TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/admissible_pair_20261028/EXTERIOR-TRANSFER-AND-E8-HODGE-20261102.txt
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000001 | FIVE-SLOT EXTERIOR TRANSPORT AND THE ORIGINAL E8 HODGE BRACKET
000002 | =============================================================
000003 | 
000004 | 2 November 2026. Conditional algebraic compatibility test on the existing
000005 | marked five-slot carrier. This is not a derivation of an original P1 event,
000006 | not a selection of a Higgs source or physical CP instrument, and not a
000007 | replacement of the full interacting generator.
000008 | 
000009 | 1. Exact source-tensor identity
000010 | ------------------------------
000011 | 
000012 | Let E=C^5 with its given orientation, H=Lambda^2 E, and use the five original
000013 | signed epsilon matrices C_m of hodge_amplitudes(). They are bilinear maps
000014 | H -> H*, not endomorphisms with an implicit identification of H* and H.
000015 | For ANY complex 5x5 matrix A put D=Lambda^2 A. In the original oriented
000016 | duad bases, the fully transported bilinear maps obey
000017 | 
000018 |    D^T C_m D = sum_n adj(A)_(n,m) C_n.                         (1)
000019 | 
000020 | Here T is transpose, NOT Hermitian adjoint. The adjugate is polynomial and
000021 | (1) also holds for singular A. Proof: expanding both sides on four original
000022 | vectors is the cofactor identity
000023 | 
000024 |  epsilon(e_m,A u,A v,A w,A z)
000025 |    = sum_n adj(A)_(n,m) epsilon(e_n,u,v,w,z).
000026 | 
000027 | Thus a correctly typed common A on both wedge legs keeps the original marked
000028 | epsilon bracket in its original 5-dimensional bracket span. For invertible A
000029 | the two spans are equal; for singular A the span can shrink. It does not say
000030 | that a given physical or interacting transfer actually has this common A, nor
000031 | that the source implements the necessary tensor-product representation.
000032 | 
000033 | An independently selected two-mode particle/hole transfer is not an A on E.
000034 | The R087 two-mode diagonal transfer must not be inserted into (1) by dimension
000035 | matching. The family leg, source scalar, reference, phase, domain, original
000036 | field insertion, reflection and time transport all require the actual map.
000037 | 
000038 | 2. Original finite 5-cycle and the trace condition
000039 | -------------------------------------------------
000040 | 
000041 | In the marked carrier chart take the cyclic slot permutation A e_j=e_(j+1).
000042 | Then A^5=I, (1) rotates the original five C_m, and the ordinary Kraus effect
000043 | sum_m K_m^* K_m=3I for K_m=D^T C_m D. Therefore the given Hodge CPTP map
000044 | constructed with K_m/sqrt(3) agrees with the original one. This is the
000045 | carrier's algebraic five-cycle; it is not an identification with physical time.
000046 | 
000047 | For an arbitrary supplied A, put B=adj(A). The original 25 matrices C_n^* C_q
000048 | are linearly independent, while sum_n C_n^* C_n=3I. Consequently the five
000049 | equally weighted K_m give a trace-preserving map after ONE common scalar
000050 | rescaling iff B B^*=r I_5 for some r>0. For invertible A this is equivalent
000051 | to A A^*=s I_5 with s>0. Thus preserving the Hodge span alone does not ensure
000052 | that a naive unweighted carrier operation is TP. A family-dependent positive
000053 | environment metric, more channels or an actual original normalization could
000054 | alter this conclusion, so this is not a no-go for the full TFPT source.
000055 | 
000056 | An exact diagnostic, not a proposed physical source: for A=diag(1,1,1,1,1/2),
000057 | the effect sum K_m^*K_m/3 equals 1/2 on duad |01> and 1/4 on |04>. No single
000058 | normalizing scalar makes both one. This distinguishes source span from source
000059 | weight and is compatible with the more general R086 joint family theorem.
000060 | 
000061 | 3. The precise interaction term that would need evaluation
000062 | -----------------------------------------------------------
000063 | 
000064 | On a finite regularized common Fock representation, suppose the ACTUAL source
000065 | generator could be written H= dGamma(h)+V, with V retaining all its original
000066 | field couplings and vertices. This decomposition itself is a premise that
000067 | must be derived from the original source; it is not asserted for S_adm here.
000068 | For a creator a*(f), put f_s=exp(-s h) f. Differentiation of
000069 | 
000070 |    F(s)=exp(-(ell-s)H) a*(f_s) exp(-sH)
000071 | 
000072 | gives the exact identity
000073 | 
000074 |  exp(-ell H)a*(f) - a*(exp(-ell h)f)exp(-ell H)
000075 |    = - integral_0^ell exp(-(ell-s)H) [V,a*(f_s)] exp(-sH) ds. (2)
000076 | 
000077 | Equation (2) says what must be evaluated before a free creator relation can be
000078 | applied to the interacting generator. Vanishing of the integral is a condition;
000079 | it need not follow from one-particle data, nor does nonzero V alone prove that
000080 | every integral or matrix element is nonzero. For time-dependent external fields
000081 | the ordered propagator and its actual source insertions must replace the
000082 | constant-h formula. The two equations concern different mathematical levels:
000083 | (1) is a finite carrier bracket identity; (2) is a conditional Fock generator
000084 | identity. They are not yet an identified common physical process.
000085 | 
000086 | The checker applies (2) to the already recorded R087 two-mode control with
000087 | ell=1, h=diag(log 4,log 64) and V=-(log 2)n1 n2. The full transfer is
000088 | diag(1,1/4,1/64,1/128), and for the first creator the exact defect in the
000089 | |second> to |pair> matrix entry is 1/256 on BOTH sides of (2). The same
000090 | matrix is the previous counterexample, not a source-selected interaction.
000091 | 
000092 | 4. Reproduction and provenance
000093 | ------------------------------
000094 | 
000095 | Run from the TFPT repository root:
000096 |  python3 -B experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/admissible_pair_20261028/check_exterior_transfer_hodge.py
000097 | 
000098 | The checker loads the existing exact SymPy original C_m builder. It checks all
000099 | five equations for each of an invertible non-diagonal integer matrix, a
000100 | singular matrix and the marked cycle, cycle order and full effect, and the two
000101 | anisotropic exact effects. Output: 15 cofactor identities, cycle order and
000102 | effect, anisotropic 1/2 versus 1/4, and the exact two-mode interaction integral
000103 | all PASS. The full polynomial identity
000104 | follows from the epsilon proof, not extrapolation from these examples.
000105 | 
000106 | Sources: source_recursion_20261028/EPSILON-WARD-SOURCE-AUDIT.txt; sibling
000107 | SHARED-HODGE-PROCESS-AND-CORRELATION.txt (25 independent products and joint TP
000108 | theorem); seam-collar-multifield-20261016/PROOF.txt (original free collar
000109 | versus interacting source and original higher vertices). No claim of a new
000110 | Lean proof, ledger promotion, physical event or completed F1.