TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
Original file SHA-256: 3537a2b080df0bb8b5d30c73da7dd83dc8e1ae5a6cfea4bb48a8b99f0080dfce
Inclusive original lines: 571-698
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.

000571 | 12. Exact non-Gaussian functional reconstruction from a complete CAR kernel
000572 | 
000573 | 12.1 What is an input, and what is reconstructed
000574 | 
000575 | P1 postulates a structured boundary kernel, not merely the number c3. We do not
000576 | replace that object by its normalization. Its active declaration at architecture
000577 | lines163-189 does not, however, define its arguments as the physical charged
000578 | CAR field variables used below. This field dictionary is an explicit premise.
000579 | The first field-dependent integral at original04:2160-2183 writes Dphi and
000580 | S_adm[phi] symbolically; its second and higher variations at2272-2302 are inputs
000581 | to the vertex expansion, not an explicit construction of that action.
000582 | 
000583 | The following is standard finite coherent-state reconstruction, implemented on
000584 | the existing search/source path. It is not a new fundamental principle. Given
000585 | an n-mode CAR dictionary and the COMPLETE parity-even matrix T in its labeled
000586 | Fock basis (n=1..5 here), it produces an exact finite polynomial field kernel,
000587 | its gluing pairing and, where defined, its full non-Gaussian link action. No
000588 | independent Gaussian/Wick or linear creator-covariance equation is imposed.
000589 | Spatial locality, Lorentz covariance and identification with the geometric P1
000590 | kernel are NOT inferred from this matrix representation.
000591 | 
000592 | 12.2 Kernel, inverse and the unique CAR gluing functional
000593 | 
000594 | Order all Grassmann bars before the unbarred variables. For occupation subsets
000595 | I,J define B_I=(-1)^(|I|(|I|-1)/2) barz_I and Z_J=z_J, with increasing indices.
000596 | Then
000597 |   K_T(barz,z)=sum_(I,J) B_I T_IJ Z_J,
000598 |   K_identity=exp(sum_j barz_j z_j).
000599 | Every coefficient recovers precisely one matrix entry, so the map is invertible.
000600 | For parity-even operators, direct Grassmann integration gives
000601 |   K_(T2 T1)(barx,y)
000602 |     = integral K_T2(barx,w) exp(-barw w) K_T1(barw,y) d(barw,w).
000603 | Our orientation is integral exp(-barw w)=1. Equivalently the Gaussian-weighted
000604 | gluing functional L obeys L(Z_J B_I)=delta_JI. The4^n products Z_J B_I are
000605 | nonzero signed distinct monomials and span the entire2n-generator Grassmann
000606 | algebra. Thus these equations uniquely determine L. This is a derivation of
000607 | the finite gluing functional from the CAR dictionary and identity/composition
000608 | requirement, NOT a derivation of the physical CAR dictionary from P1/P2.
000609 | 
000610 | Ordinary closure is
000611 |   Tr T=integral exp(-barz z) K_T(-barz,z) d(barz,z).
000612 | The minus sign is essential: omitting it gives the graded trace. An inserted
000613 | operator W is kept explicitly as K_(T W), rather than replaced by a different
000614 | commuting matrix. Unnormalized traces and scalar factors are retained.
000615 | 
000616 | 12.3 One complete link action, including higher vertices
000617 | 
000618 | Let lambda0=K_T(0,0)=T00 be nonzero. Set N=K_T/lambda0-1. The finite nilpotent
000619 | series
000620 |   S_nil=-log(1+N)=-sum_(k>=1) (-1)^(k+1) N^k/k
000621 | terminates and satisfies K_T=lambda0 exp(-S_nil) EXACTLY. This reconstructs all
000622 | vertices, including degrees4,6,...,2n. Lambda0 remains an explicit scalar; no
000623 | logarithm branch for it is selected. Importantly lambda0 is the EMPTY coherent
000624 | vacuum entry, not generally the reference eigenvalue in T Omega=lambda_ref Omega.
000625 | If T00=0 this logarithm chart fails; the kernel polynomial can still be used.
000626 | Conversely T can be singular with T00 nonzero, so this finite polynomial log
000627 | can exist even when a finite Hamiltonian -log(T)/ell does not exist.
000628 | The Grassmann link log and the matrix operator log are distinct operations.
000629 | 
000630 | For a specified closed sequence of links T_l, the product of their K_Tl kernels
000631 | with the above Gaussian gluing weights is an exact finite field functional for
000632 | Tr(T_L...T_1). Every matrix field insertion and chronological product is then
000633 | computed in the same fixed CAR dictionary. The chosen trace/initial boundary
000634 | condition remains part of the problem; an arbitrary physical initial state is
000635 | not inferred from T alone. A family T[Phi] gives a family S_nil[Phi] by the same
000636 | procedure, but the actual original Phi-dependence is not generated by a change
000637 | of representation. No interacting 1PI inverse is equated to a bare kinetic term.
000638 | 
000639 | 12.4 Interacting control retains actual quarter time and the absolute sea term
000640 | 
000641 | Reuse the supplied lowest charged particle/hole pair and ell=8log2. Its free
000642 | transfer is diag(1,1/4,1/64,1/256). For an explicitly supplied interacting control
000643 | replace only the pair entry by1/65536. This yields
000644 |   H_int=diag(0,1/4,3/4,2)=H_free+N_p N_h,
000645 |   exp(2pi i H_int)=diag(1,i,-i,1).
000646 | Thus this control verifies the actual time-loop equation, not merely [T,W]=0.
000647 | Its reconstructed S_nil has quartic coefficient -255/65536 in bars-first order;
000648 | ordinary trace is82945/65536. Its vacuum and one-particle columns coincide with
000649 | the free source; their full kernels and pair responses do not coincide.
000650 | The entire prescribed sea factor lambda_sea=2^(1/12) is retained in the kernel
000651 | body. It leaves S_nil unchanged but changes the absolute amplitude. With
000652 | H_tot=H_int-I/96 the full loop is exp(-i pi/48) diag(1,i,-i,1).
000653 | The separate angular zeta determinant1-i is not identified with this transfer.
000654 | This finite control is NOT a spatially local source selection or a P1/P2 model.
000655 | 
000656 | 12.5 Existing search outputs now expose their actual full field actions
000657 | 
000658 | The CLI consumes the existing source-clustering min/max finite-completion files.
000659 | It safely parses rational entries, independently checks diagonal Hermitian PSD,
000660 | matches the stored full matrix, and verifies exact kernel/action roundtrips.
000661 | It does not trust the stored success flags alone. The adapter is deliberately
000662 | limited to these diagonal five-mode certificates; other matrices are not
000663 | silently certified. Reference occupations and realization assumptions persist.
000664 | 
000665 | Both have Omega=|31>, lambda_ref=1, and action degree10, but T00 is1/4 versus3/4.
000666 | Ordinary traces are35/4 and87/4. Their ACTUAL internal W=i^N has traces
000667 |   min: -1-i/4, max: -11/2-11i/4.
000668 | The normalized GNS R=i^(N-N_Omega) instead has traces
000669 |   min: -1/4+i, max: -11/4+11i/2.
000670 | They are not replaced by alternating particle/hole charges just because the
000671 | diagonal matrices commute with both. Neither internal action proves a physical
000672 | quarter-time equation for these search outputs, which supply no physical ell.
000673 | 
000674 | The min-trace certificate simplifies exactly. Put B=sum barz_j z_j and let M
000675 | be the product of all five bars followed by all five fields. Then
000676 |   T_min=(I+3|31><31|)/4,
000677 |   K_min=exp(B)/4+3M/4,
000678 |   lambda0=1/4, S_nil=-B-3M.
000679 | The degree10 term exposes the historical extremal-search construction. It is
000680 | not an independently selected TFPT interaction, nor evidence that degree10 or
000681 | five-body support is physically required. Current feasibility defaults remain.
000682 | 
000683 | 12.6 Consequence and independent verification
000684 | 
000685 | This removes an unnecessary separation between kernel, finite CAR gluing and
000686 | full link vertices: a COMPLETE kernel in the actual field dictionary supplies
000687 | them together. The physical dictionary and original field-dependent kernel
000688 | must still be derived; P2's internal16 is not silently replaced by this5-mode
000689 | Fock space. Bare Berezin orientation is a representation convention, not a
000690 | separate physical probability measure generated by numerical dimension matching.
000691 | No new physical gate is closed. The existing scalar/profile, geometry and CAR
000692 | identification conditions in sections9-11 remain in force.
000693 | 
000694 | Independent review checked all1364 duality values for n=1..5,83 random even
000695 | matrix-pair roundtrips/compositions/traces and the10 targeted kernel tests.
000696 | The combined new/existing source run passed20 targeted tests. Primary coherent
000697 | state background: Ohnuki/Kashiwa, Progress of Theoretical Physics60(1978)548,
000698 | https://doi.org/10.1143/PTP.60.548; the finite identities above are proved directly.