TFPT scientific source excerpt Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/RADIAL-SPECTRAL-HESSIAN.txt Original file SHA-256: 59987f7fa72e36b3559e03496050bfbd6f84f97b52086f1a8f30c9871fa347dc Inclusive original lines: 1-167 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 ORIGINAL RADIAL HESSIAN OF THE UNTRUNCATED SPECTRAL TRACE 000002 | 3 November 2026. Original v252/v254 family; no new field model or profile. 000003 | 000004 | Result and scope 000005 | ---------------- 000006 | The actual full96 radial source family Phi(r)=r D_Y+D_M has an explicitly 000007 | evaluated spectral Hessian at r=0 for EVERY compatible C2 spectral profile. 000008 | It preserves all original complex Yukawas and symmetric Majorana entries. 000009 | On the existing original quarter-product circle its entire mode sum is 000010 | also explicit. This is a bosonic spectral-action coefficient before field 000011 | integration; it does not substitute the low-curvature Gamma coefficient, 000012 | select a physical profile, or identify the entire primitive S_adm strip. 000013 | 000014 | This reuses, and does not recount as new, R028's exact full96 radial 000015 | determinant and its first u logdet coefficient. The extension here is its 000016 | untruncated arbitrary-profile radial Hessian and whole-circle evaluation. 000017 | R112's gauge-orbit trace defects are different variations. CARRIER-SOURCE 000018 | already gives the radial Cauchy-graph jet; that is not this spectral-action 000019 | Hessian. The scoped original source searches found no previously printed 000020 | formula for this arbitrary-profile radial Hessian. 000021 | 000022 | 1. Original family and exact coefficient 000023 | --------------------------------------- 000024 | Use unchanged verification/v252_full_finite_triple.py:dirac_block,full_D 000025 | and the v254 common Higgs doublet direction established in R028: 000026 | Phi(r)=full_D(r Yu,r Yd,r Ye,r Ynu,MR)=r D_Y+D_M, 000027 | MR=MR^T, r real, u=r^2. 000028 | The Majorana term is NOT scaled with the SM Higgs radius. Define 000029 | A_ch=Tr_3(3 Yu*Yu+3 Yd*Yd+Ye*Ye), 000030 | M2=MR MR*, Qnu=Ynu*Ynu. 000031 | Stars denote adjoints. For any scalar profile g C2 near the nonnegative 000032 | finite spectrum (holomorphic profiles suffice for the proof below), 000033 | 000034 | d/du Tr_96 g(Phi(r)^2)|_(u=0) 000035 | =4 g'(0) A_ch+4 Tr_3[g'(M2) Qnu], (1) 000036 | d^2/dr^2 Tr_96 g(Phi(r)^2)|_(r=0) 000037 | =8 g'(0) A_ch+8 Tr_3[g'(M2) Qnu]. (2) 000038 | 000039 | No simultaneous diagonalization of MR and Ynu is assumed. Singular MR 000040 | is included by continuous functional calculus. The unsuppressed g'(0) 000041 | neutrino contribution cancels against the original linear Majorana cross 000042 | term; the remaining neutrino response is filtered at the actual M2. 000043 | 000044 | Proof using the actual whole determinant already evaluated in R028. 000045 | That result is, for z>0, 000046 | d/du log det_96(z+Phi(r)^2)|_0 000047 | =4 A_ch/z+4 Tr_3[(z+M2)^-1 Qnu]. (3) 000048 | For charged sectors this follows directly from the exact multiplicities 000049 | of the original up/down quarks (three colors, particle and antiparticle) 000050 | and charged leptons. The Majorana-coupled neutrino block in R028 supplies 000051 | the second term including its cross-term cancellation. Differentiating 000052 | (3) with respect to z gives the resolvent trace response. Holomorphic 000053 | functional calculus and its Cauchy formula then give (1). The derivative 000054 | with respect to r is twice the u coefficient, giving (2). Smooth finite 000055 | matrix profiles follow by the corresponding divided-difference calculus. 000056 | 000057 | This is an exact source identity for arbitrary original matrices, not a 000058 | claim that diagnostic fixture entries are physical Yukawa values. 000059 | 000060 | 2. Entire untruncated profile and actual circle modes 000061 | ---------------------------------------------------- 000062 | For the ACTUAL original squared spectral convention 000063 | Tr f((p^2+Phi(r)^2)/chi^2), 000064 | with its original chi and p, put g_p(x)=f((p^2+x)/chi^2). Then (2) gives 000065 | 000066 | Hess_r(p)=8/chi^2 [ 000067 | f'(p^2/chi^2) A_ch 000068 | +Tr_3[f'((p^2+M2)/chi^2) Qnu] ]. (4) 000069 | 000070 | For the already supplied original quarter-product circle p=n+alpha, 000071 | sum (4) over ALL original modes, on the actual trace/domain convention. 000072 | The formula is untruncated in p and f. Where f has the actual Bernstein 000073 | representation f(x)=int exp(-s x) dnu(s), it becomes 000074 | 000075 | Hess_r= -8/chi^2 int s Theta_alpha(s/chi^2) 000076 | [A_ch+Tr_3(exp(-s M2/chi^2) Qnu)] dnu(s), (5) 000077 | Theta_alpha(t)=sum_(n in Z) exp[-t(n+alpha)^2]. 000078 | 000079 | Convergence and passage of derivatives under the original profile/mode 000080 | integrals require their actual trace-class bounds; (5) is not asserted 000081 | when those bounds fail. At a finite original regulator this is ordinary 000082 | finite functional calculus. For a nonzero positive Bernstein profile with 000083 | finite required integrals, the NUMERATOR squared spectral trace Hessian 000084 | in this chart is nonpositive, and strictly negative when the surviving 000085 | Yukawa response is nonzero. This sign is not the sign of the entire 000086 | physical boson Hessian: actual reference, eta, domain and other terms 000087 | remain to be subtracted/added as their original caller specifies. 000088 | 000089 | For the UNSQUARED primitive convention Tr f(D/chi), use that actual f 000090 | and its own divided differences. If an actual symmetric-spectrum product 000091 | chart is proved, one can use the even profile 000092 | g(x)=[f(sqrt(x)/chi)+f(-sqrt(x)/chi)]/2; 000093 | no automatic squared/unsquared or residue/profile identification is made. 000094 | 000095 | 3. General full radial Hessian away from zero 000096 | -------------------------------------------- 000097 | For the actual finite operator D(r)=D_M+r D_Y, let D(r)=U diag(lambda) U* 000098 | and E=U* D_Y U. For F(x)=f(x/chi), 000099 | d^2/dr^2 Tr F(D(r)) 000100 | =sum_(a,b) F'^[1](lambda_a,lambda_b) |E_ab|^2, (6) 000101 | where F'^[1](x,y)=(F'(x)-F'(y))/(x-y) for x!=y and F''(x) 000102 | on the diagonal. Degenerate eigenspaces are included exactly. 000103 | For a squared profile use F(x)=f(x^2/chi^2). This retains every original 000104 | noncommuting Yukawa/Majorana matrix entry; the finite radial family is 000105 | affine, so there is no additional D'' term. For a nonlinear original 000106 | field path add Tr F'(D) D'' as usual. (6) is the full profile Hessian, 000107 | not its low-curvature moment or an assumed inverse boson covariance. 000108 | 000109 | 4. Bound reference, eta and the actual primitive comparison 000110 | ---------------------------------------------------------- 000111 | For a genuine fixed-domain original relative spectral source with its 000112 | original radial variation, the corresponding coefficient is 000113 | Hess_r S_prim=Hess_r Tr f(D_rel(r)/chi) 000114 | -Hess_r Tr f(D_ref(r)/chi) 000115 | +(i pi/2) d_r^2 Delta eta_Sigma. (7) 000116 | The reference derivative vanishes ONLY if that actual reference is fixed 000117 | in this variation. If it carries the same radial response, use its own 000118 | whole version of (2), (4), or (6). No new reference or zero eta derivative 000119 | is chosen. Moving domains, changing scales/marks and actual boundary 000120 | terms add their original derivatives. A time-dependent field variation 000121 | also has the actual spacetime operator response, beyond the constant 000122 | internal family evaluated here. 000123 | 000124 | Equation (7) can be supplied to the original strip comparison as a 000125 | concrete radial matrix/action coefficient. It does not itself give the 000126 | short-time strip generator: the original field integration, full source 000127 | weights and time sewing must still implement that coefficient. In 000128 | particular it must not be exponentiated anew if already integrated in 000129 | the actual omega caller. 000130 | 000131 | 5. Direct independent numerical identity check on unchanged builder 000132 | ------------------------------------------------------------------ 000133 | Used the literal v252 rng252 fixture: its four complex Yukawas generated 000134 | in original order and its original real symmetric MR. Let D0=D_M and 000135 | E=D_Y. Evaluated the full96 divided-difference Hessian (6) for 000136 | F_t(x)=exp(-t x^2), independently of the 3x3 closed formula (2). 000137 | Profiles t=.1,.4,1 are diagnostic functional-calculus probes, not a 000138 | physical profile selection or fit. Observed results: 000139 | 000140 | t=.1: full96 -68.20789868823312; closed -68.20789868823309; 000141 | relative error 4.1669234761673105e-16. 000142 | t=.4: full96 -258.94726594687614; closed -258.9472659468761; 000143 | relative error 2.1951735482879988e-16. 000144 | t=1: full96 -616.8705924102873; closed -616.8705924102874; 000145 | relative error 1.842960891966166e-16. 000146 | 000147 | No eigenvalue finite differences, altered original code, new model, 000148 | fullsuite, physical profile/Yukawa selection, or ledger promotion. 000149 | 000150 | 6. Source-type finding retained 000151 | ------------------------------- 000152 | QFT source45 section007:5-29 defines S_adm as the BARE INPUT to exp W_k; 000153 | :115-142 explicitly distinguishes it from Gamma_k and its loop terms. 000154 | Boundary source001:302-317 explicitly defines S_prim as the relative 000155 | spectral seed, with subsequent gauge/ghost/boson fluctuation introduction. 000156 | Metrology002:135-228 prints a profile-free residue functional Gamma_D^(4) 000157 | and its Z_H=1/(16pi^2); Metrology004:47-80 labels the displayed common 000158 | field Lagrangian a low-curvature effective action. The text does not 000159 | provide an evaluated equality that substitutes those truncated/retained 000160 | coefficients for the full original bare strip kernel. 000161 | 000162 | The first unjustified shortcut would therefore be to use Z_H alone to 000163 | set the primitive exact strip diffusion while discarding the original 000164 | full profile, reference, higher field response and domain data. This 000165 | does not deny the existing geometric residue or effective-action result; 000166 | the actual full-profile coefficient (1)-(7) is the constructive alternative 000167 | within the already supplied original radial chart.