TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/FULL-FINITE-ENERGY-AND-SECTOR-MENU.txt
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Inclusive original lines: 1-117
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | Full finite-energy carrier and the actual sector menu
000002 | 22 October 2026. Continuation of UR.SEAM.MIXED_CURRENT_ORIGIN.01.
000003 | 
000004 | Verdict: source-specific spectral exhaustion; no microscopic P1 E8 extension.
000005 | This strengthens the earlier fixed-channel argument to the full finite-energy
000006 | Hilbert-space inventory in the same quarter sea and source time. It does NOT
000007 | prove uniqueness of every possible scaling net from a spectral calculation.
000008 | The original P1 source, its geometrically marked V16, the known conditional
000009 | E8 field algebra and all native 4/5/20 structures are not replaced.
000010 | 
000011 | 1. Actual source, not a new parent
000012 | 
000013 | Use the CURRENT verification/v1033_charged_disorder.py builder at mass1,
000014 | width8, ONE complex CAR copy, periodic length N, quarter sectors r1 or3.
000015 | Let alpha=r/4, p_j=2pi(j-alpha)/N, P_N=1_(h_N<0), and
000016 |   H_N=N/(2pi)[dGamma(h_N)-Tr(P_N h_N)].
000017 | The vacuum is the full filled negative sea, including both edges and bulk.
000018 | The older channel proof and microscopic-charged-car-limit already identify
000019 | the two edge branches and their quasimodes. No novelty is claimed for that
000020 | inventory or for free-fermion continuum theory. The additional statements
000021 | below are the explicit uniform bound and full bounded-energy compactness.
000022 | 
000023 | Current direct controls do not silently re-pin the historical adapter chain:
000024 | the old mixed-current checker stops on its changed v1033 source SHA. The
000025 | new small calculation imports the current builder directly and checks the
000026 | precise source identities on which this proof depends. Historical receipts
000027 | and their verdicts remain historical.
000028 | 
000029 | 2. A uniform gap for every remaining one-body mode
000030 | 
000031 | The actual 16x16 Bloch matrix obeys
000032 |   h(p)=H0-sin(p)X+rho Z, rho=1-cos(p),
000033 |   H0^3=H0, rank H0=14, ||H0||=1,
000034 |   X^2=Z^2=I, {X,H0}={X,Z}=0.
000035 | Consequently
000036 |   h(p)^2=sin(p)^2 I+(H0+rho Z)^2.
000037 | The two zero vectors at p0 are the original top and bottom row vectors;
000038 | the restriction of h'(0) is diag(-1,+1).
000039 | 
000040 | For 0<=rho<=1/2, fourteen singular values of H0+rho Z are at least
000041 | 1-rho>=1/2, by the singular-value perturbation inequality. Hence at most
000042 | two eigenvalues of h have absolute value below1/2. For 1/2<=rho<=3/2,
000043 | sin(p)^2=rho(2-rho)>=3/4, so none has absolute value below1/2.
000044 | For 3/2<=rho<=2, ||(H0+rho Z)v||>=(rho-1)||v||>=||v||/2,
000045 | so again none does. These cover the entire Brillouin circle.
000046 | 
000047 | Thus the third and every later absolute one-body eigenvalue are uniformly
000048 | >=1/2 at every momentum. These modes cost at least N/(4pi) in H_N.
000049 | This is an all-p inequality, not a numerical grid conclusion.
000050 | 
000051 | 3. Full spectral exhaustion at every fixed excitation energy
000052 | 
000053 | Choose principal labels with -N/2<j-alpha<=N/2. Fix E. If N>4pi E,
000054 | an excitation costing <=E must belong to the two
000055 | small branches and to rho<1/2. The squared identity gives
000056 |   |sin p|<=2pi E/N.
000057 | Here |p|<pi/3 on the principal circle. Since |sin p|>=2|p|/pi,
000058 |   |j-alpha|<=pi E/2.
000059 | Hence a fixed, N-independent FINITE set of Fourier labels contains every
000060 | possible excitation below E. This is not just a chosen smooth-field window.
000061 | 
000062 | For each such fixed label the two eigenvalues are
000063 |   lambda_top(p)=-p+O(p^2), lambda_bottom(p)=+p+O(p^2).
000064 | Their eigenlines converge to the two distinct original edge vectors.
000065 | This follows from finite-dimensional isolated-subspace perturbation of H0;
000066 | the first-order restricted matrix has distinct slopes for p!=0. The known
000067 | source quasimodes give sharper error bounds, not needed for this conclusion.
000068 | After particle/hole conversion relative to the ACTUAL sea, both positive
000069 | excitation energies tend to |j-alpha|. Every remaining mode escapes to
000070 | infinite rescaled energy. Quart sectors have no zero Fourier label.
000071 | 
000072 | There is a uniform positive excitation gap for all sufficiently large N:
000073 | the preceding finite label reduction, with any fixed E greater than1,
000074 | leaves finitely many limits >=1/4; choose N large enough to retain a bound
000075 | 1/8 for those modes. Modes above E are already bounded below by E.
000076 | Thus at fixed E the excitation number is uniformly bounded as well.
000077 | 
000078 | Diagonalize h_N, fill its complete negative sea and form the normalized
000079 | particle/hole occupation basis. Sending each of the finitely many low
000080 | eigenmodes to its limiting top/bottom excitation gives exact finite-Fock
000081 | isometries. Additive free energies converge on every finite window. For E
000082 | outside the discrete limiting many-body spectrum, the ENTIRE spectral rank
000083 | of 1_[0,E](H_N) is eventually equal to that of
000084 |   H_infty=sum_(j in Z)|j-alpha|(n_(j,top)+n_(j,bottom)).
000085 | The graded dimension is
000086 |   prod_(j in Z)(1+t^|j-alpha|)^2.
000087 | Each coefficient in a bounded energy interval is finite. Both edges stay.
000088 | Equal-time space and charges are those of the existing CAR construction;
000089 | the opposite propagation directions have not been converted into flavours.
000090 | 
000091 | For normalized arbitrary source vectors with <H_N><=C, the spectral tail
000092 | above E has squared norm <=C/E. Combining this with the finite-rank window
000093 | identification gives subsequential norm compactness in the limiting Fock
000094 | space. It does NOT say an arbitrary sequence converges without selecting
000095 | a subsequence. It does say no independent finite-energy Hilbert multiplicity
000096 | is hidden in bulk modes or in increasingly complicated source words under
000097 | these fixed-vacuum/time comparison maps.
000098 | 
000099 | The heat traces converge as well, for every beta>0:
000100 |   Tr exp(-beta H_N) -> prod_j(1+exp(-beta|j-alpha|))^2.
000101 | Indeed every potentially small mode has rescaled energy at least
000102 | (2/pi)|j-alpha|, by the same sine bound, with at most two per label.
000103 | The other at most16N modes contribute at most
000104 | 16N exp(-beta N/(4pi)) to log Z_N. The summable label tail and this massive
000105 | tail give dominated convergence. No additional hole/antiparticle factor
000106 | is to be attached to the squared product: each original one-body mode
000107 | already supplies exactly one particle OR hole excitation.
000108 | 
000109 | 4. What the spectral result excludes, and what it does not
000110 | 
000111 | The result concerns the same vacuum and positive source energy. It excludes
000112 | an ADDITIONAL independent finite-energy carrier in that Hilbert limit.
000113 | Spectrum alone does not reconstruct a local net and does not exclude an
000114 | abstract Hilbert-space isomorphism or a different stress/locality dictionary.
000115 | For a local E8 CURRENT identification preserving the established conformal
000116 | grading, the known single-edge CAR theory has only one chiral weight1
000117 | state, while E8_1 requires248. The two original counterpropagating edges