TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-paper-source-reconciliation-20261102/shorttime_20261105M/static_hamilton/STATIC-ACTUAL-YHAM-STRICT-CONNECTED-SIGN-REVIEW.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | Independent M review: strict connected sign for the ACTUAL static Y_ham
000002 | 5 November2026. Original source derivation only. No new model, no numerical run, no construction/enumeration of a2^96 physical Fock space. Frozen L report left unchanged.
000003 | 
000004 | Verdict
000005 | PASS within the ACTUAL existing UNPROJECTED static conditional Clifford/exterior heat source. Its HYPF Hamiltonian Yukawa mark, rather than a global Euclidean Euler alias, has a strictly negative connected norm response under any actually supplied positive static H weight with a nondegenerate radius and the required finite marked moment. No value of the static scalar H_B/rest weight is needed to determine this sign. This does not evaluate the full primitive physical P and dynamical history source.
000006 | 
000007 | 1. Same actual operator and mark
000008 | R113 gives the existing conditional chart h_H=p gamma_F-i gamma_F Phi(H), Phi(H)=DM+L(H). HYPF defines Ta=-i gamma_F Va and the actual Hamiltonmark
000009 | Y_ham(H)=dGamma(-i gamma_F L(H)).
000010 | The scalar/body/reference/sea contribution is not differentiated into a different mark. The normalized conditional state is the original
000011 | rho_F(H)=Gamma(exp(-t h_H))/det(I+exp(-t h_H)), t>0,
000012 | as recorded in REFERENCE-FIBER-EVALUATION section5. Its given scalar prefactor cancels only in this conditional normalized state. That prefactor is retained in the actual H mixture weight.
000013 | For H=kappa theta, |theta|=1, set h_kappa=h0+kappa hY, with h0=p gamma_F-i gamma_F DM and hY=-i gamma_F L(theta). These are the actual finite Hermitian matrices, with arbitrary original complex Yukawas and symmetric Majorana, all ranks retained. No separate diagonalization of Yukawa and MR is assumed.
000014 | 
000015 | 2. Actual weak center, not an invented grading
000016 | The existing v252 slot list distinguishes LEFT and RIGHT fields on both conjugate sides. Define W=-1 on all weak LEFT doublets, +1 on all RIGHT singlets, using the same signs on their conjugates. This is the actual weak SU2 center action already used in the whole96 fourvertex parity proof. Directly from the original LR/nu_R slots,
000017 | W L(theta) W=-L(theta), W DM W=DM, [W,gamma_F]=0.
000018 | The Majorana entries are between right neutrino singlets and hence retain sign+. Grading gamma_F is diagonal in these same slots and commutes with W. W is not silently identified with gamma_F on the entire conjugate space. Thus
000019 | W h_kappa W=h_(-kappa).
000020 | With
000021 | F(kappa)=log det(I+exp(-t h_kappa))=Tr96 f_t(h_kappa),
000022 | f_t(x)=log(1+exp(-t x)),
000023 | unitary similarity gives F(-kappa)=F(kappa). This is an ordinary positive exterior TRACE/determinant identity. No Pfaffian orientation sign is introduced by conjugating an operator; no new Berezin basis or physical fermion multiplicity is chosen.
000024 | The older whole weak SU2 orbit proof also gives h_H=V(q(H)/|H|) h_(|H|,radial) V^dagger, because V commutes with gamma_F and fixes DM. Therefore F and the conditional Y_ham expectation are functions of kappa=sqrt(u) alone, independently of an angular dependence in the static H weight.
000025 | 
000026 | 3. Strict trace convexity without Fock enumeration
000027 | The scalar function satisfies
000028 | f_t''(x)=t^2 exp(tx)/(1+exp(tx))^2>0
000029 | for every finite real x at fixed t>0. In an orthonormal spectral basis of the actual h_kappa the exact second trace derivative is
000030 | F''(kappa)=sum_(a,b) f_t'^[1](lambda_a,lambda_b) |(hY)_(a,b)|^2,
000031 | where the divided difference equals
000032 | [f_t'(lambda_a)-f_t'(lambda_b)]/(lambda_a-lambda_b)
000033 | for distinct eigenvalues and equals f_t''(lambda_a) at equal ones. Every coefficient is strictly positive. Consequently F''(kappa)>0 whenever hY!=0, regardless of noncommuting MR or level degeneracies. This is the finite96 matrix form of the usual faithful thermal Kubo convexity, proved directly here. It neither enumerates an enormous Fock basis nor requires full Yukawa rank.
000034 | Evenness gives F'(0)=0. Thus F'(kappa)>0 for kappa>0. The ACTUAL raw mark is
000035 |  y_raw(kappa)=Tr96[kappa hY (I+exp(t h_kappa))^(-1)]
000036 |              =-kappa F'(kappa)/t.
000037 | Its derivative is
000038 |  y_raw'(kappa)=-[F'(kappa)+kappa F''(kappa)]/t<0,
000039 | and equivalently
000040 |  d y_raw(u)/du=-[F'(kappa)+kappa F''(kappa)]/(2t kappa)<0
000041 | for u>0. It extends continuously to y_raw(0)=0. If every original Yukawa vanishes, hY=0 and the mark is identically0 instead. For the actual nonzero Higgs/Yukawa map, the original weak orbit ensures hY!=0 for every nonzero H direction.
000042 | This derivative acts ONLY on the actual fermion h family to evaluate its mark. It does not add the derivative of E_sea, a(H), H_B or a reference to Y_ham. Those are different marked experiments, as the earlier L source check already established.
000043 | 
000044 | 4. Actual connected static source result, independent of the unknown positive weight value
000045 | Let the ACTUAL source be the static mixture of these original conditional fibers, with its original positive scalar/body/sea/reference weight retained. Require that its normalized H measure exists, its radius u is not almost surely constant, and E[u |y_raw(u)|]<infinity. No shape, coupling value, Gaussian prior, IR regulator or quartic approximation for that weight is selected.
000046 | Since U=u(H)I in the static fiber, its conditional covariance with Y_ham is0. The whole static mixture instead has
000047 | Cov(U,Y_ham(s))=Cov_mu(u,y_raw(u))
000048 | =1/2 E_(mu tensor mu)[(u-u')(y_raw(u)-y_raw(u'))]<0.    (M1)
000049 | Strict negativity follows from the proved strict monotonicity and nondegenerate u. The stated marked moment guarantees the needed integrability: on u>=1 strict monotonicity gives |y_raw(u)|>=|y_raw(1)|>0, so both u and |y_raw| are controlled by u|y_raw|; on the bounded interval all quantities are bounded.
000050 | Fixed insertion time s creates no Euler alias in(M1). The actual conditional rho_F commutes with its generator, so the conditional one point expectation of Y_ham(s) is stationary by trace cyclicity. U is a scalar conserved static coordinate in this source. Thus the covariance is the same for any fixed original insertion times within THIS static mixture. No derivative of a global g is being substituted for a fixed time word.
000051 | This is an actual qualitative jointness result for the original STATIC Y_ham source, valid for all supplied Yukawa ranks and any complex symmetric MR. It is stronger than the old fixed background null covariance: that old result conditioned on one H, whereas(M1) integrates an actual positive nondegenerate static H weight. The strict sign is independent of the numerical value of the retained static scalar H_B/rest weight.
000052 | 
000053 | 5. Exact scope of the result
000054 | The static conditional state and a positive scalar mixture are essential. The general R113 H_B is a BOSON/REST OPERATOR; dynamical Higgs kinetic terms and histories need not be a mixture of frozen scalar H fibers. This review does not turn that operator into a free pointwise scalar or declare its original history integral performed.
000055 | The whole physical selector P remains outside the source. It can change conditional trace weights and need not be the exterior power of a one particle projector. No transfer of(M1) to that full projected source is asserted. Signed/complex eta or reference weights cannot be replaced by positive probabilities, and their actual whole marked source must be evaluated. Additional fermionic vertices can invalidate the conditional exterior class. Nonconstant H and gauge connections need not preserve the original constant free product orbit. The actual density/domain/support and source normalization are retained.
000056 | The exact t=0 conditional trace has covariance I/2 and y_raw=0. Strictness here is for t>0. No physical short time limit, primitive OS vacuum or interchange of t with field/regulator limits is inferred. The earlier Euclidean polynomial Euler values and the raw heat mark are both preserved in their distinct actual source classes.
000057 | No fundamental physical C_UY(t,s), primitive whole spectrum or full TFPT closure is claimed. The computed sign(M1) is the actual bounded scope contribution, not another chosen helper model.
000058 | 
000059 | Original proof anchors
000060 | - verification/v252_full_finite_triple.py:GEN_FIELDS/build_index,full_D,grading: actual weak LR, conjugate, Majorana and gamma slots.
000061 | - ORIGINAL-HIGGS-FIBER-PROOF.txt:17-80: whole original weak left orbit, no custodial replacement, full MR retained.
000062 | - ORIGINAL-LOOP-FOURVERTEX-20261105.txt:25-29 and its unchanged original-builder checker: existing WDYW=-DY, WDMW=DM.
000063 | - PROJECTED-JOINT-SOURCE-GENERATOR-20261103.txt:43-100,107-146: actual conditional generator, body and whole physical selector.
000064 | - HYPF ORIGINAL-NEUTRAL-COMPOSITE-IDENTITY-20261105.md:sections1-4: U and actual raw Y_ham, original time/Clifford/pairing distinctions.
000065 | - REFERENCE-FIBER-EVALUATION.txt:144-188,196-225: actual raw/sea trace and normalized finite conditional state.
000066 | Current source paths/SHA256 are recorded in STATIC-YHAM-SOURCE-EVIDENCE.json. Frozen L source bytes were not changed.