TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/CURRENT-ARGUMENT-MAP.txt Original file SHA-256: d53d2cfc44cdfc84f3f36ab6a09248e97ab79f57c2d8c199cbaab29dab2d00f2 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 | THE EXISTING SEED PACKAGE TO GEOMETRIC OPERATOR MAP 000002 | L20261102G, 2 November2026. Targeted original-source continuation. 000003 | Canonical sources unchanged. This is a positive source-version correction, 000004 | not another bounded-D_rel obstruction or a new global IR-source condition. 000005 | 000006 | RESULT 000007 | 000008 | The route already exists in the originals: 000009 | 000010 | organizing seed master 000011 | A_Sigma,seed=B_rel directsum nabla_chi_seed 000012 | -> geometric package 000013 | A_Sigma,geo=B_rel directsum nabla_chi_geo, 000014 | chi_geo=Lambda exp(sigma), D_geo=D_ref+D_rel=D 000015 | -> actual geometric operator 000016 | K_sigma=exp(-sigma/2)D_geo exp(-sigma/2)+B_rel. 000017 | 000018 | The full U transport is attached to THIS existing operative geometric 000019 | caller. Its exact result is 000020 | 000021 | K_sigma^U=exp(-sigma/2)mathfrakD_h exp(-sigma/2)+B_rel^U, 000022 | D_ref^U=mathfrakD_href, (1) 000023 | 000024 | on the transported original graded bundle/domain, with every original 000025 | reference, scale, trace and integration factor retained. The whole S/Z 000026 | change-of-coordinates theorem already proved then applies to this caller. 000027 | There is no new reference operator, bosonic measure or global IR map to 000028 | invent for its transported copy. 000029 | 000030 | A_Sigma is NOT correctly classified, merely from this notation, as a 000031 | bounded inner one-form. The actual original definition is a package of 000032 | relative bosonic/bundle data and a covariant derivative. Its geometric 000033 | realization uses an explicitly given elliptic D_geo. It is also not 000034 | correct to infer that the raw symbol A_Sigma is by itself a specified 000035 | elliptic differential operator with a printed universal principal symbol. 000036 | The organizing package is evaluated by the later geometric definition. 000037 | 000038 | The distinction to retain is between that legitimate existing realization 000039 | and a universal operator identity D_rel+A_Sigma=K_sigma. The originals 000040 | give the former route; their statements need not be strengthened to the 000041 | latter identity on every historical field space or scale convention. 000042 | 000043 | 1. FIRST DECISIVE ORIGINAL: THIS IS EXPLICITLY A PACKAGE ROUTE 000044 | 000045 | tfpt42 BigPicture002:73-82 prints 000046 | S_TFPT=Tr f((D_rel+A_Sigma,seed)/chi_seed) 000047 | -Tr f(D_ref/chi_seed)+i*pi*Delta_eta/2, 000048 | A_Sigma,seed:=B_rel directsum nabla_chi_seed. 000049 | Its explanatory paragraph103-110 calls the master equation an 000050 | ORGANIZING closed form. It explicitly says that on the later geometric 000051 | branch the package is reorganized as A_Sigma,geo with 000052 | chi_seed -> chi_geo=Lambda exp(sigma). 000053 | 000054 | Exact freshly hash-read source: 000055 | paper-source:5a8fff1024b97ed6e6df8826:65-116, 000056 | documents/archive--paper-latex--tfpt-42/sections/ 000057 | 002-section-big-picture-what-tfpt-actually-claims.tex, 000058 | SHA256 eaade8d840e3d3e2aeac752e33e7645bef90035979190bda334418f4a792303d. 000059 | 000060 | The older tfpt31 guide independently contains the same master/package 000061 | at205-214 and the explicit organizing-form/geometric-route statement 000062 | at235-241. It calls later closure theorems internal statements on the 000063 | admissible reduced datum, not additional primitive ontology. 000064 | paper-source:5101e9f1ea8b4317c73f3b5c:195-247, 000065 | documents/archive--paper-latex--tfpt-31/sections/ 000066 | 001-section-reading-guide-and-claim-map.tex, 000067 | SHA256 5edaeddfffe99f9e20795cc96343f091cd399001b7e1c3bc4132e640bc01ae34. 000068 | 000069 | These are actual source definitions, not an interpretation generated by 000070 | an Atlas edge. They correct a reading that treated every A_Sigma as a 000071 | bounded zero-order perturbation and consequently declared the geometric 000072 | principal symbol absent from the whole original theory. 000073 | 000074 | 2. THE EXPLICIT EXISTING GEOMETRIC EVALUATION 000075 | 000076 | tfpt31 occupancy/bridge007:44-58 defines the graded geometric bundle and 000077 | mathbbA_Sigma=(D_geo,D_rel,A_Sigma,geo), 000078 | A_Sigma,geo:=B_rel directsum nabla_chi_geo. 000079 | It explicitly says these bundle data reorganize the primitive seed 000080 | package. Its next theorem identifies D_geo, not a compressed D_rel, as 000081 | the geometric Dirac anchor with the stated Clifford principal symbol. 000082 | 000083 | Crucially the local-spectral-scale definition109-126 gives a REAL 000084 | profile-dependent action on that branch: 000085 | chi_geo=Lambda exp(sigma), 000086 | D_sigma,geo=exp(-sigma/2)D_geo exp(-sigma/2), 000087 | S_rel[sigma,D]=Tr f((D_sigma,geo+B_rel)/Lambda) 000088 | -Tr f(D_ref/Lambda)+i*pi*Delta_eta/2. (2) 000089 | 000090 | This is the concrete original operator realization to use; no new 000091 | choice of an A_Sigma operator is needed to manufacture it. 000092 | paper-source:e9244069a9db317a185d62c7:44-141, 000093 | documents/archive--paper-latex--tfpt-31/sections/ 000094 | 007-subsection-occupancy-and-boundary-to-bulk-bridge.tex, 000095 | SHA256 9c441d14de6a6c9586a4d2b077b160a4c5ce020d82243ff832300ff9554e6397. 000096 | 000097 | The later tfpt42/45 source retains the explicit SAME geometric argument 000098 | K_sigma but changes the functional reading to canonical residues: 000099 | Zcal_rel(s;sigma)=Tr_rel(|K_sigma|^(-2s)-|D_ref|^(-2s)) 000100 | +i*pi*Delta_eta/2, 000101 | E_D=Res_(s=1) Zcal_rel, Gamma_D^(4)=FP_(s=0) Zcal_rel. 000102 | It explicitly states D_geo=D_ref+D_rel=D, the geometric-anchor role, 000103 | and the same seed-to-geometric package reorganization. These are the 000104 | source45 metrology001:15-17,30-66,98-128 definitions. 000105 | paper-source:10c47d481fd025831567f94b:30-150 and 000106 | research-source:20271554929f9198008f7754:1-146, 000107 | documents/archive--tfpt-45--source_extracts--05_metrology_source/ 000108 | sections/001-subsection-from-bridge-to-geometric-completion.tex, 000109 | SHA256 f03609effe2abf6629a0057fa024c29fba0c59176072588599bfa1646f48939e. 000110 | The same wording appears in carrier006:286-406, 000111 | paper-source:0718d9ef29b886f7d99dba6d, 000112 | SHA256 179b2d568e897e059c2d1ef804bb9b06ee053483d366fc9d24189f1df44253b2. 000113 | 000114 | Hence the original geometric route is supplied in both a profile-trace 000115 | version and a later profile-free residue version. Sharing K_sigma does 000116 | not make these two FUNCTIONALS equal. Keep the selected original version 000117 | and its trace, subtraction, eta and scale. Neither one is inferred from 000118 | the word relative alone. 000119 | 000120 | 3. WHAT THIS SAYS ABOUT A_SIGMA AND THE BOSONIC DATA 000121 | 000122 | A_Sigma,seed/geo is presented as bundle/superconnection package data, 000123 | not as a definition A=sum a[D,b] of a bounded represented inner one-form. 000124 | In particular it contains nabla_chi, which is covariant-derivative data. 000125 | A covariant derivative and its contracted operator realization cannot 000126 | be reduced to its local connection coefficient one-form by notation. 000127 | The original geometric reconstruction theorem fixes the full elliptic 000128 | principal symbol through D_geo before the later K_sigma is evaluated. 000129 | 000130 | The same bosonic source has explicitly first-order objects in it: 000131 | B_Ea=sigma^mu nabla_mu^(a)+Phi_a on the compact normal S2, 000132 | with the stated E2/E3 determinant classes and indices. 000133 | Freshly read original metrology001:246-280, 000134 | paper-source:a2928c70a5770cb4a52e801d:231-280, 000135 | SHA256 f03609effe2abf6629a0057fa024c29fba0c59176072588599bfa1646f48939e. 000136 | This is sufficient to reject a blanket claim that all symbols in the 000137 | package are bounded multiplication operators. It does NOT identify 000138 | B_rel with a chosen sum of the B_Ea, count a second Dirac kinetic term, 000139 | or derive the physical B_Ea vertex by equal dimensions. The operative 000140 | K_sigma already keeps the actual B_rel on its declared bundle. 000141 | 000142 | 4. ACTUAL WHOLE-U ATTACHMENT TO THAT GEOMETRIC CALLER 000143 | 000144 | Use the already proven full chart U for the same old pair, including its 000145 | actual tau and reference, and extend it by the identity on other original 000146 | bundle/field components where required. Let B_rel^U denote conjugation 000147 | of the actual B_rel with that full map, not a replacement boson term. 000148 | U is constant in the fixed H/collar chart and commutes with scalar 000149 | exp(-sigma/2). Then exactly 000150 | 000151 | U*K_sigma U=exp(-sigma/2)(U*D_geo U)exp(-sigma/2)+B_rel^U 000152 | =exp(-sigma/2)mathfrakD_h exp(-sigma/2)+B_rel^U. 000153 | 000154 | The actual reference is mathfrakD_href from PAIR-TYPING/FUNCTOR, with 000155 | Phi_ref,S=L(m) or Phi_ref,T=L(p)+D_M for the supplied old lift. The 000156 | corresponding complete M1 reader and inverse already carry these 000157 | coefficients. No new R_ref is selected at the geometric step. 000158 | 000159 | Applying the ORIGINAL trace in(2) therefore gives the identical full 000160 | geometric action in the new chart. Applying the ORIGINAL residue/FP 000161 | functional instead gives its identical residue action. The original 000162 | boson/fermion integration, state, support, selector, all external source 000163 | legs and reference scalar are transported exactly as in 000164 | FULL-FUNCTOR-REVIEW.txt. Thus this is an actual S/Z attachment to the 000165 | existing geometric realization, not another missing-map diagnosis. 000166 | 000167 | In a moving geometric frame U_g, its derivative connection and domain 000168 | transport also go with this equality. It is not legitimate to omit 000169 | those terms while differentiating the geometry. They are determined 000170 | coordinate terms, not an additional interaction choice. 000171 | 000172 | 5. KEEP SCALES, THE REFERENCE AND THE CURRENT IR SCOPE EXACT 000173 | 000174 | For constant sigma0 the ORIGINAL numerator in(2) can be rewritten 000175 | K_sigma/Lambda=(D_geo+exp(sigma0)B_rel)/chi_geo, 000176 | chi_geo=Lambda exp(sigma0). 000177 | This explicitly retains the full anchor D_geo in the geometric 000178 | realization. It is a numerator identity, not permission to rescale the 000179 | reference: the printed reference in(2) remains D_ref/Lambda. Replacing 000180 | it by D_ref/chi_geo changes the subtraction unless the original 000181 | reference is transported accordingly. R032 already distinguishes 000182 | scaled and unscaled references; that known result is reused. 000183 | 000184 | The current compact tfpt2 section010:279-318 writes D_rel+A_Sigma and 000185 | calls its D_rel the modular matter/gauge Dirac, with the finite v258 000186 | implementation. Its final paragraph expressly calls the delivered 000187 | object a BOUNDARY relative object and the real4D ambient redundancy 000188 | conditional on SEAM.EQUIV.01 plus Bisognano-Wichmann. 000189 | paper-source:2f8a823f065f3d3a1c82d4dc:273-318, 000190 | SHA256 3a31f272d999f1e0b027f25a877068576c6cbdc83150c7b5c0f8a6e17bed1904. 000191 | 000192 | Read that compact formula with the existing organizing-package and 000193 | geometric-realization route above. Do not turn its omitted package 000194 | evaluation into a theorem that the original geometric map was absent. 000195 | Equally, do not claim that a universal literal operator identity between 000196 | every historical D_rel+A_Sigma and K_sigma was proved by scalar profile 000197 | moments. These statements concern different levels of the given source. 000198 | 000199 | The finite modular reader supplies the internal coefficient through the 000200 | already established complete M1 inverse. The existing geometric/product