100 | 
000101 | All known E8, clocks, flavor, alpha, full96 coefficients and original
000102 | CARRIER-SOURCE/R113 results remain in their established scopes. The actual
000103 | common source identification, physical H_tr,pos and full spectrum remain
000104 | unresolved by this historical response calculation.