TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/W8-ORIGINAL-STRESS-RECONSTRUCTION.txt
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000001 | W8 IN THE FULL ORIGINAL STRESS CYCLE
000002 | ==================================
000003 | 26 October 2026
000004 | Claim: MATH.ORIGINAL-SEAM.W8-ORIGINAL-STRESS-RECONSTRUCTION.01
000005 |        MATH.ORIGINAL-SEAM.W8-TANGENTIAL-RESPONSE.01
000006 | Verdict: exact source reconstruction; actual local P1 image remains OPEN.
000007 | 
000008 | 1. The original generator is larger than its folded subalgebra
000009 | -------------------------------------------------------------
000010 | Retain the actual one-channel negative sea CAR source, quarter alpha=1/4 or
000011 | 3/4, all adjoints, and the old positive excitation generator H0. Its stress
000012 | raising modes on the finite-energy core are
000013 | 
000014 |  Lcal_-n = sum_r (r+alpha+n/2) :a*_(r+n) a_r: .           (1)
000015 | 
000016 | For n=8 this is exactly the old CAR Ward formula. The c=1 ground momentum
000017 | and conformal weight in the fixed NS subtraction are
000018 | 
000019 |  p=alpha-1/2=plus or minus1/4,       h=p^2/2=1/32.        (2)
000020 | 
000021 | The eightfold current presentation instead has
000022 | 
000023 |  Lhat_n=Lcal_(8n)/8+(21/64)delta_(n,0),
000024 |  N=Lhat_0-85/256=H0/8.                                  (3)
000025 | 
000026 | Only the multiples of eight of the original modes appear in (3). These are
000027 | not all original stress operations. A failure to span a state using Lhat
000028 | does not exclude a reconstruction with all Lcal modes.
000029 | 
000030 | 2. Positive theorem at every original source grade
000031 | --------------------------------------------------
000032 | The fixed relative-charge-zero CAR sector is the one-current Heisenberg
000033 | Fock sector F_p. Its excitation character is product_(n>=1)(1-t^n)^(-1),
000034 | equivalently the Maya-diagram/partition count. On this SAME source the
000035 | Sugawara modes have c=1, lowest weight (2), and the given adjoints.
000036 | 
000037 | The c=1 Kac determinant has zeros only at
000038 | 
000039 |  h_(r,s)=(r-s)^2/4,       r,s positive integers.          (4)
000040 | 
000041 | The actual weight1/32 is not a zero. The natural Verma-to-Fock map is
000042 | therefore injective at every grade. Both sides have dimension p(k) at
000043 | grade k, so this map is onto at each grade. Hence
000044 | 
000045 |  span{Lcal_-lambda Omega_alpha : |lambda|=k}=(F_p)_k,
000046 |  closure span{all Lcal_-lambda Omega_alpha}=F_p.          (5)
000047 | 
000048 | This uses the original state and current algebra, not a new bosonic state.
000049 | The classification is standard Feigin-Fuchs theory, not claimed as new.
000050 | A primary account is Bouwknegt, McCarthy and Pilch, hep-th/9108023, section2:
000051 | the type-I Verma/Fock modules are isomorphic and irreducible. At c=1, Q=0,
000052 | the exceptional momentum condition is sqrt2 p integral; p=plus or minus1/4
000053 | is nonexceptional. This is also the elementary specialization (4).
000054 | 
000055 | 3. An explicit exact CAR certificate on the first W8 response
000056 | ------------------------------------------------------------
000057 | The earlier complete two-point calculation gives
000058 | 
000059 |  z^8 Y_d(W8,z)|d> = sum_(ell>=0) v_ell z^ell,
000060 | 
000061 | where v_ell has folded N level ell and original H0 grade8ell. In particular
000062 | v_1 has original grade8, not original grade1. There are22 partitions of8.
000063 | 
000064 | The new checker constructs the ACTUAL22 states
000065 | 
000066 |  Lcal_-lambda Omega_alpha,       |lambda|=8,
000067 | 
000068 | by the CAR bilinear (1), on modes -8,...,7 with the original filled sea.
000069 | This finite window is exact for grade at most8: any particle at r>=8 or
000070 | hole at r<=-9 already forces neutral excitation energy at least9. No finite
000071 | CCR representation, cyclic UV identification, or continuum cutoff is used.
000072 | Intermediate words retain the whole original neutral CAR sector. They are
000073 | not projected back into the smaller residue-preserving folded-current sector.
000074 | 
000075 | The target is also built literally in CAR. In the old standard E8 chart,
000076 | the eight coefficients of v_1 are in the previous two-point JSON. With the
000077 | SAME old orthogonal M, source-channel coefficients are M^T v_1. The target
000078 | is their sum of a*_k a_(k-8)Omega, including the exact fermionic signs.
000079 | 
000080 | For BOTH original quarters the22 by22 coefficient matrix is invertible,
000081 | with exact determinant
000082 | 
000083 |  210367012400956593540483748829474773276980423821618625375
000084 |  /87112285931760246646623899502532662132736.
000085 | 
000086 | All22 coefficients of the actual W8 target in this stress-word basis are
000087 | given in w8-stress-cyclic-results.json. Direct substitution has residual0.
000088 | This is a reconstruction of the target, not a dimension-only argument.
000089 | 
000090 | [OMITTED: non-scientific conversation or execution metadata]
000091 | [OMITTED: non-scientific conversation or execution metadata]
000092 | [OMITTED: non-scientific conversation or execution metadata]
000093 | [OMITTED: non-scientific conversation or execution metadata]
000094 | [OMITTED: non-scientific conversation or execution metadata]