TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ROOT-C58-WHOLE-WEAK-CAP-AND-REFERENCE.txt
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000001 | WHOLE ACTUAL WEAK CAP AND THE ORIGINAL REFERENCE
000002 | 30 October 2026. Independent root proof and original-source review.
000003 | 
000004 | 1. Same existing source, not a new field choice
000005 | Use h_p=p gamma+V, V=tQ gamma-i gamma D_F, p in Z+alpha,
000006 | alpha=1/4 or3/4, with the existing full96 real weak lift Q.
000007 | V=V* is finite and independent of p. The old source endpoint is
000008 | P0,p=1_(p gamma<0), rank r=48. This is the given source comparison,
000009 | not a deduction that original APSdomprov has selected this endpoint.
000010 | No source parameter, physical time or new state is fitted.
000011 | 
000012 | For each mode, the exact finite exterior formula from C57 is
000013 |  E_p=Tr(h_p 1_(h_p<0)),
000014 |  H_exc,p=dGamma(h_p)-E_p I >=0,
000015 |  Z_p(ell)=exp(ell E_p) det_RanP0(P0 exp(-ell h_p)P0).
000016 | It equals the old Slater vector expectation of exp(-ell H_exc,p).
000017 | Consequently 0<Z_p(ell)<=1 for every finite ell>=0. No ground
000018 | overlap or Thouless graph assumption is needed for this finite formula.
000019 | 
000020 | 2. An explicit whole-circle bound, independently checked
000021 | Write g=|p| and v=||V||. For g>2v the coupled negative projection Ph
000022 | has the same rank r as P0 by the minmax gap bound. For any normalized
000023 | negative eigenvector u with eigenvalue lambda<=-g+v, projection on
000024 | the positive h0 space yields
000025 |  (g-lambda) Pplus u = -Pplus V u,
000026 |  ||Pplus u|| <= v/(2g-v).
000027 | Summing over an orthonormal negative eigenbasis gives
000028 |  d_p=Tr(Ph Pplus)=Tr(P0(I-Ph)) <= r v^2/(2g-v)^2.
000029 | The squared old/coupled ground overlap is
000030 |  w_p=det_RanP0(P0 Ph P0).
000031 | All eigenvalues of this compression lie in[0,1]. The elementary product
000032 | inequality 1-product a_j<=sum(1-a_j) proves
000033 |  0<=1-w_p<=d_p.
000034 | The spectral resolution of H_exc,p now gives Z_p(ell)>=w_p. Hence
000035 |  0<=1-Z_p(ell)<=r v^2/(2g-v)^2,
000036 | uniformly in ell>=0 on the high modes.
000037 | 
000038 | The right side is summable over the original quarter circle. Outside
000039 | finitely many p it is <1/2, so -log Z_p<=2(1-Z_p). Therefore
000040 |  Z_cap(ell)=product_p Z_p(ell)>0
000041 | exists for every finite ell>0; every finite low-mode factor is positive.
000042 | This closes the previously finite-only weak cap on the given source.
000043 | It does not assert the older scalar C_p matrix formula for noncommuting Q.
000044 | 
000045 | At zero modes the negative spectral projection does not choose a unique
000046 | ground vector. Nevertheless E_p is unchanged by adding any subprojection
000047 | of ker h_p, because h_p vanishes there. Thus H_exc,p and the cap above
000048 | with fixed P0 are independent of that zero-mode occupation. No new
000049 | zero-mode state choice is required to evaluate this cap. A particular
000050 | vacuum ray or Thouless chart can require such data; distinguish them.
000051 | A large-ell cap limit can vanish if an old low mode is orthogonal to the
000052 | whole ground space. Strict positivity here is for finite ell.
000053 | 
000054 | Constant weak fields preserve Fourier p. What fails is reduction into
000055 | the earlier anticommuting scalar singular pairs, not Fourier factorization.
000056 | 
000057 | 3. Actual free reference is not coupled normal ordering
000058 | At finite cutoff let E_ref=Tr(h0 P0). The same exterior amplitude A_fixed
000059 | and the actual old free reference amplitude obey
000060 |  A_fixed/A_ref=exp[-ell(E_p-E_ref)] Z_p(ell).
000061 | Dividing by the free reference does not select E_p as energy zero.
000062 | This is exactly the bulk factor already retained in FIXED-CAP-DETERMINANT
000063 | section4, not a new ambiguity. On the full circle its subtraction and
000064 | phase must follow the original prescription. The convergent cap product
000065 | does not supply that prescription.
000066 | 
000067 | A numerical control used the existing v252 rep, Kswap, grading, full_D
000068 | on all96 slots, with the same diagnostic inputs as the prior weak moment
000069 | test. At p1/4,t7/10,ell1/8:
000070 |  aligned log cap=-14.174830473420858,
000071 |  aligned log free-relative amplitude=5.096812329424876,
000072 |  their difference=19.271642802845733 is exactly the retained bulk term
000073 |  to floating precision. The old t0 scalar formula agrees within4e-15.
000074 | High-mode p100.25 occupation deficit0.01755337 is below the explicit
000075 | bound0.07501711. These numerical controls do not prove convergence;
000076 | section2 does. C58-ACTUAL-WEAK-CAP-CHECK.json records the full controls.
000077 | 
000078 | 4. Original identifications remain typed
000079 | Original boundary001:353-377 chooses an adapted boundary operator
000080 | A_partial and a modified APS domain. It does not set A_partial=p gamma.
000081 | Original001:65-135 already defines the whole D_+ Calderon source before
000082 | its D_ref/D_rel split. Degeneracy of a later particular split does not
000083 | remove that whole source. QFT004:111-155 retains the separate given
000084 | T_V/Perron source. QFT008's actual positive-time OS caller also remains.
000085 | No blanket first-order relative-bound-zero claim is used; that original
000086 | error is already excluded in the Atlas.
000087 | 
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000094 | 
000095 | SOURCES
000096 | Actual weak source: ACTUAL-WEAK-CONNECTION-AND-JOINT-SOURCE.txt.
000097 | Original lines and hashes: C58-ORIGINAL-APS-AND-SEWING-REVIEW.txt.
000098 | Independent mathematical review: C58-THOULESS-AND-CAP-REVIEW.txt.
000099 | Existing cap: FIXED-CAP-DETERMINANT.txt and FIXED-CAP-STATE.txt.