TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/external_candidate_review/PROOF.txt
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000001 | REVIEW OF THE PAIRED-QUARTER / LOWEST-BLOCK SOURCE CANDIDATE
000002 | 29 October 2026
000003 | 
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000011 | 
000012 | 1. Exact original source, not an inferred low-energy convention
000013 | Read archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/
000014 | 001-section-primitive-core-and-carrier-interface-handoff.tex, lines 192-289.
000015 | The actual displayed formula is
000016 | 
000017 |  Pprim = P_(Sigma,+) P_(prim,gap),
000018 |  P_(prim,gap) = 1_[epsilon_adm^2/4,infinity)(T),
000019 |  T = P_(Sigma,+) Dprim^* Dprim P_(Sigma,+).
000020 | 
000021 | The cutoff retains HIGH squared singular values of this compressed operator.
000022 | It is not a projection onto the first eigenvalue of |B_Sigma|. Neither the
000023 | operator identity T=|B_Sigma|^2 nor Hcar=ker(|B_Sigma|-lambda1) is supplied by
000024 | this formula. The occurrence of lambda1 in chi_seed is a separate definition.
000025 | Even if the proposed operator identification is granted for a diagnostic,
000026 | lambda1 < epsilon_adm/2 <= lambda2 REMOVES the lambda1 block and retains the
000027 | lambda2 block. At lambda1=1/2, lambda2=3/2, epsilon_adm/2=3/4 the selected
000028 | squared values are 9/4 and higher, whereas 1/4 is removed.
000029 | This is a countercheck of the proposed identification, not a full P1 model.
000030 | The old positive-only compression result is unchanged: on an actual carrier
000031 | inside Ran(Pprim) subset Ran(P_C), (2P_C-I) is the identity. A different
000032 | coefficient-factor carrier map must be supplied with its own type.
000033 | 
000034 | 2. What the quarter-pair conjugation actually proves
000035 | For A_a=-i partial_theta+a on periodic H1(S1), with 0<a<1, define
000036 | 
000037 |  B_a = 2(A_a directsum A_(1-a)),
000038 |  j(f,g)(theta) = exp(-i theta) (conj(g(theta)),conj(f(theta))).
000039 | 
000040 | Multiplication by exp(-i theta), conjugation, and summand exchange preserve
000041 | the full H1 domain. On each Fourier basis vector j exchanges (n,a) with
000042 | (-n-1,1-a); their B_a eigenvalues are opposite. Thus j^2=I and j B_a j=-B_a
000043 | as equalities of closed operators, not only inside a finite Fourier window.
000044 | For P=1_(B_a>0), j P j=I-P. For a=1/4 the absolute minimum is 1/2 in each
000045 | paired copy. Equal internal multiplicity d gives positive and negative
000046 | minimum eigenspaces of complex dimension d each, if that block is chosen.
000047 | The choice of that block, its multiplicity and its physical field map have
000048 | not followed from section 1.
000049 | On a single quarter operator A_(1/4), no antiunitary j with j A j=-A exists,
000050 | because its spectrum Z+1/4 is not invariant under negation. This excludes
000051 | that specified selfdual dynamical realization. It does NOT exclude a CAR
000052 | algebra on that Hilbert space, the original complex charged source, or an
000053 | alternative properly derived field conjugation.
000054 | 
000055 | 3. The full two-face graph and its oriented conjugation
000056 | Let B be selfadjoint and j an antiunitary involution with jBj=-B. For ell>0,
000057 | 
000058 |  a=(I+exp(-2ell B))^-1, c=(2cosh(ell B))^-1,
000059 |  G=[[a,c],[c,I-a]].
000060 | 
000061 | The bounded spectral calculus gives c^2=a(I-a) and hence G^2=G=G^*.
000062 | It also gives jaj=I-a and jcj=c. Therefore the naive doubled conjugation
000063 | j directsum j obeys
000064 | 
000065 |  (j directsum j) G (j directsum j) = [[I-a,c],[c,a]],
000066 | 
000067 | whose difference from I-G is [[0,2c],[2c,0]]. For finite spectral values c
000068 | is nonzero. Correct separate face covariances therefore do not certify this
000069 | unmodified full-graph conjugation. With the oriented choice
000070 | 
000071 |  C = j directsum (-j)
000072 | 
000073 | one has C^2=I and CGC=I-G exactly, including both off-diagonal correlations.
000074 | This is the standard purification of a selfdual CAR covariance, not a new
000075 | general theorem. See H. Araki, On Quasifree States of CAR and Bogoliubov
000076 | Automorphisms, Publ. RIMS 6 (1970/71), 385-442, Definition 4.4 and Lemmas
000077 | 4.5-4.6, p.391. DOI 10.2977/prims/1195193913.
000078 | Primary full text: https://ems.press/content/serial-article-files/41673
000079 | The one-face operator a is a mixed covariance: a^2-a=-c^2. It is not a
000080 | basis projection at finite ell and finite spectral values. Its purification
000081 | G is a basis projection for C. Applying the existing projector-state theorem
000082 | here does not add an independent quasifree premise to a given pure source.
000083 | But identifying this selfdual CAR functor and C with the physical original
000084 | field/adjoint map remains a separate actual-source identification.
000085 | 
000086 | 4. No internal 3+2 no-go from a changed field type
000087 | If an invariant finite-dimensional complex space H has an involution F and
000088 | an antiunitary j with jFj=-F, then j bijects its two F eigenspaces. Their
000089 | dimensions are equal. This standard lemma applies to that full selfdual
000090 | one-particle space. It does not apply automatically to an internal P2
000091 | coefficient factor, a positive-polarization space E, or a spinor Fock module.
000092 | In particular an independent 3+2 internal involution on E need not equal F.
000093 | The D8 real vector (10,1)+(1,6) and its spinor sectors remain possible. No
000094 | new particle multiplicity is proved by a formal selfdual doubling.
000095 | 
000096 | 5. Necessary next original-source step
000097 | The original J, normal Clifford multiplication, domain, reference and
000098 | reflection must be transported together to the actual two-face field map.
000099 | The algebraic C above is not by itself an OS reflection, a physical time
000100 | generator, or a proof that the original gap selects a P2 carrier. A common
000101 | operative source already includes its state; another arbitrary measure is
000102 | not demanded universally. Stationarity or absence of a displayed map alone
000103 | does not prove general underdetermination: any claimed full countermodels
000104 | must satisfy all the original operator, domain and action conditions.
000105 | This review retains both the useful conjugation repair and these limits.
000106 | 
000107 | Verification
000108 | check_candidate.py uses exact rational symbolic algebra for the paired
000109 | spectral graph with q=exp(ell b)>0. The analytic domain and full-spectrum
000110 | arguments are given above. The four-by-four check is a control of these
000111 | identities, not an all-mode locality or physical origin proof. Assertions
000112 | are not used as the only enforcement of the checks.