TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ACTUAL-ONEFORM-COUPLED-SOURCE-RESPONSE.txt
Original file SHA-256: 0336d7c900705f05959769c8dc5060a76cebc38325d2fbc5938bc566ceb4f838
Inclusive original lines: 1-54
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 | THE ORIGINAL INNER-FLUCTUATION FAMILY HAS A FULL COUPLED TWO-FACE RESPONSE
000002 | 30 October 2026. Source-bound analytical continuation and direct v252 control.
000003 | Verdict: ESTABLISHED_SCOPED. No original, ledger or Atlas change.
000004 | 
000005 | 1. Actual family; distinguish reuse from the new calculation
000006 | Use the actual full96 coefficient of verification/v252_full_finite_triple.py:
000007 | full_D:102-116, grading:119-121, rep:139-168, control inputs:202-212.
000008 | The inner-one-form prescription is printed in v254_inner_fluctuations.py:6-12.
000009 | The explicit Higgs witness is already proved in original_insertions/PROOF.txt:
000010 | 29-49 and seam-joint-source-origin-audit-20261016/HIGGS_VARIATION_PROOF.txt:
000011 | 135-182. Its existence is reused, not a new origin result.
000012 | 
000013 | Set a=pi(0,I2,0), b=pi(1,0,0), A=a[D_F,b]. The actual real selfadjoint
000014 | completion A+A*+J_F(A+A*)J_F^-1 is D_Y=full_D(Yu,Yd,Ye,Ynu,0).
000015 | Consequently the existing real radial Higgs direction gives
000016 |   Phi_s=D_F+s D_Y=(1+s)D_Y+D_MR.                         (1)
000017 | All four Yukawa matrices are still the original supplied coefficient inputs;
000018 | the Majorana block is retained unchanged. No new field family is introduced.
000019 | 
000020 | The NEW calculation is the response of BOTH faces and their crossing block
000021 | along (1), at the already coupled background D_F, including its Majorana
000022 | coupling. It does not replace D_F by zero to obtain a commuting derivative.
000023 | 
000024 | 2. Whole source, actual domains and noncommutative derivative
000025 | Reuse graded_boundary_functor/FINITE-COUPLED.txt:10-68 and the existing
000026 | pfaff_source_review/NORMAL-KINETIC-PAIRING.txt:sections1-4. In the constant
000027 | common-quarter product chart, p=n+alpha, alpha=1/4 or3/4, let ell>0 and put
000028 |   h_s=p gamma_F-i gamma_F Phi_s,
000029 |   V=-i gamma_F D_Y, T_s=p^2+Phi_s^2,
000030 |   h_s^2=T_s, dot(T)_0={D_F,D_Y}=:Q.
000031 | The finite coefficient is bounded. Thus the original H1 operator domain is
000032 | unchanged, and both total Clifford grades and the original adjoint slots
000033 | remain in the full T_faces chart. No finite carrier or chirality is removed.
000034 | 
000035 | The given charged two-face covariance is the ENTIRE projector
000036 |   G_s=[[a_s,X_s],[X_s,1-a_s]],
000037 |   a_s=(I+tanh(ell h_s))/2, X_s=(2cosh(ell h_s))^-1.
000038 | The conjugate block is I-bar(G_s), as already proved for T_faces.
000039 | Define, for u>0,
000040 |   q(u)=tanh(ell sqrt(u))/sqrt(u),
000041 |   x(u)=1/(2cosh(ell sqrt(u))).
000042 | At s=0 the exact whole response is
000043 |   dot(a)=1/2 [V q(T)+h Dq_T(Q)],
000044 |   dot(X)=Dx_T(Q),
000045 |   dot(G)=[[dot(a),dot(X)],[dot(X),-dot(a)]].              (2)
000046 | Here h=h_0 and T=T_0. In an eigenbasis of T,
000047 |   (Df_T(Q))_ij=f[1](t_i,t_j) Q_ij,
000048 | where f[1](u,v)=(f(u)-f(v))/(u-v) and f[1](u,u)=f'(u).
000049 | It is generally WRONG to replace Df_T(Q) by f'(T)Q.
000050 | 
000051 | The first diagonal response is nonzero whenever D_Y is nonzero: equivalently
000052 | dot(a)=D(tanh(ell h))[V]/2. Every divided difference of tanh(ell u) on the
000053 | real spectrum, including its diagonal derivative, is strictly positive for
000054 | finite u and ell>0. This finite-matrix Frechet map is injective. This proves