TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/REFERENCE.txt Original file SHA-256: 9f99952b6a75c28d65de9de9d45d2c409ac6135d2b82f39ba4f4bee35620e328 Inclusive original lines: 1-200 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 | E CONTINUATION: ORIGINAL REFERENCE, VACUUM CONNECTION AND ACTUAL SOURCE JET 000002 | 2 November 2026. Read-only canonical source review and analytic calculation. 000003 | 000004 | Outcome 000005 | 000006 | There is a constructive way to determine the null-source factor together 000007 | with the normalized source: integrate the SAME original action/measure 000008 | score and the SAME reference score, from the actual reference datum. 000009 | The coupled equations are given explicitly below. This extends the already 000010 | established full regulator pair rule, rather than repeating the old claim 000011 | that normalized moments lose a scalar. 000012 | 000013 | For the newly constructed local force reader there is one concrete source 000014 | comparison before this transport can be used: an exponential of the 000015 | time-smeared Wick force is not automatically the Dyson source obtained by 000016 | varying the original time-local Dirac operator. Their second-order real 000017 | parts agree, but their phase differs by an explicitly computable ordered 000018 | commutator integral. In the actual quark36 class its logarithmic coefficient 000019 | is fixed below. The reference's actual source/background two-jet, not its 000020 | name or an arbitrary normalization, decides whether the original relative 000021 | quotient removes that phase. This is the next single concrete calculation. 000022 | 000023 | 1. Original provenance and existing results preserved 000024 | 000025 | Fresh Atlas revision0456ee27bbe55730e8b4de57, shared plan81, R101 preserved. 000026 | L20261102C remains controlling: P1 is the structured seam kernel and the 000027 | common S_adm/Z_rel/Gamma/OS chain is original. The local variable-background 000028 | CAR construction is an application theorem, not a new global TFPT gate. 000029 | No E8, clock, Y/Flavor/alpha, common state/time or physical result is reopened. 000030 | 000031 | Original current S_rel definition: 000032 | research-source:131751903c6895be3e43ab55:273-318, 000033 | documents/tfpt_2_standard_model/sections/010-section-the-explicit-mass-gap- 000034 | closes-dynamics-and-cluste.tex, hash 000035 | 3a31f272d999f1e0b027f25a877068576c6cbdc83150c7b5c0f8a6e17bed1904. 000036 | It is Tr f((D_rel+A_Sigma)/chi)-Tr f(D_ref/chi)+i pi Delta_eta/2, 000037 | with the modular/covariance source and its stated96-dimensional implementation. 000038 | This is not automatically the older profile-free -6chi^2 Res1+FP0 contract. 000039 | 000040 | Original relative generating functional: 000041 | paper-source:35c4c41ef6cded2368b36527:265-278, freshly read with235-318, 000042 | Source45 QFT002, hash 000043 | 61ecee7727bdec47936bbffc028d3dbac5260a025b6dbcce30124a60d635fc9c: 000044 | Z_rel[J,eta,eta_bar]=Z_adm[J,eta,eta_bar]/Z_ref[0,0,0]. (1) 000045 | The reference is a fixed normalizing denominator in the SOURCE arguments. 000046 | That does not assert independence from a varying geometry/background. 000047 | 000048 | R025 full flow is already established and reused: 000049 | research-source:7816b374a7515b08d57a537b:1-102 and 000050 | research-source:11363fc8655cdd14c912d9ce:177-232, 000051 | FULL-SOURCE-REGULATOR-PROOF.txt, hash 000052 | 589a9054dd0af97934ff572ad3a5e1b18d8a0e3ed9cfb848f450b509dbf48df0. 000053 | It retains q=Z/Z[0] AND ell=log Z[0], all exterior field legs, conditional 000054 | integrals and the denominator. The exact pair is not a newly missing rule. 000055 | 000056 | R032 variable bound-reference calculation is fully retained: 000057 | research-source:039eb8d38b194a51c6147af4:1-108 and 000058 | research-source:f76af118431de9a886f088be:109-208, 000059 | ORIGINAL-HIGGS-BOUND-REFERENCE-PROOF.txt, hash 000060 | 13e79f2be052ca918fd2be8faf8cf6f1669fc3ad535ccf9a3fdff5b7c0b1b3c5. 000061 | Its two explicit reference jets are used only in their supplied product 000062 | contracts; neither is selected as the current modular source's reference. 000063 | 000064 | R080 original local force source was reread in full: 000065 | research-source:8f6a57abe438221d13e6e0d7:1-161. 000066 | Its S2/det2/centering and whole ordered-unitary source are reused. The new 000067 | variable-background proof discharges the constant-H restriction in this 000068 | reader; it does not assert an equality to every original S_adm insertion. 000069 | 000070 | R101 original source application was read directly at the exact known path 000071 | experiments/theory-contracts/seam-positive-kernel-search-20261015/ 000072 | PRIMITIVE-SOURCE-APPLICATION-20261102.txt. Its primitive n4 glue is already 000073 | present, not a new missing condition. research-source:27e6dc401dd06d30a44a5c3e 000074 | GENESIS-SYNTHESIS-20261102.txt:1-177 was hash-read, including the original 000075 | quarter/eta and Quillen inference controls. No optional glue theorem is 000076 | substituted for the common source's reference connection. 000077 | 000078 | 2. Full background connection, including the actual integration anchor 000079 | 000080 | Work first in the original finite regulator or any domain where the given 000081 | integrals and derivatives exist. The same identities are formal otherwise; 000082 | no new base measure is selected. Let b_s be a path in the ORIGINAL allowed 000083 | background/domain family. Transport its measure to one common chart, 000084 | dmu_s(phi)=m_s(phi)dmu_0(phi), 000085 | and keep the full original action S_s and source insertion O_s. Write 000086 | Z_adm,s[J]=int exp[-S_s(phi)+J.O_s(phi)]m_s(phi)dmu_0(phi), 000087 | r_s=partial_s log Z_ref,s[0], 000088 | K_s=partial_s S_s-partial_s log m_s. 000089 | Jacobians, gauge/Berezin conventions, changing domains and source frames 000090 | are part of these original transported data, not silently set to zero. 000091 | For an even background parameter and a fixed source chart, differentiation 000092 | of THIS integrand yields 000093 | partial_s log Z_rel,s[J] 000094 | =-_(s,J)-r_s. (2) 000095 | 000096 | Put q_s[J]=Z_adm,s[J]/Z_adm,s[0] and ell_s=log Z_rel,s[0]. Then 000097 | partial_s log q_s[J] 000098 | =-_(s,J)+_(s,0), 000099 | partial_s ell_s=-_(s,0)-r_s. (3) 000100 | Thus a full original normalized field source and its original action score 000101 | determine the vacuum variation, not just the field responses. A polynomial 000102 | K_s can be inserted by the corresponding full source derivatives; a 000103 | nonpolynomial spectral K_s requires its actual defined insertion/limit. 000104 | The local-force theorem controls its stated force words, not every 000105 | unbounded action insertion without checking that insertion. 000106 | 000107 | With the actual base value ell_0, the unique nonzero-branch solution is 000108 | Z_rel,1[J]=exp(ell_0) 000109 | exp[-int_0^1 (_(s,0)+r_s)ds] q_1[J]. (4) 000110 | Equation(4) is anchored thermodynamic integration on the original datum. 000111 | It reconstructs the lost vacuum scalar together with q; no free c(b) 000112 | has been added. At zeros use Z itself, and carry the original eta/kernel 000113 | branch rather than pretending one logarithm exists globally. 000114 | 000115 | The EXPLICIT identity-reference anchor is legitimate when the numerator 000116 | at b_0 is exactly the same full integral as the denominator reference: 000117 | same field domain, measure, regulator, boundary condition, orientation and 000118 | counterterms. Then Z_adm,b0[0]=Z_ref,b0[0], so ell_0=0. 000119 | For the fermion determinant quotient this follows immediately from equal 000120 | operators on the same determinant/Pfaffian contract. It is not a newly 000121 | chosen vacuum. If the denominator denotes only a sector, its identity 000122 | anchor normalizes that sector, not all remaining bosonic integrals. 000123 | The equation S_rel(b0)=0 alone does not establish equality of two integral 000124 | contracts. The original given Z_ref is the anchor object to bind, not an 000125 | arbitrary point H=0 or a dimensionless scale renamed as a normalization. 000126 | 000127 | At finite fermion cutoff, the particularly concrete version is 000128 | d log[det M_b/det M_ref,b] 000129 | =Tr(M_b^-1 dM_b)-Tr(M_ref,b^-1 dM_ref,b). (5) 000130 | Its integral from the identical reference operator gives the exact 000131 | determinant ratio1 at the base. A Pfaffian uses one half of the trace and 000132 | its actual orientation, so96 doubled chart slots must not be counted as 000133 | 96 independent determinant species. The separate Berezin/KMS source audit 000134 | owns which M and domain actually implement the original96 reader. 000135 | 000136 | 3. How the ORIGINAL spectral reference and eta enter the score 000137 | 000138 | For the current displayed spectral action set 000139 | X_s=(D_rel,s+A_Sigma,s)/chi_s, Y_s=D_ref,s/chi_s. 000140 | Whenever its supplied trace/domain calculus permits differentiation, 000141 | partial_s S_spec 000142 | =Tr[(partial_s f_s)(X_s)+f_s'(X_s)partial_s X_s] 000143 | -Tr[(partial_s f_s)(Y_s)+f_s'(Y_s)partial_s Y_s] 000144 | +(i pi/2)partial_s Delta_eta_s. (6) 000145 | Here partial_s X includes the actual chi derivative, connection/domain 000146 | transport and field frame; similarly for Y. The trace simplification in(6) 000147 | uses the ordinary cyclic trace actually displayed, not an arbitrary 000148 | weighted trace. Along a changing APS domain its additional boundary 000149 | transport is not represented merely by a bounded dD. 000150 | 000151 | The numerator's reference subtraction in(6) and the denominator score r_s 000152 | in(2) are separately named original data. They must not be merged or 000153 | double-counted on the assumption that their common word 'reference' 000154 | makes them the same functional. Eta is retained as a phase and can have 000155 | spectral-flow/kernel contributions on the original branch. It is not 000156 | replaced by a generic Quillen number or a nonzero-total-eta requirement. 000157 | 000158 | For a linear finite matrix variation X+epsilon T at fixed f,chi, the 000159 | noncommuting quadratic trace jet is explicitly computable without fitting: 000160 | d_epsilon^2 Tr f(X+epsilon T)|0 000161 | =sum_ij f'[lambda_i,lambda_j] |T_ij|^2, (7) 000162 | where f'[x,y]=(f'(x)-f'(y))/(x-y), with value f''(x) on the diagonal. 000163 | Add Tr f'(X)X'' for a nonlinear original path, subtract the SAME Y jet, 000164 | and include the eta, measure and denominator jets. This is the concrete 000165 | original operator computation for the common phase; a new counterterm is 000166 | not authorized by requiring the two answers to agree. 000167 | 000168 | 4. Source variation and background variation have different reference jets 000169 | 000170 | If epsilon f is an EXTERNAL SOURCE argument J in(1), with original b fixed, 000171 | the literal Z_ref[0,0,0] has zero epsilon derivative to every order. It cannot 000172 | cancel a J-dependent quadratic phase by a supposed reference variation. 000173 | Such a source contact term, if part of the original numerator, must be 000174 | carried by that numerator's actual source prescription. 000175 | 000176 | If epsilon f is instead an actual BACKGROUND H variation, then every 000177 | original bound D_ref(H), chi(H), eta(H), measure/domain and Z_ref[0;H] 000178 | must be varied by(2),(6). This is a coupled background connection; it is 000179 | not equivalent to holding the reference fixed in the source experiment. 000180 | This distinction is forced directly by the original definition(1). 000181 | 000182 | Ward/Schwinger-Dyson field equations are inherited by the same q because 000183 | their source differential equations are homogeneous in Z. The background 000184 | score(3), together with its reference anchor, carries the vacuum part. 000185 | Where the full scalar functional is smooth and nonzero, mixed derivatives 000186 | of its log commute; thus the actual score connection is locally integrable. 000187 | Any proposed replacement score must pass this integrability check and the 000188 | original eta/line holonomy, not merely reproduce a two-point function. 000189 | 000190 | 5. Actual new local test: Dyson source versus exp of the smeared force 000191 | 000192 | Use precisely R080's constant circle block h_p=p gamma+V*, positive-energy 000193 | occupation C_p=expit(2ell h_p), and one original Hermitian odd Ta. The 000194 | centered field is K(t)=dGamma(e^(ith)Ta e^(-ith))-its actual mean. 000195 | For a real smooth compact pulse f define A_f=int f(t)e^(ith)Ta e^(-ith)dt. 000196 | The variable-background theorem controls exp(i epsilon K(A_f)). 000197 | The actual time-local operator perturbation instead generates its Dyson 000198 | source. Its log coefficient at epsilon^2 is 000199 | B_N=-sum_(p cutoff) int_(t>s) f(t)f(s) 000200 | Tr[C_p T_p(t-s)(I-C_p)Ta]dt ds. (8)