TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-PHASE-ENERGY-REFERENCE.txt
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000001 | WHOLE ORIGINAL PHASE SOURCE: ENERGY DOMAIN AND THE RETAINED SEA BODY
000002 | 3 November 2026. Anonymous analytical continuation, actual source only.
000003 | 
000004 | RESULT
000005 | The actual phase lift is implementable, yet can give infinite energy
000006 | relative to a vacuum reference artificially held fixed. Both statements
000007 | are compatible. The existing complete source already prescribes transport
000008 | of its reference, sea body, domain and insertions. Following that source
000009 | removes the apparent contradiction without choosing a new vacuum or a
000010 | free counterterm. This evaluates the phase transport of the given source;
000011 | it does not identify it with primitive interacting S_adm.
000012 | 
000013 | 1. EXACT CHECK WITH THE FULL ORIGINAL MASS COEFFICIENT
000014 | Keep the actual constant full96 coefficient sector already contained in
000015 | the original source, with its arbitrary symbolic Yukawa/Majorana blocks.
000016 | This is an exact specialization of that source, not a replacement model.
000017 | For p=n+alpha write M=-i gamma Phi and h_n=p gamma+M.
000018 | Then {gamma,M}=0, M*=M, M^2=Phi^2, h_n^2=p^2+Phi^2.
000019 | Let E_n=(p^2+Phi^2)^(1/2); it commutes with h_n, Phi, gamma and U.
000020 | The original negative projection is P_n=(I-h_n/E_n)/2.
000021 | For U=exp(i chi gamma), h_chi=U*hU, P_chi=U*PU,
000022 | (M_chi-M)^2=4 sin(chi)^2 Phi^2 exactly, with no matrix diagonal fit.
000023 | Consequently the energy of the transported sea relative to the old h is
000024 | E_P(chi)=Tr(|h|(P_chi-P)^2)
000025 |  =sin(chi)^2 sum_n Tr_F[Phi^2/(p^2+Phi^2)^(1/2)].
000026 | At the original Fourier cutoff |n|<=N this equals
000027 | 2 sin(chi)^2 Tr_F(Phi^2) log N+O(1).
000028 | Thus it diverges when Phi sin(chi) is nonzero, although the unweighted
000029 | sum of squared projection changes is finite. It is the SAME full source
000030 | and its original mass coefficient, not an independently fitted fermion.
000031 | 
000032 | The actual original coefficient gives the explicit positive norm
000033 | Tr_F(Phi^2)=12Tr(Yu*Yu+Yd*Yd)+4Tr(Ye*Ye+Ynu*Ynu)+2Tr(MR*MR),
000034 | where stars on Y/MR denote Hermitian adjoints. Color multiplicity gives
000035 | the quark factor12; all generations and the symmetric Majorana block
000036 | are retained. The direct symbolic control checks this full matrix identity.
000037 | No numerical Yukawa value, Planck calibration or flavor spectrum is fitted.
000038 | 
000039 | 2. LOCAL COEFFICIENT FOR THE ACTUAL SMOOTH VARIABLE FIELDS
000040 | The same leading divergence has a local expression. On the circle of
000041 | coordinate length2pi, the sign projection of the original smooth h has
000042 | the large covariable expansion, away from xi=0,
000043 | P(x,xi)=(I-gamma sign(xi))/2+i gamma Phi(x)/(2|xi|)+O(|xi|^-2).
000044 | The commuting a term has no offdiagonal contribution at order-1. This
000045 | also follows by inserting the bounded perturbation in the same sign
000046 | resolvent integral used in the implementation proof: its first offdiagonal
000047 | term is -M/(2|xi|), and subsequent orders have two inverse powers.
000048 | Smooth coefficients control the Fourier remainder at that order.
000049 | 
000050 | For multiplication U(x), derivatives of the leading sign symbol vanish
000051 | at xi!=0. Thus DeltaP=P_chi-P has leading symbol
000052 | i gamma(Phi_chi-Phi)/(2|xi|), and (DeltaP)^2 has leading symbol
000053 | sin(chi)^2 Phi^2/xi^2. Multiplication by |h| gives the positive energy
000054 | operator |h|^(1/2)(DeltaP)^2|h|^(1/2) with leading symbol
000055 | sin(chi)^2 Phi^2/|xi|. The order-2 remainder is trace class in1D.
000056 | Both spectral tails of the original Fourier cutoff therefore give
000057 | E_P,N(chi)=(log N/pi) integral_0^(2pi) sin(chi(x))^2 Tr_F(Phi(x)^2)dx
000058 |  +O(1).
000059 | The constant-coefficient identity above independently fixes this factor.
000060 | No new spatial dimension, state or regulator scale is selected by this
000061 | asymptotic statement; it describes removal of the specified Fourier cutoff.
000062 | 
000063 | 3. IMPLEMENTABILITY DOES NOT IMPLY PRESERVATION OF THE ENERGY DOMAIN
000064 | In a finite regulator, a sea with occupation P_chi has excitation energy
000065 | Tr[h(P_chi-P)]=Tr[|h|(P_chi-P)^2]>=0. This follows by writing its two
000066 | diagonal occupation blocks in the spectral P/(I-P) split of h.
000067 | In the limit the expectation is the positive quadratic-form energy,
000068 | possibly infinity. Thus the implementer need not preserve the form domain
000069 | of the old positive H=dGamma(|h|), even though it is a unitary on Fock.
000070 | 
000071 | It does NOT follow that the conjugated operator W*H W is unbounded below.
000072 | It is positive selfadjoint on W*Dom(H), with form domain W*Dom(H^1/2).
000073 | An unrenormalized fixed-sea quadratic may instead fail to act on the old
000074 | vacuum: its pair-creation coefficient is not energy weighted S2. These
000075 | are different assertions; an AI claim equating them is not adopted.
000076 | 
000077 | 4. THE ORIGINAL WHOLE SOURCE SUPPLIES THE ACTUAL RESOLUTION
000078 | At the compatible finite original regulator let
000079 | H_P(h)=a^dagger h a-Tr(P h), h_chi=U*hU, P_chi=U*P U.
000080 | The whole original source transport gives exactly
000081 | W*H_P(h)W=H_(P_chi)(h_chi)=H_P(h_chi)+E_counter,
000082 | E_counter=Tr[h_chi(P-P_chi)]
000083 |  =Tr[|h_chi|(P-P_chi)^2]>=0.
000084 | E_counter has the same leading logarithmic coefficient as in section2.
000085 | One cannot omit this scalar and then compare unnormalized sources.
000086 | 
000087 | The original SEED-FIELD source already uses its own E_sea=Tr(P h) in the
000088 | finite exterior kernel, then the positive excitation definition in the
000089 | continuum, never an infinite difference of absolute sea energies. Its
000090 | transport retains E_sea(h_chi)=Tr(P_chi h_chi)=E_sea(h).
000091 | This is precisely the body needed in the last identity. The continuum
000092 | operator is therefore the transported positive excitation operator with
000093 | its transported domain. There is no new discretionary energy constant.
000094 | Transport of D_ref and the complete relative source follows the same rule.
000095 | This resolves the phase energy issue for that GIVEN source, without a
000096 | second fermion integration or an extra Fujikawa term.
000097 | 
000098 | 5. WHY THIS DOES NOT YET SUPPLY THE FUNDAMENTAL INTERACTION
000099 | The complete phase orbit, including Phi_chi, is a coordinate transport
000100 | and is isospectral. Its bosonic spectral weight cannot acquire a new
000101 | stiffness merely from the rotation. The historical geometric axial field
000102 | path holds Phi fixed before the inverse fermion rotation. It is a distinct
000103 | path in the original field space. On that path the actual bosonic source
000104 | body, including the contact term and fixed regulator, must be evaluated.
000105 | The R113 source already distinguishes this body from its numerical h;
000106 | do not replace it by an average, normal-ordering choice or new rotor.
000107 | 
000108 | The recovered restricted chiral dictionary and the full chart diagram
000109 | remove the naive gamma_F/gamma5 identification obstacle in their stated
000110 | scopes. The fixed-reference logarithm fixes what the WHOLE source must
000111 | retain. It is not a new theta potential, a primitive S_adm derivation,
000112 | the full H_tr,pos or the common interacting physical spectrum. Later E8,
000113 | alpha, flavor and clocks retain their own established conditional scopes.