TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/GEOMETRIC-SOURCE-RETURN.txt
Original file SHA-256: 61d8e5b176353204958f6a6851c108471cb4e5c2846a39893485231c900bb71a
Inclusive original lines: 1-115
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 | GEOMETRIC SOURCE RETURN FROM THE EXISTING PIN DIRAC OPERATOR
000002 | 26 October2026. Explicitly authorized source-first continuation.
000003 | No primitive physical gate or full TFPT completion is claimed.
000004 | 
000005 | 1. Original comparison object and the pure-source bridge
000006 | Use exactly D0=-i(sigma1 partial_x+sigma2 partial_w), w in[-1,1],
000007 | psi(x+1,-w)=c sigma1 psi(x,w), c=+i or-i, and sigma1 psi=b psi
000008 | at w=+/-1, b=+1 or-1. These are the original source/domain conventions
000009 | already reviewed in NS-COVER-FULL-DIRAC-RP.txt and PIN-WINDING-DESCENT.txt.
000010 | Keep its pure D-sea CAR quantization SEPARATE from the free Green
000011 | quantization with x as Euclidean time and spatial H=i sigma3 partial_w.
000012 | 
000013 | The existing normal zero vector is phi0=(1,b)/2. On the same unit-length
000014 | Pin fundamental region, V f=phi0 f is an isometry from the circle domain
000015 | f(x+1)=q f(x), q=c*b. Direct differentiation gives
000016 |     D0 V=V B_b, B_b=b(-i partial_x).
000017 | The same holds with the uniform normal coefficient a>0 in front of
000018 | sigma2 partial_w. The normal zero sector is reducing, so spectral calculus
000019 | gives 1_(D0<0)V=V 1_(B_b<0). Therefore a(f)->a_D(V f) is a CAR embedding
000020 | preserving the given pure quasifree state and its B_b/D0 time on all words.
000021 | Theta_x=K R_x satisfies Theta_x V=V theta_x and preserves the same q domain.
000022 | For b=-1 the orientation B_b is essential, not an optional sign change.
000023 | If working on the length2 orientation cover instead, include its actual
000024 | fundamental-region norm factor; no second independently occupied species
000025 | is introduced. This retains and explicitly composes the known zero branch.
000026 | It is NOT an identification of the full D sea with a one-dimensional sea.
000027 | For theta=2pi*x the generator is B_b=2pi*b*(-i partial_theta); transport
000028 | this actual unit conversion,not an extra physical Clock identification.
000029 | 
000030 | V maps longitudinal support I to I times the WHOLE normal strip. Its normal
000031 | projector P0=VV* is not a projection onto an arbitrarily thin local collar.
000032 | For Q=I-P0, <phi0,w phi0>=0 and ||Q w phi0||^2=1/3. Thus this spectral
000033 | subnet is not automatically the P1 admissibility/local seam map.
000034 | P0 is also NOT the native quartic five-code projector.
000035 | For the unvaried or uniformly rescaled comparison, Q D_a P0=0 exactly.
000036 | The existing reducing circle alone supplies no exterior-memory coupling.
000037 | 
000038 | 2. Actual conformal geometry supplies the missing KIND of coupling
000039 | Let h(w) be smooth, real and even. This is a declared geometric probe,
000040 | not a geometry/history selected by P1. Use g_e=exp(2e h)g0. Standard Dirac
000041 | conformal covariance, transported unitarily to the original flat L2 density,
000042 | gives exactly
000043 |     D_e=exp(-e h/2) D0 exp(-e h/2).
000044 | Reference: Ginoux, The Dirac Spectrum, Proposition1.3.10,
000045 | https://nicolas-ginoux.perso.math.cnrs.fr/spectrumDirac6309.pdf .
000046 | Even h preserves the original Pin relation. Scalar multiplication preserves
000047 | the b boundary lines. The common elliptic first-order domain is unchanged.
000048 | The numerically preferred probe below is zero near the known physical
000049 | boundaries AND w=0, retaining their original product collars. Identifying
000050 | these with any abstract P1 seam still requires the actual geometric map.
000051 | 
000052 | The first derivative is fixed, V_h=-(h D0+D0 h)/2, with NO learning rate.
000053 | For psi=phi0 exp(ikx), phi_perp=(1,-b)/2, and
000054 | <h>=(1/2) integral_-1^1 h(w)dw, direct differentiation gives
000055 |     V_h psi=b[-k h phi0+(h_prime/2)phi_perp]exp(ikx),
000056 |     P0 V_h psi=-b k<h> psi,
000057 |     Q V_h psi=b[-k(h-<h>)phi0+(h_prime/2)phi_perp]exp(ikx).
000058 | The two spinor directions are pointwise orthogonal and both normalized
000059 | in the original normal L2 measure. Hence the ALL-PROFILE identity is
000060 |     ||Q V_h psi||^2=k^2 Var(h)+(1/4)<h_prime^2>.                 (1)
000061 | It vanishes exactly for constant h. The polarized form is
000062 | k^2 Cov(h,g)+(1/4)<h_prime g_prime>. This is a source-derived first
000063 | return form, not a primitive spectral-action Hessian or selection law.
000064 | 
000065 | For h=w^2, (1) is 4k^2/45+1/3. In the actual Pin sector of phi0 the allowed
000066 | transverse indices are the even H indices, conveniently m inZ with energy
000067 | pi*m. The complete exterior amplitudes for m!=0 are
000068 |     -2b(-1)^m(k+i*pi*m/2)/(pi^2*m^2).
000069 | Their full squared-norm sum is
000070 | (4/pi^4)[2k^2 zeta(4)+(pi^2/2)zeta(2)]=4k^2/45+1/3,
000071 | independent of any finite normal cutoff. For a general even cosine profile
000072 | h=a0+sum_m>=1 a_m cos(pi*m*w), the sum is
000073 |     sum_m>=1 a_m^2[k^2/2+pi^2*m^2/8].
000074 | These second responses are quadratic forms on first-order-domain inputs.
000075 | No unproved assertion that psi lies in Dom(D_e^2), or that V_h^2 psi is
000076 | a defined boundary-domain vector, is used. Source propagation below uses
000077 | bounded unitary operators, for which the exact memory elimination is valid.
000078 | 
000079 | 3. Actual source evolution and derived memory, not a fitted recurrence
000080 | The controlled run uses b=+1,c=+i,q=i,k=pi/2. Its smooth even profile is
000081 | h(w)=exp(1-1/(1-r^2)) for r=(|w|-.5)/.25, |r|<1, zero otherwise.
000082 | The chosen location,width and e=.1 are diagnostic inputs, NOT TFPT outputs.
000083 | For the actual normal basis phi_m=(exp(-i*pi*m*w),b exp(i*pi*m*w))/2,
000084 | integration of the differential expression and integration by parts give
000085 |     (D_e)_(mn)=b[k+i*pi*(m-n)/2] A_(m+n),
000086 |     A_j=integral_0^1 exp(-e h(w))cos(pi*j*w)dw.                 (2)
000087 | Boundary terms vanish since sin(pi*j)=0. Formula(2) is Hermitian and is the
000088 | actual Galerkin matrix, not a learned coupling or product-truncation ansatz.
000089 | At e=0 it is b(k-i*pi*n)delta_(m,-n). The original negative D0 projection
000090 | C0 is used as initial covariance. Evolution gives U* C0 U on test functions,
000091 | so all free CAR word responses follow by Wick from the SAME initial state.
000092 | No claim of a common infinite-dimensional Fock implementer for a changed
000093 | metric is required; bounded CAR automorphisms and the finite computations
000094 | are the stated objects. No intermediate normal/code projection is executed.
000095 | 
000096 | For the bounded actual sampled step U=exp(i dt D_e), dt=.01, put
000097 |  K=P0 U P0, B=Q U P0, C=P0 U Q, E=Q U Q.
000098 | For initial exterior component0, exact block elimination gives
000099 |  x[n+1]=K x[n]+sum_(j=0)^(n-1) C E^(n-1-j) B x[j].            (3)
000100 | This holds for the full Hilbert-space unitary as well as its Galerkin runs.
000101 | Since U is unitary and NOT selfadjoint, C must NOT be replaced by B*.
000102 | The earlier positive L=S* S source-response identity is a different object.
000103 | 
000104 | 33,65,129 and257 normal modes were run400 actual steps, through t=4 in
000105 | the source time convention. With257 modes the reduced memory recurrence
000106 | and full propagation differ by at most2.11e-13 in amplitude. Replacing C
000107 | by B* produces error.0208424 and is a firing negative control. Norm error
000108 | is below3e-15. The129/257-mode curves differ by less than3e-10 in normal
000109 | survival and2e-10 in the actual initial-sea test-mode occupancy.
000110 | The largest tested normal excursion is.00262562, occupancy.000260247.
000111 | For this given profile the full first-return form is.03132457684635;
000112 | 257 modes retain.03132456291111. The residual tail is reported, not hidden.
000113 | The e=0 control has no excursion or occupancy beyond numerical roundoff.
000114 | No claim of a proved infinite-cutoff long-time convergence rate is made.
000115 |