TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/ADMISSIBLE-ACTION-DERIVATION-20261103.txt Original file SHA-256: d3b7ea73d59c1c671af0fec4a4f9b1048dfc297eb26a932fe18fda2af03b972d Inclusive original lines: 131-271 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. 000131 | 5. EXPLICIT VARIATIONAL PRIMITIVE AND THE SAME ORIGINAL BOUNDARY FLOW 000132 | 000133 | Use the actual ORIGINAL whole Clifford chart, not a finite KO6 pairing. 000134 | The existing normal symbol N and conjugation C0 give 000135 | L_b=-N D_b=partial_r+B_b, K_b=partial_r+h_b, 000136 | beta_L(x,y)=integral _new dr. 000137 | Its exact Green identity is 000138 | beta_L(x,y)+beta_L(y,x)=[_new]_0^ell. 000139 | On the actual matching closed domain this boundary pairing vanishes; 000140 | for the original open polarization it must instead be retained explicitly. 000141 | No new conjugation, extra fermion copy or normal multiplier is selected. 000142 | 000143 | Let C be the matrix of this original symmetric bilinear pairing and 000144 | M_b=C L_b its skew kernel on the actual domain. The inverse variational 000145 | homotopy for the EXACT off-shell Euler operator E_b(Psi)=M_b Psi gives 000146 | S_F,bulk=integral_0^1 dt Psi^T E_b(t Psi) 000147 | =1/2 integral Psi^T M_b Psi. (8a) 000148 | The integral in t is the scaling homotopy, not a new physical time. 000149 | For a fixed closed domain its left variation is exactly M_b Psi. 000150 | For an open collar the even variation retains the original surface form 000151 | delta S_F,bulk=integral deltaPsi^T M_b Psi 000152 | +1/2[Psi^T C deltaPsi]_0^ell. 000153 | In the original charged coordinates this same functional is 000154 | S_F,sym=integral barpsi K_b psi dr 000155 | -1/2[barpsi psi]_0^ell. 000156 | With the ORIGINAL input psi_0 and output dual barpsi_ell fixed, the 000157 | polarization adds -1/2(barpsi_ell psi_ell+barpsi_0 psi_0), hence 000158 | S_F,open=integral barpsi K_b psi dr-barpsi_ell psi_ell. (8b) 000159 | Its boundary variation is -delta barpsi_ell psi_ell-barpsi_0 delta psi_0, 000160 | which vanishes for exactly these fixed incoming/outgoing variables. 000161 | On the actual classical solutions, 000162 | S_F,open,on-shell=-barpsi_ell R_b psi_0, 000163 | R_b=exp(-ell h_b), (8c) 000164 | or the already given ordered evolution when the original h varies. 000165 | Thus the same original whole R, not only its occupations, is recovered 000166 | from an explicit boundary action. Exponentiating this classical boundary 000167 | action gives the already known exp(barpsi_ell R_b psi_0) factor. No 000168 | determinant scalar or physical quantum measure is inferred by this step. 000169 | 000170 | The exact variational primitive is unique up to B[b] and the actual fixed 000171 | boundary convention IF the full off-shell Euler operator M_b Psi is the 000172 | given one. Merely having the same classical solution space does not impose 000173 | that stronger identification. No CAR or Schwinger-Dyson assumption is 000174 | needed for this classical variational integration. Promoting E_b to the 000175 | physical source's Euler operator is still a physical identification. 000176 | 000177 | If the actual UNINTEGRATED zero-fermion restriction equals the given 000178 | bosonic source S_B,original[b], B[b] is fixed by that SAME source value, 000179 | not fitted anew. An already fermion-integrated logZ/Gamma is not this body. 000180 | This supplies the explicit existing-branch variational representative 000181 | S_*[b,Psi]=S_B,original[b]+1/2 Psi^T M_b Psi+S_boundary. (8d) 000182 | The printed S_prim supplies its relative spectral contribution, including 000183 | reference and eta. Other actual gauge/ghost weights are not silently 000184 | identified with S_prim or discarded. Neither (8d) nor a bosonic trace alone 000185 | proves the full primitive quantum source identification. 000186 | 000187 | Crucially the SAME action now gives the coupled bosonic equation, 000188 | delta_b S_*=delta_b S_B,original+1/2 Psi^T(delta_b M_b)Psi 000189 | +delta_b S_boundary. (8e) 000190 | This is the actual fermionic backreaction current with coefficient fixed 000191 | by (8a). It is not an independently chosen coupling. On the actual fixed 000192 | invertible regulated fermion fiber, integration gives 000193 | S_B,eff=S_B,original-log Pf(M_b), 000194 | delta_b S_B,eff=delta_b S_B,original 000195 | -1/2 Tr(M_b^-1 delta_b M_b). (8f) 000196 | Retain the matching original reference Pfaffian and eta/orientation phase, 000197 | and their derivatives where relevant. At zero modes keep the polynomial 000198 | Pfaffian/source minors rather than its logarithm. At a moving domain carry 000199 | the original domain variation. This is one actual fermion integration, 000200 | not an extra full96 determinant multiplying an already integrated source. 000201 | 000202 | 6. WHAT THE FERMIONIC QUANTUM SOURCE RULE WOULD FORCE 000203 | 000204 | At the original finite Grassmann regulator let Psi denote the actual 000205 | physical field coordinates, with original orientation and boundary 000206 | prescription. For an actual skew kernel M_b, the linear source identity 000207 | for all homogeneous polynomials F is 000208 | int(partial_a^L F) w_b 000209 | =(-1)^|F| int F (M_b Psi)_a w_b. (9) 000210 | Integration by parts and nondegeneracy of Berezin pairing imply 000211 | partial_a^L w_b=-(M_b Psi)_a w_b. 000212 | On a nonzero-body logarithmic branch this becomes 000213 | partial_a^L S_b=(M_b Psi)_a. 000214 | Grassmann Euler degree by degree now forces, exactly, 000215 | S[b,Psi]=S_B[b]+1/2 Psi^T M_b Psi. (10) 000216 | Every conditional fermion vertex of degree4 or higher is then zero. 000217 | At singular fibers keep the polynomial weight identity rather than 000218 | dividing by its body. This theorem does not choose a new Berezin measure. 000219 | 000220 | The existing fixed-domain Clifford/Berezin kinetic presentation satisfies 000221 | (9) with its actual boundary prescription; its K_b=partial_r+h_b pairing 000222 | is already derived. Applying this to the graph/primitive physical source 000223 | requires the original physical field, boundary and integration identity. 000224 | A boundary Cauchy projector alone is not an invertible bulk Berezin kernel. 000225 | The classical Dirac Cauchy equation alone is weaker; this is explicitly 000226 | separated in the original multifield source proof. No such physical 000227 | identity is declared proved by this report. 000228 | On a valid TOTAL KO2 product realization, the corresponding corrected 000229 | fermionic term would include 1/2. 000230 | Finite KO6 alone is not that physical Grassmann pairing. 000231 | 000232 | 7. DYNAMIC BOSONS: THE DERIVED INTERACTION HIERARCHY 000233 | 000234 | Even under(10), integrate the actual ORIGINAL bosons, without freezing 000235 | them or choosing Gaussian weights. At a fixed finite original regulator 000236 | write M_b=M0+sum_a b_a M_a where that actual affine source chart applies, 000237 | J_a=1/2 Psi^T M_a Psi. With the original boson weight at this integration 000238 | stage (NOT a second copy of an already integrated fermion determinant), 000239 | assume its actual Z_B!=0 and that the needed moments exist. Then 000240 | W_eff[Psi]=Z_B exp(-S_F0) _B. 000241 | Its exact nilpotent logarithm is 000242 | S_eff[Psi]=-log Z_B+S_F0 000243 | +sum_(n>=1) (-1)^(n+1)/n! kappa_B(b_a1,...,b_an) 000244 | J_a1...J_an. (11) 000245 | The sum terminates by Grassmann degree. For spacetime fields the labels 000246 | include the actual positions and all original contractions. Signed or 000247 | complex weights give algebraic cumulants, not positive probabilities. 000248 | If Z_B=0 retain the unnormalized polynomial, not this logarithmic chart. 000249 | The absolute scalar and the original external reference are retained. 000250 | Equation(11) fixes effective multi-fermion coefficients from the SAME 000251 | bosonic source, rather than introducing independent couplings. It does 000252 | not evaluate its unknown cumulants or a primitive mirror vertex. In 000253 | particular a16-fermion action coefficient is an eighth-bilinear cumulant 000254 | in this chart; it is not identified with R109's sixteen-step time process. 000255 | 000256 | 8. WHY SPECTRAL STATIONARITY ALONE DOES NOT SUPPLY(9) 000257 | 000258 | For the actual printed spectral variables, differentiating the trace gives 000259 | delta Tr f(D_b/chi)=Tr[f'(D_b/chi) delta D_b/chi], 000260 | plus actual reference, eta, scale and domain derivatives. The original 000261 | stationarity variables are alpha,chi,e,omega; Psi is not one of them. 000262 | For a squared heat argument the numerator derivative is 000263 | -2/chi^2 int t dnu(t) Tr[D_b exp(-t D_b^2/chi^2)delta D_b]. 000264 | This derives a spectral bosonic current. Identifying its cotangent with a 000265 | spinor bilinear would produce the weighted spectral gradient, not D_b. 000266 | No such new cotangent identification is adopted here. 000267 | The known original4D relevant/marginal basis excludes dimension6+ terms 000268 | only in that truncation, not in the whole microscopic source. Its scalar 000269 | stationarity does not prove(9). These facts preserve both the known 000270 | bosonic action and the exact conditional fermionic reconstruction. 000271 |