TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/QP-SOURCE-FUNCTORIALITY-ADDENDUM.txt
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000001 | q TO p IMAGE SOURCES: ORIGINAL FUNCTORIALITY REMOVES A SECOND INPUT
000002 | Independent original-source addendum, 2026-10-30.
000003 | Clarifies QP-WHOLE-SOURCE-PULLBACK-REVIEW.txt without changing that file.
000004 | 
000005 | DECISION
000006 | 
000007 | An independently specified Phi_p is NOT REQUIRED for the transported image
000008 | source. The original Seed category and inner-fluctuation construction
000009 | already transport the ENTIRE datum. Once the actual unitary source/bundle
000010 | map is given, the p image has its representation, operator, grading,
000011 | Real structure, background, domain, state, reflection, time and reference
000012 | fixed by that map. It is the same covariant construction in the transported
000013 | chart, not a new freely chosen p theory.
000014 | 
000015 | The previous review's request to compare an independently prescribed p
000016 | background applies only if one additionally claims equality with such a
000017 | SEPARATELY FIXED target. It must not become a prerequisite or a new Phi_p
000018 | rule for the positive image-source arrow. That stronger requirement was
000019 | unnecessary for the image interpretation and is removed here.
000020 | 
000021 | The remaining substantive test is the actual global comparison of the
000022 | geometric spin/source pullback with the normal/collar realization and any
000023 | assertion that originally fixed marks are unchanged. None of those tests
000024 | requires a second choice of p controls.
000025 | 
000026 | 1. EXACT ORIGINAL SEED CONTRACT
000027 | 
000028 | Original path:
000029 |  documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000030 |  sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex
000031 | 
000032 | Lines103-119 define
000033 | 
000034 |  Seed=(A_loc,tau_t,Theta,omega,[u_Sigma],D_coll)
000035 | 
000036 | and explicitly make morphisms unitary intertwiners preserving these
000037 | displayed structures. Lines122-143 declare the collar-completion FUNCTOR
000038 | C_seed:Seedop->SBord_adm and its boundary tuple
000039 | (A_+,H_+,D_+,J,Gamma,B_Sigma).
000040 | 
000041 | Consequently, given a unitary source isomorphism U, the image is the tuple
000042 | with EVERY datum pushed forward: the represented net/algebra and state,
000043 | time, Theta, seam, collar, domains and declared reference. U is then an
000044 | intertwiner to that image by construction. Reflection positivity,
000045 | clustering and relative data are transported as the same structures.
000046 | This does not state that any arbitrary U is a symmetry of the fixed tuple,
000047 | or prove the original existence/reconstruction theorem beyond its stated
000048 | scope. It identifies the actual equivalence notion used by the originals.
000049 | 
000050 | In particular the finite values of a supplied Y or MR are not independently
000051 | retuned by changing their matrix presentation. The image matrices are
000052 | fixed by conjugating the complete operator and its representation.
000053 | 
000054 | 2. EXACT INNER-FLUCTUATION NATURALITY, INCLUDING THE REPRESENTATION
000055 | 
000056 | Original verification/v254_inner_fluctuations.py:6-7 explicitly defines
000057 | 
000058 |  Omega1_D(A)=span{pi(a)[D,pi(b)]},
000059 |  D_A=D+A+epsilon' J A J^-1.
000060 | 
000061 | Put pi'=U pi U^-1, D'=U D U^-1, J'=U J U^-1,
000062 | gamma'=U gamma U^-1. For each generator, exactly
000063 | 
000064 |  U pi(a)[D,pi(b)] U^-1=pi'(a)[D',pi'(b)].
000065 | 
000066 | Hence U Omega1_D U^-1=Omega1_D' and, with A'=U A U^-1,
000067 | 
000068 |  U D_A U^-1=D'+A'+epsilon' J' A' J'^-1.
000069 | 
000070 | The statement includes every supplied finite block, including Majorana.
000071 | It does not require U to normalize the OLD represented copy pi(A), or to
000072 | belong to its fixed inner gauge group. If one holds pi fixed instead,
000073 | one has asked the stronger stabilizer question. That is not the question
000074 | of covariance under a simultaneous change of the entire datum.
000075 | 
000076 | The actual finite operator is v252_full_finite_triple.py:100-116,
000077 | grading119-121, and Kswap/Japply124-132. The previous whole-operator
000078 | calculation used precisely those blocks. It already proves the target
000079 | matrix Phi_p=-transport(Phi_q) in the compared source chart. Naturality
000080 | makes this the induced background, not a new background selection.
000081 | 
000082 | 3. ACTUAL D5/A3 OPPOSITION AND E8 ROOT TEST
000083 | 
000084 | The existing real signed96 wedge dictionary gives
000085 | 
000086 |  J_F=C_comp, kappa_split=coefficient conjugation,
000087 |  K_F=J_F kappa_split=C_comp kappa_split.
000088 | 
000089 | Thus K_F is the LINEAR signed Hodge complement on the exterior carrier.
000090 | It flips (-1)^N and exchanges the two inequivalent D5 half-spinors. The
000091 | already checked D5 matrices obey
000092 | 
000093 |  K_F H_i K_F^-1=-H_i,
000094 |  K_F E_ij K_F^-1=-E_ij^*,
000095 |  K_F F_ij K_F^-1=-F_ij^*.
000096 | 
000097 | These are actual opposition formulas for all D5 Cartan/root generators,
000098 | not a dimension match. On the family factor the corresponding operation
000099 | is dual transport 4<->bar4; one must include it on the full four slots.
000100 | 
000101 | On the E8 root lattice the simultaneous operation is -I8. It is in
000102 | W(D8): for each coordinate pair (i,j), the product
000103 | s_(e_i-e_j) s_(e_i+e_j) negates both coordinates; four disjoint pairs
000104 | negate all eight. It preserves the roots +/-e_i+/-e_j and the even-parity
000105 | half roots. On D5+A3 it induces opposition on BOTH factors, taking the
000106 | actual signed (16,4) half-root branch to (bar16,bar4).
000107 | 
000108 | A compact Weyl normalizer representative exists by these root reflections.
000109 | Equivalently in a compatible Chevalley basis the full linear opposition
000110 | is H->-H, E_alpha->-E_-alpha. This preserves Lie brackets, has square one
000111 | and commutes with compact conjugation. Lifts differing in root-vector
000112 | phases must be compared in the retained original cocycle/frame; a lattice
000113 | permutation alone does not fix those phases. This phase comparison is
000114 | separate from, and unnecessary for, inner-fluctuation functoriality.
000115 | 
000116 | On the actual marked torus this normalizer sends kappa to kappa^-1 and
000117 | sigma0 to sigma0^-1. It lies outside the fixed K in which kappa is central,
000118 | but it normalizes that D5+A3 structure. Therefore the previous fixed-K
000119 | obstruction is not an obstruction to the full dual source transport.
000120 | The compact Real structure commutes with this real compact operation;
000121 | the split coefficient conjugation and J_F remain distinct operators.
000122 | 
000123 | 4. GLOBAL GEOMETRIC PULLBACK AND ASSOCIATED FIELD DATA
000124 | 
000125 | The actual R=sc^2 covers z->1/z and already sends the q sign/spin packet
000126 | to the p packet with its fixed label shift and spin lift. On the existing
000127 | joint cover its coefficient maps are genuine global maps, not point
000128 | permutations. The spin lift retains r^6=-I. Its endpoint/frame factors
000129 | are precisely the H(x) and twist factors retained in the previous review.
000130 | 
000131 | For a bundle with transitions g_ij, pullback by R replaces them by
000132 | g_ij composed with R. A simultaneous normalizer dual transport replaces
000133 | them by n(g_ij composed with R)n^-1. The associated field map satisfies
000134 | n rho(g)=rho(n g n^-1)n, so it glues globally. Connections, determinant
000135 | powers, all associated representations and products transport with it.
000136 | In particular the D^(-1-m_j) factors on the odd conjugate branch cannot
000137 | be dropped; functorial transport carries them rather than identifying
000138 | them with untwisted copies by hand.
000139 | 
000140 | This proves a global IMAGE construction in the associated bundle category
000141 | once the actual source lift is specified. It does not assert that the
000142 | resulting target transitions equal a separately frozen target trivialization,
000143 | or that the normalizer has identity action on native clocks. Inversion
000144 | also transports the seam orientation and the normal/collar sides. The
000145 | existing source reflection, actual time and reference are carried along,
000146 | not replaced or declared missing.
000147 | 
000148 | 5. CONSTRUCTIVE CONSEQUENCE AND THE REMAINING COMPARISON
000149 | 
000150 | The previous review supplies the exact all96 h/B/charged-chart arrow with
000151 | the actual K_F, including MR, the inverse quarter domain and the required
000152 | normal factor iN. The present original-source test says that its p image
000153 | does NOT wait for an independent Phi_p formula: that formula is the same
000154 | inner-fluctuation/source rule transported with the datum. Its whole graph,
000155 | state, reflection and time are the corresponding pushed-forward structures.
000156 | 
000157 | An assertion that this is the original NORMAL-source realization of the
000158 | geometric q/p spin family must still exhibit that specific global source
000159 | comparison, including its spin and collar maps. An assertion of a
000160 | fixed-background/fixed-clock symmetry must additionally pass the relevant
000161 | stabilizer tests. These are concrete different claims. Neither changes
000162 | the fact that the positive covariant p image is already defined, with no
000163 | new p source law, masses, metric or cover.
000164 | 
000165 | No existing physical source-origin theorem is promoted by this addendum;
000166 | the unnecessary second-target-input condition is the point corrected.
000167 | Earlier registered reports, Atlas and original files are unchanged.