TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-paper-source-reconciliation-20261102/shorttime_20261105L/connected_source/ACTUAL-YUKAWA-MARK-AND-CONNECTED-IR-COEFFICIENT-20261105.txt Original file SHA-256: 96b35f53b591e87ab212a42934aab4065cc4df28aee87b6bab45b43eec6270f8 Inclusive original lines: 1-156 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 | Actual Yukawa mark and connected IR coefficient 000002 | 5 November2026. Original source/type review and analytic calculation. No model, physical input choice, numerical rerun or canonical edit. 000003 | 000004 | Result 000005 | The positive radius divergence of the preceding K report does not decide the connected UY response. The actual connected subtraction can cancel its entire leading divergence. An exact finite connected coefficient is derived below for the genuinely matching polynomial Euclidean caller. The actual HYPF target is a local, time ordered Hamiltonmark, however, so neither a global Euler degree nor an isolated96 mass determinant may be substituted for it. This first identification has been checked directly against R113 and the existing full96 fiber formulas. 000006 | 000007 | 1. Original mark, including its time/locality 000008 | HYPF ORIGINAL-NEUTRAL-COMPOSITE-IDENTITY:section1-3 defines 000009 | H=(x1+i x2,x3+i x4), U(x)=sum_a Xa(x)^2, 000010 | Ta=-i gamma Va, Kb=dGamma(Tb), Y_ham(s)=sum_b Xb(s) Kb(s). 000011 | The whole spatial labels/test functions and original time are retained. The affine whole coefficient is Phi=DM+sum Va Xa. Therefore varying only the actual H linear fermion coefficient scales the Yukawa insertion. DM/MR, geometry, gauge vertices, profile, scalar body and reference factors are not automatically rescaled with it. 000012 | R113:43-100 places this in its actual conditional bilinear generator class H_joint=H_B+sum Q0[h_ji] c_j^dagger c_i. The physical selector belongs outside the whole word; its charged intermediate legs are not individually projected. 000013 | A coupling source g multiplying Y_ham in the actual Hamiltonian gives an integrated Euclidean Duhamel insertion. For a Gibbs/ordered trace this is integral_0^beta Y_ham(s) ds with the actual time order. It does not give Y_ham at one fixed s without the corresponding local coupling profile. The original HYPF text explicitly distinguishes its Ta=-i gamma Va Hamiltonmark from a fully contracted Euclidean SF insertion and retains the original pairing, halving and time/Clifford map. Those original data cannot be replaced by a guessed sign. 000014 | 000015 | 2. Exact Euler identity for the matching Euclidean insertion 000016 | On an actual finite real original coordinate/domain chart let D_H=sum h_i partial_hi, U=sum h_i^2, and let the real source mark be Y_E=D_H SF, with SF's actual H dependence affine. Other action/density dependence is retained in 000017 | B=D_H(S_bos+S_extra)-D_H log m, 000018 | including H dependent inner reference, eta, moving scale and actual density terms. With the original regulator rU/2 and no boundary flux, integration by parts gives exactly 000019 | =d-r-, 000020 | < U Y_E>=(d+2)-r-, 000021 | C_(U,Y_E)=2-r Var(U)-Cov(U,B). (1) 000022 | No positivity is required for the algebra when the actual normalized complex source exists and Z is nonzero. An original boundary/domain flux contributes its own retained term; it cannot be discarded to impose(1). For a local cell s, replace D_H by its actual local field dilation D_s, d by the number of real cell coordinates and D_s u_l by2 delta_sl u_l. Thus the local equation retains cross gradient, gauge, reference and density forces in B_s. 000023 | Equation(1) already shows why of order1/r need not imply divergent connected UY. For a pure degree m fermion factor the conditional Yukawa score is constant and its connected covariance is exactly0, even though the radius diverges. 000024 | 000025 | 3. Legitimate whole polynomial coupling mark, without dividing by Pfaffian zeros 000026 | For the actual finite Euclidean Berezin caller, the following scaling identity requires that ALL actual H dependence of the polynomial called P lies in the marked affine fermion part. Additional fermionic vertices must be H independent for this identity, or must explicitly have the same marked scaling. Under that precise condition, if g multiplies the original H linear fermionic vertices while the rest is held fixed, its whole unnormalized coefficient satisfies 000027 | P(h;g)=P(g h;1), 000028 | partial_g P|g=1=D_H P=-integral_Berezin Y_E exp(-SF). 000029 | These are polynomial identities through zero determinant/Pfaffian fibers. No P inverse is required. A bosonic H dependent w may be held fixed without invalidating this identity. But an additional H dependent FERMIONIC S_extra held fixed under g contributes to the total Berezin polynomial and prevents P_total(h;g)=P_total(g h;1) in general. It cannot be hidden in a scalar w while Grassmann variables remain. In that case retain the actual marked polynomial Q_Y=-partial_g P_total and use the exact general identity 000030 | C_(u,Y)=[I(u Q_Y) I(P_total)-I(u P_total) I(Q_Y)]/I(P_total)^2, 000031 | with I the original held fixed bosonic/rest integration. Degrees j P_j may replace Q_Y only after the scaling identity has actually been established. If g instead changes underlying Yukawa parameters also inside the bosonic profile, MR, sea, reference or density, it is a different derivative and its extra original insertions must be included. 000032 | Subject to that exact affine scaling condition, decompose the actual polynomial into homogeneous components P=sum_j P_j. Set 000033 | A_j(r)=int P_j(h) w(h) exp(-rU/2) dh, 000034 | B_j(r)=int U P_j(h) w(h) exp(-rU/2) dh, 000035 | where w is precisely the original held fixed remainder. Then the connected global mark is EXACTLY 000036 | C= [-(sum_j j B_j)(sum_k A_k)+(sum_j B_j)(sum_k k A_k)]/(sum_j A_j)^2. (2) 000037 | This is useful before any asymptotic expansion and requires only the actual unnormalized coefficients. Whole polynomial trace/degree data of the appropriate actual pairing are needed; dimension96 alone gives no such degree. 000038 | 000039 | 4. Finite subleading connected value, when the actual asymptotics apply 000040 | Assume the ACTUAL polynomial has the affine scaling property just established, its even parity is actually established (not inferred from dimension96), P=P_m+P_(m-2)+..., and its held fixed remainder is uniformly bounded and has an actual ray limit 000041 | w(y/sqrt(r))->w_infinity(y), 000042 | with w_infinity homogeneous of degree0. The original whole domain in this statement is R^d; other domains retain their own scaled domain. Assume 000043 | A=int P_m(y)w_infinity(y)exp(-|y|^2/2)dy !=0, 000044 | B=int P_(m-2)(y)w_infinity(y)exp(-|y|^2/2)dy. 000045 | Bounded convergence with the Gaussian polynomial envelope gives 000046 | A_m(r)~r^(-(d+m)/2) A, 000047 | A_(m-2)(r)~r^(-(d+m-2)/2) B. 000048 | Gaussian radial homogeneity gives the corresponding U integrals with factors d+m and d+m-2. Substituting in(2) yields the actual connected subtraction coefficient 000049 | lim_(r down0) C_(U,Y_E)=-4 B/A. (3) 000050 | No convergence rate of w to its ray limit is needed for this conclusion: the exact pair formula(2), not a subtraction of separately uncontrolled finite terms, is used. Higher even degrees contribute vanishing powers. A is not assumed nonzero from positivity of a different conditional fiber. With signed/complex weights its leading angular integral may cancel; then the actual first nonzero order must be analyzed. If degree m-1 terms survive, the analogous contribution can be of order r^(-1/2), so parity cannot be guessed. 000051 | This finite number is fixed by the ORIGINAL connected subtraction. It is not a freely chosen subtraction of the divergent radius or a new renormalization condition. Its coefficients have not yet been evaluated for the whole actual primitive pairing/domain here. 000052 | 000053 | 5. The HYPF local word requires its actual local coefficient 000054 | For local u_l, let 000055 | A_l=int u_l(y)P_m(y)w_infinity(y)exp(-|y|^2/2)dy, 000056 | B_l=int u_l(y)P_(m-2)(y)w_infinity(y)exp(-|y|^2/2)dy. 000057 | The connected coefficient against the GLOBAL integrated mark is 000058 | lim C_(u_l,Y_E)=2[B_l/A-A_l B/A^2]. (4) 000059 | Only summing all local u_l recovers(3). 000060 | For an actual local coupling mark at s, the leading polynomial must also be checked. If P_m is multihomogeneous in the relevant original cell variables, D_s P_m=m_s P_m, define the actual deficit polynomial 000061 | R_s=m_s P_(m-2)-D_s P_(m-2). 000062 | Then 000063 | lim C_(u_l,Y_s)=int u_l R_s w_infinity gamma/A 000064 | -(A_l/A)int R_s w_infinity gamma/A. (5) 000065 | If the leading local derivative is not proportional to P_m, a connected order1/r term can remain. Neither global degree nor the old finite transfer recurrence decides it. The target fixed time Y_ham(s) additionally needs its actual local time source and the existing original time/pairing map. A time integrated scalar coefficient is not claimed as that fixed time word. 000066 | 000067 | 6. Concrete check against the ACTUAL full96 conditional producer 000068 | REFERENCE-FIBER-EVALUATION:144-180 supplies the existing whole96 coefficient, including all MR mixing. On its original constant circle background H=kappa H0, with whole grading block B(kappa), let 000069 | E_a(kappa)=sqrt(p^2+sigma_a(B(kappa))^2), a=1,...,48. 000070 | The actual coordinate Fock raw trace is 000071 | Z_raw=det(I+exp(-t h))=product_a4 cosh^2(t E_a/2), 000072 | E_sea=-sum_a E_a, 000073 | Z_heat=exp(t E_sea)Z_raw=product_a(1+exp(-t E_a))^2. 000074 | These are entire heat/exterior transfer expressions, not a polynomial determinant of the isolated96 Higgs mass matrix. They retain the same known global scalar/body once. No new physical96 independent fermion integration is performed. 000075 | For the actual raw Hamiltonmark Y_ham=dGamma(kappa partial_kappa h), exact differentiation of this already computed raw trace gives 000076 | _conditional=-kappa sum_a tanh(tE_a/2) partial_kappa E_a. (6) 000077 | Every original MR cross term remains inside B(kappa); no separate neutrino mass choice appears. This is a direct mark formula for the existing UNPROJECTED CONDITIONAL coordinate Fock fiber, before dynamic H integration and before the whole physical selector. It is not the physical mixed UY value. 000078 | Differentiating the SEA CORRECTED trace while also changing E_sea instead gives the DIFFERENT mark 000079 | 000080 | =kappa sum_a[1-tanh(tE_a/2)] partial_kappa E_a. (7) 000081 | Thus treating the actual raw Y_ham as a logarithmic degree of a substituted finite mass polynomial, or silently varying its scalar sea/body factor, changes the observable. The source's actual H_B and reference/body derivative must follow the original marked experiment. 000082 | A compatible finite Euclidean Berezin time regulator can have an actual polynomial P and make sections2-5 applicable, but its degree, kinetic entries and scalar normalization belong to that regulator; its time limit must be controlled with the ORIGINAL mark. Replacing exact exp(-delta t h) by a polynomial time discretization without that control is not an exact identity on all unbounded H amplitudes. 000083 | 000084 | 7. Actual current consequence 000085 | The leading radius divergence does not exclude a finite original connected response. Formula(2) is the exact unchanged connected operation and(3)-(5) identify its first candidate subleading coefficient where the actual polynomial/time/domain assumptions hold. The native whole pairing coefficients, full held fixed weight/domain and local time mark must be inserted, not replaced by a96 mass determinant or the global Euler alias. Formula(6)-(7) already decide that tempting substitution at the existing original producer level. 000086 | No physical value for C_UY(t,s), new primitive state, missing global dictionary, independent ambient gate, physical coupling/profile or freely chosen measure is claimed. Parent/route3 are inspecting the actual physical Pfaffian/kinetic coefficients separately. The source maps and actual original ordering, physical P, MR, ref/eta, scalar body and limit prescription remain retained. 000087 | 000088 | Direct original sources 000089 | - PROJECTED-JOINT-SOURCE-GENERATOR-20261103.txt:17-100,107-146,160-175. 000090 | - HYPF ORIGINAL-NEUTRAL-COMPOSITE-IDENTITY-20261105.md:sections1-5, especially actual Yham definition and Euclidean pairing warning. 000091 | - REFERENCE-FIBER-EVALUATION.txt:144-188,196-225: raw/sea whole96 formulas and actual fiber state. 000092 | - DYNAMICAL-HIGGS-SOURCE-PROOF.txt:88-144: whole unnormalized Berezin polynomials, complementary minors, actual physical pairing and no inverse through zeros. 000093 | - transverse-whole-force-and-first-mixed-contact.txt:sections1,5-7: actual whole Euler force and both reference roles. 000094 | 000095 | Final scope and a genuinely connected norm check 000096 | The preceding K bare norm proof DOES NOT imply divergence of C_UY. Sections2-5 explicitly show its leading cancellation and the precise extra assumptions needed for a finite subleading coefficient. No Gaussian/Pfaffian degree calculation is identified with the full fixed time R113 Hamiltonmark. The actual evaluated Y value(6) is ONLY the existing unprojected conditional heat/exterior fiber, not the joint physical P and history integration. The full requested physical C_UY(t,s) is not computed here. 000097 | There is a separate robust connected norm test within the same narrow POSITIVE ordinary volume class. Let p be a nonzero nonnegative original polynomial of degree m, gamma_(r,p) its normalized Gaussian polynomial weight, and mu_(r,pw) its bounded positive reweighting e^(-C)<=w<=e^C. Then 000098 | dmu/dgamma_(r,p) is between e^(-2C) and e^(2C), 000099 | Var_mu(U)=1/2 integral (U(h)-U(h'))^2 dmu(h)dmu(h') 000100 | >=e^(-4C) Var_gamma_(r,p)(U). 000101 | The same homogeneous Gaussian leading term gives 000102 | Var_gamma_(r,p)(U)~2(d+m)/r^2. 000103 | Consequently a connected UU divergence is present in this particular positive bounded weight/unbounded ordinary volume realization, even if its connected UY leading term cancels. Subtracting a scalar from U or changing an overall vacuum normalization does not change this variance. No positive inequality is transferred to a signed/complex Pfaffian/eta caller. No free field rescaling or new counterterm is introduced, and other actual source supports, density factors, definitions of composites and limiting prescriptions are not excluded. 000104 | 000105 | Decisive raw heat mark test, with its actual original time and source scope 000106 | For the existing UNPROJECTED conditional circle heat fiber, keep its original t>0 fixed, set H=kappa theta, and write h_kappa=h0+kappa hY(theta), hY=-i gamma L(theta). Let C_kappa=(I+exp(t h_kappa))^-1. The actual raw Hamiltonmark has conditional value 000107 | y_raw(kappa theta)=Tr[kappa hY C_kappa]. 000108 | The existing full96 paired spectrum gives 000109 | Tr[h_kappa C_kappa]=-sum_a E_a(kappa)tanh(tE_a(kappa)/2). 000110 | Since E_a(kappa)/kappa tends to the singular values of the actual leading block BY(theta), and |Tr[h0 C_kappa]|<=96||h0||, 000111 | lim_(kappa to infinity) y_raw(kappa theta)/kappa 000112 | =-c(theta), c(theta)=sum_(a=1)^48 sigma_a(BY(theta)) 000113 | =Tr96|hY(theta)|/2. (8) 000114 | This argument needs no invertibility assumption: leading zero singular values contribute0. It needs the actual finite paired heat chart; it is not a generic physical Fock identification. The entire MR and kinetic h0 remain in(6); their bounded fixed contribution drops from this LEADING coefficient, not from the original whole source. 000115 | The old whole weak SU2 rotation map for a single constant C^2 Higgs fiber makes the leading singular values independent of the unit direction theta. In that actual chart, 000116 | c=2 Tr3(3|Yu|+3|Yd|+|Ye|+|Ynu|), |Y|=sqrt(Y^dagger Y). 000117 | It is strictly positive when the actual Yukawa map is nonzero. No values of Y or MR have been selected; all four original sectors and their correct multiplicities are retained. This is the original leading matrix invariant, not a determinant degree. 000118 | 000119 | Now consider exactly the stated conditional coefficient volume class, not a new proposed physical measure: h in R^d, U=||h||^2, original regulator exp(-rU/2), and its actual positive bounded corrected weight w. The already evaluated whole finite spectral factor and sea corrected heat factor are bounded in their proved fixed rank/product classes. The remaining original scalar/body/reference factor must ALSO have the needed actual bound and degree0 ray limit; it is not set to1. Assume the ACTUAL w obeys 0w_infinity(theta)>0 with degree0 ray limit. 000120 | Under the resulting leading measure, radial R and angle theta factor, with R distributed as chi_d and angular weight proportional to w_infinity(theta). Equation(8) and the uniform bound |y_raw(h)|<=96||hY(h)|| supply dominated convergence for the marked moments. Therefore 000121 | Cov_r(U,Y_raw)~ -r^(-3/2) _(angular) Cov_chi_d(R^2,R) 000122 | =-r^(-3/2) _(angular) E_chi_d R, (9) 000123 | since E R^3=(d+1)E R and E R^2=d. The coefficient is strictly negative for a nonzero actual leading Yukawa map. For the actual single constant Higgs doublet d=4 and its weak rotation invariant c, 000124 | Cov_r(U,Y_raw)~-[3 sqrt(2pi)/4] c r^(-3/2). (10) 000125 | Thus the polynomial Euler leading degree cancellation does not occur for THIS actual raw heat Hamiltonmark. It would have changed the observable and the time/source caller. There is no new numerical toy example, fitted coefficient or supplied physical field value in this calculation. 000126 | 000127 | Limits and remaining original factors are decisive. If the original scalar H_B/rest adds an unbounded confining tail or changes the ray limit, (9) does not apply; no such factor is deleted. A signed/complex eta/Pfaffian weight cannot use the positive angular probability argument without its actual nonzero denominator/asymptotics. Whole physical P, dynamic bosonic histories and the source density have not been integrated. This calculation is a direct check of the existing unprojected conditional heat trace when combined with that explicitly stated volume/IR class, not the primitive physical C_UY(t,s). 000128 | It is essential that t>0 is held fixed. At exact t=0, C_kappa=I/2 and Tr hY=0 give y_raw=0 for every kappa. The large H and zero time limits are therefore not interchangeable here. The original R113 physical short time derivative and the complete state/time/P/history limiting prescription must be kept. No raw heat divergence is promoted to a theorem about the full primitive short time jet. 000129 | 000130 | Additional exact original for the coefficient in(8)-(10): ORIGINAL-HIGGS-FIBER-PROOF.txt:17-80, the full weak left rotation and the charged/neutral determinant multiplicities. Current SHA is retained in original-mark-source-evidence.json. 000131 | 000132 | Actual original384 counter mode coefficient integrated 000133 | The independent whole product pairing calculation in 000134 | work/shorttime20261105K_pfaffian/WHOLE-ORIGINAL-MASS-PFAFFIAN-AND-SUBLEADING-KINETIC.md 000135 | has now supplied the actual two coefficients, rather than leaving a kinetic trace to insert. For the genuine distinct original free product modes p and -p, the physical oriented skew matrix has size384; a single192 matrix at p!=0 is not a standalone skew Pfaffian. The whole original pair polynomial, with every internal Yukawa/Majorana slot retained, is 000136 | P_192(H)=C_Y^2 u^96, 000137 | P_190(H)=4p^2 C_Y^2 Sigma u^95, 000138 | P_191=0, 000139 | C_Y=|detYu|^12 |detYd|^12 |detYe|^4 |detYnu|^4, 000140 | Sigma=Tr3[3(Yu^daggerYu)^(-1)+3(Yd^daggerYd)^(-1) 000141 | +(Ye^daggerYe)^(-1)+(Ynu^daggerYnu)^(-1)]. 000142 | ALL FOUR supplied Yukawa matrices must be invertible for this displayed leading degree and inverse Sigma formula. Merely A>0 or one nonzero Yukawa is insufficient. At singular Y the actual polynomial/minors remain valid, but its actual first nonzero homogeneous degree must replace192; no division by C_Y is made. 000143 | 000144 | On the actual single constant Higgs coefficient chart H in C^2=R^4 with u=|H|^2 and original IR insertion exp(-r u/2), the radial integral is explicitly 000145 | int_R4 u^k exp(-r u/2)d^4H=pi^2 Gamma(k+2)(2/r)^(k+2). 000146 | Equivalently the two standard Gaussian moments needed by the exact connected pair formula have the ratio 000147 | E_chi4[u^95]/E_chi4[u^96]=Gamma(97)/(2Gamma(98))=1/194. 000148 | If the ACTUAL held fixed remaining source weight has the previously stated bounded, nonzero degree0 ray limit, its angular factor multiplies BOTH radial moments equally, even if that factor depends on the unit Higgs direction. It cancels from this ratio; no angular probability or new weight is selected. Thus the actual coefficients yield 000149 | B/A=(4p^2 Sigma)/194=2p^2 Sigma/97, 000150 | and the already exact connected subtraction gives 000151 | lim_(r down0) C_(u,S_Y,global,Euclidean) 000152 | =-8p^2 Sigma/97. (11) 000153 | This is an evaluated original mode source integration coefficient, not another unevaluated B/A placeholder. No numerical physical Yukawa or momentum value was chosen. It is nonzero and negative for a genuine nonzero p and invertible supplied Y. The entire symmetric M_R was retained in the source algebra; the whole original block proof shows that it first enters u^94 or below in this384 pair, so it does not contribute to(11). 000154 | 000155 | The source g for(11) multiplies only the H linear fermion vertices in this actual affine product branch, with all other original factors held fixed. The resulting mark is the complete Euclidean coupling/Euler action insertion for that branch, not the fixed time HYPF raw Y_ham(s). This384 pair is one ACTUAL free product mode factor, not the entire spatial/time regulator, physical selector or history integral. Nonconstant H mixes modes; whole mode products, actual cutoff/domain, scalar sea/body, reference, eta and extra fermionic vertices cannot be inferred from this one factor. In particular an actual scalar/body tail violating the bounded degree0 premise changes the limiting integration and(11) cannot be used. A selfdual zero mode has its original192 mass pairing and is not doubled into the384 distinct pair. The original order of regulator and time limits remains unchanged. 000156 | The inference of(11) uses the exact connected pair formula and leading asymptotics of its separate homogeneous contributions. It does not assume that the entire Z has a remainder of order r beyond its leading coefficient; a slower held fixed weight convergence does not by itself invalidate the cancellation proof.