TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/COLLECTIVE-HIGGS-SOURCE-PROOF.txt Original file SHA-256: 260e663a93282f41bdcdf41352384853175173640c162a264425d332436b604d Inclusive original lines: 102-202 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. 000102 | 000103 | Put m=grad Wcol and Gamma_col(m)=sup_j [j.m-Wcol(j)]. 000104 | The Gibbs variational principle gives exactly 000105 | Gamma_col(m)=min_{rho>=0,Tr rho=1,Tr rho Ka=ma} D(rho||rho0). (5) 000106 | This uses ALL density matrices as a variational domain. It does not choose 000107 | new physical states. Compactness and continuity of entropy in finite 000108 | dimension make the constrained value closed and convex; Fenchel duality 000109 | proves (5), including the boundary of the feasible domain. 000110 | 000111 | For every unit n, the eigenvalues of T(n) are the ± singular values of Yu 000112 | and Yd, each three times. The maximal Fock eigenvalue of dGamma(T(n)) is 000113 | s=3[sum sv(Yu)+sum sv(Yd)]. (6) 000114 | It is independent of n, since q(n) is unitary. Hence the convex joint 000115 | expectation set {Tr rho K:rho density} has support function s|j|, and is 000116 | EXACTLY the closed four-dimensional ball |m|<=s. This is a mean-value set, 000117 | not a measure or a classical random joint spectrum of Ka. 000118 | 000119 | Assume Yu or Yd nonzero. For each nonzero v, dGamma(T(v)) is not scalar: 000120 | it vanishes on the Fock vacuum and is nonzero on the one-particle sector. 000121 | Every finite-j Gibbs state is faithful. Its Kubo variance vanishes only 000122 | for scalar operators, so Hess Wcol is strictly positive definite everywhere. 000123 | For |m|= (s-|m|)|j|+lambda_min(log rho0) 000125 | follows from lambda_max(log rho0+j.K)>=lambda_min(log rho0)+s|j| and 000126 | Tr exp A>=exp(lambda_max A). It does NOT use operator monotonicity of exp. 000127 | Thus the minimizer in j exists, is unique and grad Wcol maps R4 bijectively 000128 | onto the OPEN ball. Boundary means are limits, not attained at finite j. 000129 | Gamma_col>=0 with minimum0 at m0=Tr rho0 K. The original given background 000130 | already selects m0; this is no independent symmetry-breaking calculation. 000131 | 000132 | The background/source map is ALREADY explicit here, not a new unknown: 000133 | for the given h_H and C=(I+tanh(ell h_Hstar))/2, kappa=-2ell h_Hstar, 000134 | j=2ell delta x gives -kappa+j.T=2ell h_(Hstar+delta H). 000135 | Thus (4) is precisely the known ORIGINAL fermionic determinant response 000136 | under the whole Higgs shift, now with its collective mean and entropy dual. 000137 | It does NOT identify the new mean coordinate m with the old H coordinate. 000138 | The full geometric/primitive bosonic action, references, eta and measure 000139 | are not replaced by Gamma_col or by exp(-Gamma_col) dm. 000140 | 000141 | 5. Executed native controls and exact scope 000142 | check_collective_higgs_source.py imports the unchanged original builder, 000143 | with its own seed252 structural inputs. Quarter p=1/4,ell=1 are existing 000144 | arithmetic controls, not selected physical scales. All 29 unique checks passed 000145 | with Python assertions disabled. They include actual SU2, hypercharge and 000146 | color maps, reduction, commutator (1), native Hessian finite differences, 000147 | entropy dual, source order and independent 16-dimensional Fock traces 000148 | for four supported original quark insertions in their actual CAR marginal. 000149 | The small Fock check is a trace-identity control, not a truncated Higgs model. 000150 | The entire-class theorems and bijection follow from the proofs above; 000151 | finite representative checks alone do not prove them. 000152 | 000153 | Native numbers: (1)=1968.238966710276. Connected radial/phase words are 000154 | -0.7955187242405932 i and +0.7955187242405977 i (real residual<9e-15). 000155 | The original h time point .3 gives 0.8464385019759293+0.1478585261660290 i. 000156 | The same radial equal-time variance2.579668225693019 differs from the 000157 | Gibbs Hessian2.050593904482716. s=25.09648561603111, log Z_F0=57.318713020624486. 000158 | At source j=(.1,.2,.3,.4), Gamma_col=.9874295284614973 and the independently 000159 | evaluated quasifree relative entropy=.9874295284614832. 000160 | These are structural control numbers, no physical mass/coupling predictions. 000161 | 000162 | The genuine next origin task remains the primitive supported field/weight 000163 | producer. A collective route must identify its original insertion in the 000164 | same S_adm/Wk or CSigma and relate its mean field to the original H with 000165 | the entire action/source/measure, not merely impose H=mean(force). 000166 | If stationarity is used, variation of the FULL given bosonic part, fermion 000167 | determinant, bound reference, eta, H0 and Z0 must all be retained. Stationarity 000168 | alone does not construct its fluctuation measure or a physical local net. 000169 | The full original source and R026/R035/R037/R043/R054/R063/R067/R068/R069/R071, 000170 | charged intermediate paths, memory, State/Time/RP,E8/clocks,Y/flavor/alpha 000171 | remain intact. Physical common3+1 and universal spin2 remain the goal. 000172 | 000173 | 6. Complete conditional H/mean coordinate and its measure transport 000174 | The previous bijection plus the ALREADY GIVEN j=2ell(x-xstar) proves 000175 | F:x in R4 -> m(x)=grad Wcol(2ell(x-xstar)) in open B_s 000176 | is itself a diffeomorphism for this finite constant-H family. Thus its 000177 | conditional H/mean COORDINATE MAP is now explicit in both directions; 000178 | it must not be requested again as if absent. It is nonlinear, not m=H. 000179 | The real Jacobian is dm/dx=2ell Hess Wcol, and therefore 000180 | d^4x=d^4m/[(2ell)^4 det Hess Wcol(j(m))]. (7) 000181 | For an ACTUAL given original weight w(x), its pushforward is exactly 000182 | w(x(m))/[(2ell)^4 det Hess Wcol(j(m))] d^4m. Every other source insertion, 000183 | reference, eta, determinant, H0 and absolute Z0 is transported with it. 000184 | Compactness of the closed mean ball is not a normalized measure. In 000185 | particular, the pushforward of flat d^4x has total mass vol(R4)=infinity, 000186 | by the change-of-variables theorem; its inverse Jacobian is not integrable 000187 | at the boundary. R055/56 bounded-action nonconfinement cannot be repaired 000188 | by this coordinate change. This does NOT assert that the primitive TFPT 000189 | measure is flat or exclude a functional measure by one constant slice. 000190 | No new weight exp(-Gamma_col) dm may be substituted for this pushforward. 000191 | 000192 | The two additional native controls verify the actual original shifted 000193 | D+sum delta xa Va against the same single source exponent, then invert 000194 | the full mean map at the original four-component control source. The 000195 | recovered j differs by less than2.4e-13; det(dm/dx)=20094.078069890544. 000196 | There are29 unique checks, not an additional29 on top of the earlier27. 000197 | 000198 | The remaining primitive task is the original field/support/weight producer 000199 | for the entire theory. Neither the given constant-H coordinate map nor 000200 | its Jacobian is now missing. Extending a constant-mode consumer to full 000201 | H(x), joint measure, original time/reflection and physical3+1/spin2 cannot 000202 | be inferred from this invertible coordinate change.