TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/charge_transport/HILBERT-TYPE-TEST.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | THE EXISTING SOURCE DISTINGUISHES MANY-BODY TIME FROM GEOMETRIC DIRAC
000002 | 29 October 2026. A scoped test of a proposed identification, not a P1 no-go.
000003 | 
000004 | The source002 operational Seed already includes the full local observable
000005 | net, state, reflection and time. Its C_seed proof describes the one-sided
000006 | OS Hilbert space. The optional geometric realization in source001:83-91
000007 | instead uses L2 spinors with finite internal coefficient. These descriptions
000008 | must be related by actual maps; a full quantum Hilbert space is not declared
000009 | a one-particle space merely by giving it a geometric name.
000010 | 
000011 | There is a decisive spectral check using the already existing actual source,
000012 | without choosing a new model. toe_research_20260909.tex:67-95 gives
000013 | 
000014 |  H=sum_j |j-1/4| n_j^exc,
000015 |  Q=sum_(j<=0) n_j^b - sum_(j>=1) n_j^d.
000016 | 
000017 | For r=0,...,m-1 form the independent particle/hole pair
000018 |  b^*_(−r) d^*_(r+1).
000019 | It has charge zero, even fermion parity, and exact energy
000020 |  (r+1/4)+(r+3/4)=2r+1.
000021 | Every subset of these m pairs gives a different orthonormal Fock vector
000022 | with charge zero and total energy at most sum_(r=0)^(m-1)(2r+1)=m^2.
000023 | Consequently, even on the neutral sector,
000024 | 
000025 |  N_(H,Q=0)(m^2) >= 2^m.
000026 | 
000027 | For any finite p>0 this implies
000028 | 
000029 |  Tr_(Q=0) (1+H)^(-p) >= 2^m/(1+m^2)^p -> infinity.
000030 | 
000031 | Thus the actual many-body time generator has no finite p-summable resolvent.
000032 | The same holds on the full source space. Positive affine rescaling of H
000033 | does not change this conclusion. The proof is an exact occupation count;
000034 | it needs no asymptotic partition formula, numerical fit or new cutoff.
000035 | 
000036 | Therefore a proposed unitary identification of THIS full source time with
000037 | the absolute value of a compact finite-dimensional elliptic first-order
000038 | Dirac operator with finite-rank coefficient and its usual finite p-summable
000039 | realization is impossible. Unitary equivalence preserves the displayed
000040 | trace. Restricting merely to neutral/even states does not repair it.
000041 | Failure of finite p-summability does not exclude compact resolvent or finite
000042 | thermal heat traces. The already existing thermal source remains intact.
000043 | The standard finite-order geometric condition is explicit in Connes,
000044 | On the spectral characterization of manifolds, introduction condition (1):
000045 | https://arxiv.org/html/0810.2088v1 . The occupation bound above, rather than
000046 | that geometric theorem, proves the actual source's divergent power traces.
000047 | 
000048 | This excludes only that literal joint Hilbert-space/TIME identification.
000049 | An abstract Hilbert-space unitary alone is not excluded. A separately
000050 | specified D_coll on the same OS space need not equal H. The P1 primitive
000051 | definition does not itself explicitly impose every summability hypothesis
000052 | used in this geometric comparison. No exclusion of all P1 realizations,
000053 | all noncommutative spectral geometries, or a correct one-particle extraction
000054 | is claimed. The microscopic source and the geometric Clifford chart remain
000055 | valid in their stated roles.
000056 | 
000057 | Practical continuation: identify the one-particle/coefficient functor or a
000058 | direct observable-net embedding in the original construction before applying
000059 | another Fock quantization to its full OS Hilbert space. Prove the actual
000060 | time intertwiner separately. A positive thermal source may use a two-sided
000061 | GNS Liouvillian; positivity of that Liouvillian is not a universal physical
000062 | requirement. No new Gaussianity or independent Berezin measure is demanded
000063 | for a complete already supplied operational Seed.