TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-source-reconstruction-20261024/ORIGINAL-SOURCE-AUDIT.txt
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000001 | MATH.SEAM.ESERIES.SOURCE-RECONSTRUCTION.06 / original-source audit
000002 | 24 October 2026
000003 | 
000004 | Question and verdict
000005 | 
000006 | Does the established P1/P2 compiler and its now reconstructed canonical E8
000007 | field completion already identify the original operational source, state,
000008 | time, and local observables?
000009 | 
000010 | Verdict: NOT DERIVED BY THE CITED ORIGINAL ARGUMENTS. This is an audit of
000011 | specific implications, not a theorem that no stronger P1 reconstruction can
000012 | exist. The mathematical extension theorem in THEOREMS.txt is exact in its
000013 | declared chiral object class. It does not establish that the original source
000014 | belongs to that class. Historical physical-origin tasks remain historical;
000015 | this audit does not reactivate them or alter ledger statuses.
000016 | 
000017 | The established compiler is retained throughout:
000018 | 
000019 |   P1/P2 -> D5+A3 with marked C4 glue -> L=E8,
000020 |   c3=1/(8*pi), phi_base=(4/3)c3=1/(6*pi).
000021 | 
000022 | 1. Three different input packages
000023 | 
000024 | The current architecture document, documents/tfpt_1_architecture_e8/sections/
000025 | 001-source.tex, lines67-80, declares P1 as an oriented seam, a primitive
000026 | reflection-positive boundary kernel, unit winding, and c3 normalization.
000027 | It explicitly treats P1/P2 as axioms. Its conditional compiler result is
000028 | not disputed by the following audit.
000029 | 
000030 | The stronger spectral boundary datum is
000031 | 
000032 |   T_boundary=(A_+,H_+,D_+,J,Gamma,B_Sigma).
000033 | 
000034 | Its definition assumes the algebra and spectral geometry. It specifies a
000035 | smooth compact boundary manifold, a product collar, an adapted self-adjoint
000036 | boundary operator and reflection regularity. It does not explicitly supply
000037 | the region-indexed observable net O -> A(O), a distinguished cyclic state,
000038 | or an action of time translations on that net.
000039 | 
000040 | The still stronger operational seed is
000041 | 
000042 |   S=(A_loc,tau_t,Theta,omega,[u_Sigma],D_coll).
000043 | 
000044 | The original definition includes A_loc, omega and tau_t as inputs. Its
000045 | completion functor C_seed sends this package to a spectral boundary datum.
000046 | The displayed direction is S -> T_boundary. No inverse is constructed.
000047 | 
000048 | Original sources: boundary source section001, lines65-80; section002,
000049 | lines103-144. Full paths are in contract_index.json.
000050 | 
000051 | 2. Correct minimization theorem
000052 | 
000053 | Let C:S -> B be any functor and d:B -> N^k an isomorphism-invariant defect,
000054 | with the original lexicographic order on N^k.
000055 | Write d_seed=d o C. The unconditional conclusion is
000056 | 
000057 |   s minimizes d_seed iff C(s) minimizes d on im(C).
000058 | 
000059 | Proof. Both comparisons are exactly comparisons of values d(C(s')).
000060 | Passing from the image to all B needs an extra condition: at least one
000061 | global minimizer must lie in im(C). Even then, recovering s uniquely needs
000062 | classification of the relevant fiber of C, not just its image.
000063 | 
000064 | Consequently well-foundedness proves existence of a minimum defect vector
000065 | in any nonempty allowed image. It does not prove that its fiber has one
000066 | isomorphism class. A nonempty subset of the positive integers has a least
000067 | element k>=1; it need not contain1. Unit spectral flow requires either its
000068 | own axiom or an admissible unit-flow witness.
000069 | 
000070 | The manuscript makes the first statement correctly at section002,
000071 | lines194-216: uniqueness is reduced to classification of the last stratum.
000072 | It does not provide that classification in the later proof at269-306.
000073 | The proof at238-243 also replaces minimization over im(C_seed) by all
000074 | admissible bordisms without proving the necessary image condition.
000075 | 
000076 | This proof audit makes no independence claim about the full P1 axioms.
000077 | 
000078 | 3. Conditional spectral-rescaling obstruction
000079 | 
000080 | Assume the boundary class is closed under positive constant rescaling of
000081 | its metric, reference pair, finite-block definition and dimensional cutoffs.
000082 | Assume a compact self-adjoint elliptic realization, or a derived B_Sigma
000083 | with a first positive absolute eigenvalue, is part of the object and is
000084 | preserved by its isomorphisms. Then define for lambda>0
000085 | 
000086 |   B_lambda=(A_+,H_+,lambda D_+,J,Gamma,lambda B_Sigma).
000087 | 
000088 | Using g_lambda=lambda^-2 g and r_lambda=r/lambda gives
000089 | 
000090 |   lambda D_+=gamma_n(partial_(r_lambda)+lambda B_Sigma).
000091 | 
000092 | The Hilbert spaces for the rescaled constant densities are identified by
000093 | the natural constant unitary. Ellipticity and the collar identities persist.
000094 | Positive rescaling preserves spectral-flow crossing signs, kernels, the
000095 | structural finite ranks, determinant-line degrees and the Calderon Cauchy
000096 | solution space. Under the stated cutoff/reference covariance assumptions,
000097 | all four defect entries and essentialization are therefore unchanged.
000098 | 
000099 | Let mu_1>0 be the first positive absolute eigenvalue of the chosen compact
000100 | realization. A unitary isomorphism B_lambda -> B_nu would preserve its
000101 | spectrum and imply lambda*mu_1=nu*mu_1, hence lambda=nu. Thus any minimum
000102 | belongs to a continuum of equal-defect, pairwise nonisomorphic objects.
000103 | 
000104 | The original chi_seed does not normalize this family: it is defined by
000105 | chi_seed=(lambda_1^+(|B_Sigma|))^-1 and therefore scales as lambda^-1.
000106 | The dimensionless c3 and winding remain unchanged. The spectral action
000107 | is scale-sensitive, since D_lambda/chi_lambda=lambda^2 D/chi, but the
000108 | boundary category asks only for its finiteness, not a specified value or
000109 | a stationarity equation.
000110 | 
000111 | The rescaling conclusion is CONDITIONAL, not an unconditional counterpair
000112 | of complete P1 models. The text does not fully define rank_ess H_prim^fin,
000113 | the action domain, a global external reference, or covariance of the
000114 | freely chosen epsilon_adm spectral window. An absolute fixed cutoff or
000115 | fixed reference could prevent the rescaling, but would have to be declared
000116 | and analyzed. Nor does the proof sketch for C_seed establish that a time-
000117 | rescaled operational seed is mapped to B_lambda. No assertion about that
000118 | operational image is made here.
000119 | 
000120 | Original sources: section001 lines42-55,192-261,302-311,353-376;
000121 | section002 lines166-183,252-306; section003-subsection-relative-objects.tex
000122 | lines1-16, all in the same boundary-source directory.
000123 | 
000124 | 4. What an actual reconstruction theorem can recover
000125 | 
000126 | The complete supported graded Schwinger hierarchy, with the hypotheses of
000127 | Osterwalder-Schrader II, reconstructs its Wightman theory up to unitary
000128 | equivalence. It supplies all moments, their support and covariance, not
000129 | only a scalar kernel or a spectral tuple. Complete truncated functions
000130 | determine the moments by the partition expansion with the required graded
000131 | signs. This is the data actually used in the QFT-source sections005 and008.
000132 | 
000133 | In this paragraph R denotes linear geometric reflection on the graded word
000134 | algebra, with the prescribed fermionic signs. The antilinear OS reflection
000135 | is j(F)=R(F)^*. This R is distinct from the antilinear VOA PCT operator in
000136 | THEOREMS.txt. On the positive-time word algebra the reflection form is
000137 | 
000138 |   (F,G)_OS = omega(j(F) G) = omega(R(F)^* G),
000139 |   N_OS={F:(F,F)_OS=0},
000140 |   H_OS=completion(A_+/N_OS), Omega=[1].
000141 | 
000142 | With the translation/regularity/contraction hypotheses of OS reconstruction,
000143 | the induced positive-time translations extend to T_t=exp(-t H), H>=0.
000144 | The remaining OS hypotheses and analytic continuation reconstruct the
000145 | Wightman distributions and their common field domain. Arbitrary positive-
000146 | time multiplication is NOT asserted to act on the quotient: N_OS need not
000147 | be a left ideal for such multiplication.
000148 | 
000149 | Alternatively, Schlingemann's algebraic reconstruction starts from a full
000150 | Euclidean net with Euclidean-group action, a regular reflection-positive
000151 | functional, and the time-zero condition. It then constructs a Haag-Kastler
000152 | net using the controlled time-zero representation. A net with only tau_t,
000153 | Theta and omega as listed in Seed_op has not yet supplied these additional
000154 | hypotheses.
000155 | 
000156 | Neither reconstruction automatically recovers an arbitrary unreduced
000157 | Euclidean presentation, its collar generator or seam class. It recovers
000158 | the theory generated by the supported correlation data after the OS null
000159 | reduction. Claiming an inverse to C_seed needs a separate faithfulness and
000160 | localization theorem.
000161 | 
000162 | 5. The exact remaining source identification
000163 | 
000164 | THEOREMS.txt completely reconstructs the canonical local chiral extension
000165 | of V_D5 tensor V_A3 with the prescribed four C4 sectors, once each. To
000166 | identify that mathematical completion with the original primitive source,
000167 | one must prove that the source has this supported unitary chiral realization
000168 | and this generated sector content, and intertwines the actual products,
000169 | reflection, state and specified time action. The compiler's lattice result
000170 | alone does not state such an operator/localization map.
000171 | 
000172 | This is a single conjunctive identification problem. It cannot be replaced
000173 | by declaring a lattice, a preferred vector, or a convenient Hamiltonian to
000174 | be the source. The canonical conformal L0 also does not fix an absolute
000175 | physical time unit; on a circle of radius R the scale is L0/R.
000176 | 
000177 | References checked at the originals
000178 | 
000179 | Osterwalder-Schrader I, II (II corrects and strengthens the reconstruction
000180 | hypotheses of I):
000181 |   https://doi.org/10.1007/BF01645738
000182 |   https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-42/issue-3/Axioms-for-Euclidean-Greens-functions-II-with-an-Appendix-by/cmp/1103899050.pdf
000183 | Schlingemann, From euclidean field theory to quantum field theory:
000184 |   https://arxiv.org/html/hep-th/9802035