TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE46-GRAM-RESPONSE-AND-VADM-REVIEW.txt
Original file SHA-256: 6df3e624f781d6884de0c8c2c2ad09cd9f0aa3572048e9057ff2a4f16a89dbed
Inclusive original lines: 1-150
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 | [OMITTED: non-scientific conversation or execution metadata]
000002 | [OMITTED: non-scientific conversation or execution metadata]
000003 | 
000004 | Result
000005 | 
000006 | [OMITTED: non-scientific conversation or execution metadata]
000007 | [OMITTED: non-scientific conversation or execution metadata]
000008 | [OMITTED: non-scientific conversation or execution metadata]
000009 | [OMITTED: non-scientific conversation or execution metadata]
000010 | [OMITTED: non-scientific conversation or execution metadata]
000011 | [OMITTED: non-scientific conversation or execution metadata]
000012 | [OMITTED: non-scientific conversation or execution metadata]
000013 | [OMITTED: non-scientific conversation or execution metadata]
000014 | [OMITTED: non-scientific conversation or execution metadata]
000015 | [OMITTED: non-scientific conversation or execution metadata]
000016 | [OMITTED: non-scientific conversation or execution metadata]
000017 | 
000018 | The original EM5 contract permits an unspecified V_adm satisfying two
000019 | conditions.  It does not set V_adm=-kappa M0P0.  The condition
000020 | [Pstr,V_adm]=0 has not been checked because Pstr=P_Seam,+ P_sing P_Theta is
000021 | not the positive C6 branch selector and none of these projectors is P0.  The
000022 | existing spine/fold report explicitly leaves the coefficient-fiber pair
000023 | distinct from the physical F5 face pair.  Hence neither the original source
000024 | choice nor structural admissibility follows from the Gram response alone.
000025 | 
000026 | 
000027 | 1. Original contract
000028 | 
000029 | Original EM5, 005-source.tex:42-94, states
000030 | 
000031 |  D_f=D_0+V_adm,
000032 |  [Pstr,V_adm]=0,
000033 |  ||V_adm||<epsilon_f/2,
000034 |  epsilon_f=2/3-delta_ph.
000035 | 
000036 | EM3:91-97 defines
000037 | 
000038 |  Pstr=P_Seam,+ P_sing P_Theta,
000039 |  P_Seam,+=(1+iC)/2,
000040 |  P_Theta=(1+Q_Theta)/2,
000041 |  P_sing=(1+Z_c+Z_c^2)/3.
000042 | 
000043 | The source gives a class of gap-stable perturbations and proves positivity
000044 | for every member of that class.  It supplies no formula identifying the
000045 | new Hardy matrix moment with V_adm.  In particular, the positive/negative
000046 | three-dimensional C6 blocks of diag(M0,-M0) are not the P_Seam eigenspaces.
000047 | The statement that a block-supported matrix commutes with a block selector
000048 | therefore does not verify the printed commutator condition.
000049 | 
000050 | The marked family endpoint P0 is likewise not Pstr, P_Theta or P_sing.  Even
000051 | the forbidden substitution Pstr=P0 would fail:
000052 | 
000053 |  ||[P0,M0P0]||=1.                                      (1)
000054 | 
000055 | Thus the commutation condition must be checked only after an actual original
000056 | map into the Pstr Hilbert space has been exhibited.  P0=P_Theta must not be
000057 | inserted by notation.
000058 | 
000059 | 
000060 | 2. Exact spectrum and Fréchet derivative
000061 | 
000062 | Set
000063 | 
000064 |  y=2/3,
000065 |  delta=0.593277280133224001394066545420926568...,
000066 |  g=y-delta=0.07338938653344262...,
000067 |  D0=yI-delta M0,
000068 |  H0=D0*D0.
000069 | 
000070 | The spectrum is
000071 | 
000072 |  spec(H0)={g^2,h,h},
000073 |  g^2=0.005386002055755119...,
000074 |  h=1.191940562322203... .
000075 | 
000076 | Let f=(1,-1,1)/sqrt3, so M0f=f and |f_0|^2=1/3.  For the declared
000077 | direction V=-z M0P0, the derivative of K=(D*D)^(-1/2) is obtained from the
000078 | standard divided differences of x^(-1/2).  On the fixed line it gives
000079 | 
000080 |  <f,delta K_Re f>=1/(3g^2)=61.888824007625...,
000081 |  <f,delta K_Im f>=0.                                    (2)
000082 | 
000083 | [OMITTED: non-scientific conversation or execution metadata]
000084 | [OMITTED: non-scientific conversation or execution metadata]
000085 | 
000086 |  real direction:
000087 |    Frobenius norm 62.3267060810,
000088 |    operator norm  62.3198583277,
000089 |    rank3,
000090 |    fixed-line amplitude fraction 0.9929744069;
000091 | 
000092 |  imaginary direction:
000093 |    Frobenius norm 8.24850950745,
000094 |    operator norm  5.83257700740,
000095 |    rank2,
000096 |    fixed-line coefficient zero.
000097 | 
000098 | Therefore 62.33 and 8.25 reproduce only when double bars are interpreted as
000099 | Frobenius norms.  The 99.3 percent statement is an amplitude ratio for the
000100 | artificially real direction; its squared Frobenius share is 98.60 percent.
000101 | 
000102 | [OMITTED: non-scientific conversation or execution metadata]
000103 | 
000104 |  delta K=Re(kappa)/(3g^2) ff*+O(1)
000105 | 
000106 | has the wrong remainder.  Put Q=ff* and R=I-Q.  The divided difference of
000107 | x^(-1/2) between the two Gram eigenvalues is
000108 | 
000109 |  c(g)=[1/g-1/sqrt(h)]/[g^2-h]
000110 |      =-1/(3y^2 g)+O(1),
000111 |  h=y^2+(y-g)^2+y(y-g).
000112 | 
000113 | For V=-kappa M0P0 one has
000114 | 
000115 |  Q delta H R
000116 |    =-y bar(kappa) QP0(M0*-I)R+O(g),
000117 |  ||QP0(M0*-I)R||=sqrt(2/3).
000118 | 
000119 | Consequently, at y=2/3,
000120 | 
000121 |  ||Q delta K R+R delta K Q||_op
000122 |       =|kappa|/[sqrt(6)g]+O(1),
000123 |  ||Q delta K R+R delta K Q||_F
000124 |       =|kappa|/[sqrt(3)g]+O(1).                         (3)
000125 | 
000126 | The fixed-line identity Re(kappa)/(3g^2)Q is exact.  For nonzero
000127 | Re(kappa) it remains the leading singularity, with (3) as the subleading
000128 | term.  For purely imaginary kappa, (3) is the leading divergence.  Thus the
000129 | imaginary response is not uniformly O(1) as the gap closes.  A direct replay
000130 | at g=10^(-4) gives g times the cross operator norm 0.40827417 and g times its
000131 | Frobenius norm 0.57738688, converging to 1/sqrt6 and 1/sqrt3.
000132 | 
000133 | 
000134 | 3. The actual complex Hardy direction
000135 | 
000136 | HARDY-BMU-FIRST-NONZERO-ENTRY.txt fixes
000137 | 
000138 |  kappa=(4X_*/pi) I,
000139 |  X_*=[2cosh(ell_NS/2)]^(-1),
000140 |  arg I=-5pi/12.
000141 | 
000142 | For the unit direction z=exp(-5pi i/12), real linearity of the Fréchet derivative
000143 | gives
000144 | 
000145 |  ||delta K||_F=17.9916735432,
000146 |  ||delta K||_op=17.8809143219,
000147 |  rank(delta K)=3,
000148 |  |<f,delta K f>|=16.0180063322.
000149 | 
000150 | The fixed-line Frobenius amplitude fraction is 0.8903010770 and its squared