TFPT scientific source excerpt Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/BEREZIN-INVERSE.txt Original file SHA-256: 8715fb914a5621c894463133820bcc84374fd6ee397cb6e9593cd9f13bce05a4 Inclusive original lines: 1-180 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 | FINITE ORIGINAL SOURCE: CONSTRUCTIVE BEREZIN INVERSE AND ADMISSIBLE TESTS 000002 | 3 November 2026. Readonly source review, workspace proof note. 000003 | 000004 | Result 000005 | 000006 | On an actually supplied finite Grassmann field chart, complementary source 000007 | insertions reconstruct every coefficient of the original unnormalized 000008 | density. A single saturated insertion extracts its zero fermion body, 000009 | including at zeros of the Pfaffian. Compact group invariance lets this 000010 | inverse use invariant composites without requiring individual charged 000011 | letters to be physical observables. Applicability to the full original 000012 | Padm source still requires the saturated composite's actual field/contact 000013 | representation and admissibility. Neither is inferred from its degree. 000014 | 000015 | 1. Exact coefficient inverse, including the sign 000016 | 000017 | Use the ORIGINAL finite regulator, ordered independent odd coordinates 000018 | theta_1,...,theta_N with N even, base integration dnu(b), and its given 000019 | orientation. Put tau=theta_1...theta_N and int Dtheta tau=1. Here b denotes 000020 | every remaining original variable, not a newly chosen boson background. 000021 | Let 000022 | 000023 | w(b,theta)=sum_(I subset {1,...,N}) w_I(b) theta^I, 000024 | L[A]=int dnu(b) int Dtheta A(b,theta) w(b,theta). (1) 000025 | 000026 | All words theta^I have increasing indices. If J=I^c, define 000027 | 000028 | sigma(I)=(-1)^(#{(j,i):j in J, i in I, j>i}). 000029 | 000030 | Thus theta^J theta^I=sigma(I)tau. Every other coefficient either has 000031 | a repeated index or fails to saturate all indices, so 000032 | 000033 | L[f(b) theta^(I^c)]=sigma(I) int dnu(b) f(b) w_I(b). (2) 000034 | 000035 | Since sigma(I)^2=1, the same sign inverts the formula. This identifies 000036 | the coefficient distributions against every actually allowed test f. 000037 | For an even density, only even I occur and all complementary insertions 000038 | are even. No odd observable access is required by this inverse. 000039 | 000040 | In particular 000041 | 000042 | tau w=tau w_empty, 000043 | L[f tau]=int dnu f w_empty. (3) 000044 | 000045 | This is an explicit inverse operation on the source. It does not infer 000046 | an unknown value of L[f tau] from lower moments or from reflection positivity. 000047 | 000048 | 2. The bare body and Pfaffian zeros 000049 | 000050 | On an actual chart with w=exp(-S_B) exp(-theta^T M_b theta/2), 000051 | 000052 | L[f]=int dnu f exp(-S_B) Pf(-M_b), 000053 | L[f tau]=int dnu f exp(-S_B). (4) 000054 | 000055 | The sign in the Pfaffian is fixed here by the displayed exponent and 000056 | orientation. A convention absorbing that sign into its M gives Pf(M). 000057 | Equation(3), and therefore body extraction, is unchanged. It requires 000058 | neither an inverse of M nor division by Pf(M), so it survives singular 000059 | fermion fibers. Higher even fermion interactions also leave(3) unchanged. 000060 | 000061 | If N counts only matter fermions, w_empty still contains the original 000062 | ghost and other odd variables. A pure boson density requires saturation 000063 | of every integrated odd direction at that integration stage. This is 000064 | not automatic in a physical BRST quotient. Conversely, ghosts already 000065 | eliminated in the original caller are not integrated a second time. 000066 | 000067 | On a fixed nonzero body branch, all coefficients from(2) give the action 000068 | by a finite algebraic logarithm. Write w=w_empty(1+n), with n nilpotent: 000069 | 000070 | S=-log(w_empty)-sum_(k=1)^N (-1)^(k+1)n^k/k. (5) 000071 | 000072 | Terms vanish once their Grassmann degree exceeds N; for an even n the 000073 | sum terminates already at k<=N/2. The chosen branch and every original 000074 | boundary, determinant, eta and reference factor remain part of this 000075 | same caller. On a zero body stratum retain w itself. No logarithmic 000076 | representation is asserted there. 000077 | 000078 | If only normalized omega=L/L[1] is known, (2) reconstructs w/L[1]. In 000079 | particular omega(f tau)/omega(tau), when its denominator is nonzero, 000080 | is the normalized body functional. The missing absolute scalar is not 000081 | free: the original Z_rel[0] and reference caller fix it when supplied. 000082 | No normalization is reset in these statements. 000083 | 000084 | 3. Compact invariant quotients require fewer probes 000085 | 000086 | Let the ACTUAL compact structural group G act by even automorphisms on 000087 | this chart, preserve the COMPLETE original joint pairing 000088 | B(A,w)=int dnu(b) Dtheta A w, and leave the joint density w invariant. 000089 | At fixed b a purely Berezin pairing is valid only for actions fixing b. 000090 | If gauge transformations move b, all its original density/Jacobian data 000091 | are transported in the joint pairing. Its normalized Haar average R_G obeys 000092 | 000093 | B(R_G A,w)=int_G B(g A,w)dg=B(A,w). (6) 000094 | 000095 | Thus the averaged complementary insertions R_G(theta^(I^c)) reproduce 000096 | the same coefficient functionals. More intrinsically, the restricted 000097 | pairing separates all invariant densities: if invariant delta w pairs 000098 | to zero with all invariant A, then it pairs to zero with every A by(6), 000099 | and Berezin nondegeneracy implies delta w=0. This separates invariant joint densities; it does not assert invariance 000100 | of a conditional density at each fixed b. It applies to the even 000101 | subalgebra as well when N is even. It does not require every individual 000102 | charged field to preserve the physical Hilbert space. 000103 | 000104 | For an original paired complex chart transforming by 000105 | 000106 | psi -> U psi, barpsi -> barpsi U^(-1), 000107 | 000108 | the determinant character of the combined odd representation is 000109 | det(U)det(U^(-1))=1. Its saturated top monomial is therefore invariant 000110 | under this action, for any invertible U. Gauge and color center neutrality 000111 | of this particular paired top probe follows directly. A genuine Majorana 000112 | chart instead has its actual determinant/orientation character, which 000113 | must be checked; it is not replaced by the paired formula. 000114 | 000115 | 4. Exact restriction criterion and determinant line bookkeeping 000116 | 000117 | At a fixed b, let D be the permitted linear density class and A_adm the 000118 | linear space of actual admissible composite tests. Define 000119 | 000120 | N_adm={delta w in D: B(A,delta w)=0 for every A in A_adm}. 000121 | 000122 | The body is identifiable from admissible source data if and only if 000123 | 000124 | N_adm is contained in ker(delta w -> delta w_empty). (7) 000125 | 000126 | Proof: two densities with identical data differ by N_adm; their body 000127 | values agree exactly when(7) holds. In finite dimension the equivalent 000128 | dual criterion is that the body probe tau belongs to A_adm+D^perp. 000129 | Here D^perp denotes the annihilator of D under B. The invariant theorem 000130 | above verifies this criterion when the actual reduction is precisely 000131 | the stated compact invariant class. It is not a theorem for every 000132 | unrelated spectral restriction. 000133 | 000134 | Under a variable dependent chart theta'=R(b)theta, 000135 | 000136 | Dtheta'=det(R)^(-1)Dtheta, 000137 | w'(b,theta')=det(R)w(b,R^(-1)theta'). 000138 | 000139 | Hence w'_empty=det(R)w_empty. A global top test has the local form 000140 | T=kappa(b)tau, with kappa'=det(R)^(-1)kappa. The reconstructed 000141 | kappa w_empty is then invariant. These are density and determinant 000142 | line transformations, not an extra phase or measure choice. Use the 000143 | original trivialization/eta/orientation when it exists; do not choose 000144 | a new frame to make the extracted body take a desired value. 000145 | 000146 | 5. Applicability to the original physical source 000147 | 000148 | The original complete source and its given field/contact dictionary must 000149 | realize L[f theta^(I^c)] as the relevant ordered source derivatives. 000150 | In particular the represented saturated word must preserve Ran(Padm) 000151 | together with its adjoint. It cannot be declared physical merely because 000152 | it is even or gauge invariant. A top Grassmann insertion is not in general 000153 | a CAR occupation projector: coherent symbol, time ordering and coincident 000154 | contacts belong to the original map. 000155 | 000156 | The original whole Padm includes a spectral P_gap as well as seam, color 000157 | and determinant selectors. Gauge neutrality from section3 supplies no 000158 | commutator with P_gap. If the original caller instead supplies an actual 000159 | independent reduced Grassmann chart on its physical field domain, apply 000160 | the inverse directly in that chart. No ambient CAR saturation is required. 000161 | The notation integral_Hadm alone does not construct such a chart. 000162 | 000163 | A nonmultiplicative selector kernel must remain in the full original 000164 | kernel. Equation(3) then extracts the body of that kernel. Calling it the 000165 | unselected exp(-S_B) would require the selector's actual field action. 000166 | 000167 | First remaining original premise: a concrete admissible composite or 000168 | source coefficient that realizes this top probe on the actual common 000169 | field chart, with its prescribed contact, determinant and reference data. 000170 | The inspected original passages do not compute that source value or 000171 | prove its full Padm admissibility. No primitive omega value, actual bare 000172 | boson weight, or full source equality has been obtained by this note. 000173 | 000174 | 6. Original passages inspected 000175 | 000176 | [OMITTED: non-scientific conversation or execution metadata] 000177 | 000178 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/ 000179 | 001-section-primitive-core-and-carrier-interface-handoff.tex:319-343: 000180 | physical local net requires both A and A* to preserve Ran(Padm).