TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/GEOMETRIC-SECTOR-RESPONSE.txt
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000001 | GEOMETRIC RESPONSE OF THE RETAINED SEAM DEFECT, WITH CAUSAL MEMORY
000002 | 26 October 2026. Existing original-source contract, experiment-only PARTIAL.
000003 | No physical gate or original ledger claim is promoted.
000004 | 
000005 | 1. Question, actual source and decisive test
000006 | 
000007 | Can the fixed scalar seam factor determine a geometric response of the
000008 | retained source? Reuse the existing pair-index theorem, not another model.
000009 | For C-P trace class and the actual restricted unitary U,
000010 |  det_F[(I-zUCU*)(I-zC)^-1]=(1-z)^k, k=Tr(UPU*-P).
000011 | At fixed z and fixed k this functional has ZERO derivative on trace-class
000012 | covariance directions. It does not determine their other field responses.
000013 | Reference: PAIR-INDEX-DETERMINANT.txt, sections1-3, already proved and reviewed.
000014 | Index background: Avron,Seiler,Simon, JFA120(1994),220-237,
000015 | doi:10.1006/jfan.1994.1031. No new general index theorem is claimed.
000016 | 
000017 | Test the actual supplied Pin Dirac family from GEOMETRIC-SOURCE-RETURN.txt:
000018 |  D0=-i(sigma1 partial_x+sigma2 partial_w), w in[-1,1], b=+1,c=+i,
000019 |  q_Pin=i, k=pi/2 and all compatible normal modes. Keep the original pure
000020 | negative D0 sea C0. The old normal zero branch V f=(1,1)f/2 transports
000021 | the old quarter-circle CAR and time into this source. Its support is the
000022 | WHOLE normal strip, not a proved thin P1 seam image.
000023 | The old canonical positive defect Pi=|psi0><psi0| gives C1=C0+Pi.
000024 | Pi is the original extra particle from DIRECT-DEFECT-CLOSURE.txt, sections1-3,
000025 | transported along the established normal-zero embedding. No new state is fitted.
000026 | 
000027 | Use the SAME earlier smooth even diagnostic profile h, zero near w=0 and
000028 | w=+/-1, and D_s=exp(-s h/2)D0 exp(-s h/2). Profile and history are chosen
000029 | probes, not selected TFPT histories. The known Pin/b boundary domain remains.
000030 | Success: derive the relative geometric response from this source and match
000031 | the direct full evolution, including its first parameter derivative.
000032 | Stop criterion: do not identify this response with primitive P1 dynamics
000033 | without the actual admissible insertion, state, time and domain map.
000034 | 
000035 | 2. Same-history relative response, without changing the original sea
000036 | 
000037 | For a specified constant intervention s switched on at t0, evolve BOTH initial
000038 | covariances with exp(-itD_s). Their exact difference is
000039 |  DeltaC_s(t)=exp(-itD_s) Pi exp(itD_s)=|v_s(t)><v_s(t)|.
000040 | Thus the relative geometric readout is the rank-one form
000041 |  R_h(s,t)=Tr[DeltaC_s(t) partial_sD_s]
000042 |          =<v_s(t),partial_sD_s v_s(t)>.                         (1)
000043 | In finite Galerkin dimension it is the difference of the same neutral
000044 | quadratic observable in the two evolved CAR states. Its continuum relative
000045 | form is well-defined on the stated smooth vectors; no difference of undefined
000046 | individual infinite energy expectations is used. Bounded CAR evolution needs
000047 | no assertion of a common infinite-Fock implementer for a changed metric.
000048 | It is NOT the primitive spectral-action derivative by definition.
000049 | 
000050 | For this ACTUAL same-history pair the scalar invariance needs no additional
000051 | assumption that the evolved sea minus its original sea is trace class.
000052 | Write P_t=U_t C0 U_t*, Q_t=P_t+Pi_t, Pi_t=U_t Pi U_t*. Then P_t Pi_t=0
000053 | and Pi_t is still an orthogonal rank-one projection. Exactly
000054 |  (I-zQ_t)(I-zP_t)^-1=I-zPi_t, det_F=1-z.                       (1a)
000055 | This holds for every unitary source propagator, including the tested pulse,
000056 | in the full source Hilbert space. It is a transported pair of projections,
000057 | not a cyclic finite-mode shift incorrectly assigned nonzero index.
000058 | 
000059 | Let V_h=partial_sD_s|0=-{h,D0}/2. Direct calculation gives
000060 |  <psi0,V_h psi0>=-k<h>,
000061 |  <psi0,partial_s^2D_s|0 psi0>=k<h^2>,                  (2)
000062 | where <h>=integral_0^1 h(w)dw is the original even normal average.
000063 | For mixed scalar profiles h,g the second expectation is k<hg>.
000064 | The geometric readout itself varies: omitting this explicit contact term
000065 | would miss part of the response even before any primitive identification.
000066 | 
000067 | 3. Complete relative causal kernel
000068 | 
000069 | For coupling +epsilon V_g define the sign convention
000070 |  chi_hg(t)=-i theta(t)<psi0,[V_h(t),V_g]psi0>,
000071 |  V_h(t)=exp(itD0)V_h exp(-itD0).
000072 | This is the finite-rank relative Kubo form: normal-ordering central terms
000073 | cancel between the two states. Do not infer the existence of every separate
000074 | infinite quadratic operator on the vacuum from this relative form.
000075 | Background: Kubo1957, JPSJ12,570-586, doi:10.1143/JPSJ.12.570.
000076 | For a step profile the full first derivative of(1) is
000077 |  partial_s R_h(s,t)|0=k<h^2>+integral_0^t chi_hh(u)du.           (3)
000078 | More generally an infinitesimal prescribed history a(t) gives the contact
000079 | a(t)k<hg> and convolution integral_0^t chi_hg(t-u)a(u)du.
000080 | This is a response to a given history, not a new autonomous metric law.
000081 | 
000082 | Expand h=h0+sum_(m>=1)h_m cos(pi*m*w), likewise g. Put a_m=pi*m,
000083 |  E_m=sqrt(k^2+a_m^2). In the ordered normal pair(+m,-m),
000084 |  D_m=[[0,k+i a_m],[k-i a_m,0]],
000085 |  V_h psi0=-(h_m/2)(k+i a_m/2,k-i a_m/2)^T.
000086 | The exact positive and negative spectral weights divided by h_m^2 are
000087 |  w_m^+=(E_m+k)^3/(16E_m), w_m^-=(E_m-k)^3/(16E_m).
000088 | These sum to k^2/2+a_m^2/8. Therefore
000089 |  chi_hg(t)=2 theta(t) sum_m h_m g_m[
000090 |    w_m^+ sin((k-E_m)t)+w_m^- sin((k+E_m)t)].                    (4)
000091 | No fitted frequencies or rates enter(4). It is the actual source spectrum.
000092 | For smooth profiles the sum converges absolutely; the absolute kernel is
000093 | bounded by2 sum|h_m g_m|(k^2/2+a_m^2/8), finite already for H1 profiles.
000094 | For the stated smooth profiles termwise differentiation at zero gives
000095 |  chi_hg(0)=0, chi_hg'(0)=-k<h_prime g_prime>.                  (5)
000096 | The constant Fourier component changes(2) and the contact, but contributes
000097 | no commutator kernel. Both contributions are retained in(3).
000098 | 
000099 | 4. Explicit neutral sea response and its circle trace-class direction
000100 | 
000101 | For k>0, the original sea occupation of the normal-zero test mode after
000102 | the step intervention is zero at s0 and has coefficient
000103 |  <psi0,exp(-itD_s)C0 exp(itD_s)psi0>
000104 |   =s^2 d_k(t)+O(s^3),
000105 |  d_k(t)=4 sum_m h_m^2 A_m(k)sin^2((E_m+k)t/2),
000106 |  A_m(k)=w_m^-/(E_m+k)^2
000107 |        =a_m^6/[16E_m(E_m+k)^5].                              (6)
000108 | For the opposite sign of longitudinal momentum, the covariance change has
000109 | the opposite sign and the same formula in |k|. The first compressed
000110 | covariance derivative is zero: a projection derivative has no diagonal
000111 | matrix element in its original occupied/empty vector.
000112 | For the actual quarter-circle momenta k_n=2pi(n+1/4),
000113 |  |d_(k_n)(t)| <= S6/[128|k_n|^6], S6=sum_m h_m^2(pi*m)^6<infinity.
000114 | The computed second covariance response is therefore trace class on the old
000115 | circle. The protected index functional annihilates this response direction.
000116 | This tangent statement does NOT prove trace-class differentiability or
000117 | membership for arbitrary finite-s full 2D covariances, and does not establish
000118 | arbitrary higher local-field/point-splitting regularity.
000119 | There is no contradiction between a protected scalar factor and nonzero
000120 | source field/geometry responses such as(1),(4),(6).
000121 | 
000122 | 5. What q does and does not encode
000123 | 
000124 | The original historical sector weight, including its reviewed normalization,
000125 | is Q_full(g,alpha)=exp(-alpha*d*s_index)R(g,alpha), d=2,s_index=1.
000126 | Here R=Z_def^0/Z_triv^0 includes all residual measure/reference contributions.
000127 | At fixed alpha: delta_g log Q_full=delta_g log R. The minimal expression
000128 | q_min=delta_top exp(-2alpha) agrees only under its retained residual contract.
000129 | Do not identify either parameter with the Pin phase or the elliptic nome.
000130 | For the fixed minimal transfer T(q)=(1-q)^(-5/4), delta log T=(5/4)delta q/(1-q).
000131 | The protected determinant does not itself calculate delta_g R.
000132 | The original complete-source Ward identity remains
000133 |  delta_g log R=-<delta_g S_def^0>_def+<score_def>_def
000134 |               +<delta_g S_triv^0>_triv-<score_triv>_triv.       (7)
000135 | This is the same REWEIGHTING-REVIEW.txt contract, now applied to geometry.
000136 | Equation(1) is a particular calculated relative CAR response. To substitute
000137 | it into(7), the primitive action/measure must actually induce that CAR
000138 | insertion with the same state, time, support and domains. No free Gaussian
000139 | action or thermal/heat-to-EM identification is introduced to make this true.
000140 | 
000141 | 6. Direct finite tests and a same-final-geometry history test
000142 | 
000143 | For the earlier collar-preserving bump and k=pi/2, t1:
000144 |  baseline=-k<h>=-.47394864832401096,
000145 |  contact=k<h^2>=+.38617274219097286,
000146 |  integral_0^1 chi=-.4054682615132567,
000147 |  complete derivative=-.01929551932228385.
000148 | The source occupation coefficient from(6) is.015177917523060008 at129 modes.
000149 | The independent direct matrix commutator differs from(4) by<3.6e-13.
000150 | Symmetric derivatives of the FULL conformal matrix family, not a linearized
000151 | replacement, converge quadratically for epsilon.02,.01,.005,.0025,.00125.
000152 | Richardson comparison with(3) has max error3.26e-8 over0..4. Omitting the
000153 | contact term fails with error.3861728. The coarser initial extrapolation
000154 | missed a factor100 accuracy target; the smaller interventions resolve it.
000155 | Normal and python -OO result bytes agree.
000156 | 
000157 | For a separate control apply the same probe at strength+.01 for50 steps,
000158 | then return EXACTLY to D0 for50 steps, dt=.01. Compare the original
000159 | undeformed history and a negative probe. At t1 all final operators are D0,
000160 | but the relative geometric readouts are respectively
000161 |  no pulse: -.4739486483240111,
000162 |  earlier positive pulse: -.47012535688664536,
000163 |  earlier negative pulse: -.47803970864576567.
000164 | The100-step state agrees with the full two-piece evolution to4.74e-14.
000165 | For the same full finite pair at z=1/3, direct matrix inversion gives
000166 | det=.6666666666666671 and the exact operator identity(1a) error4.45e-16.
000167 | Here z is a diagnostic determinant argument, not a fitted EM coupling.
000168 | The pulse linear derivative is predicted by integral_.5^1 chi=.3956885828;
000169 | the symmetric finite+.01/-.01 derivative is.3957175880.
000170 | The state retains its history after the geometry returns. Pulse amplitude
000171 | and duration are diagnostic inputs, not TFPT predictions. No state reset or
000172 | projection is performed during the pulse or return.
000173 | 
000174 | 7. Actual machine integration and scope
000175 | 
000176 | source_geometric_machine.py now computes(1) in its existing400-step actual
000177 | source path. No force is fed back as a new metric selection rule. At the
000178 | earlier diagnostic epsilon.1 the129/257 normal-mode readout curves differ
000179 | by<5.4e-7. Relative energy conservation error at257 modes is<4.7e-15.
000180 | Conserved energy and the partial operator-geometry readout(1) are different:
000181 | the total energy derivative along varied histories also varies the state.
000182 | Both source states evolve together; the original sea is not replaced by
000183 | the changed-metric ground state or the unrelated thermal Green state.
000184 | The original evidence snapshots remain intact; the extended machine and
000185 | new checker/results are separately pinned in geometric_sector_response/.
000186 | 
000187 | Exact(2)-(6) and their scoped convergence are source mathematical results;
000188 | long finite histories are numerical Galerkin tests. The actual original
000189 | P1 admissible insertion/measure and primitive-action identification remain
000190 | open. Native quartic/octic words, their adjoints, charges, support, time and
000191 | domains cannot be replaced by one scalar index or by this one bilinear test.
000192 | No local3+1D completion, chosen autonomous geometry or full TFPT solution is
000193 | claimed. The new concrete response is a necessary comparison only if this
000194 | probe belongs to the actual original insertion image.