TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CURRENT-ACTION-ARGUMENT-PROOF.txt
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000001 | CURRENT ORIGINAL SPECTRAL ACTION: ARGUMENT, MOMENTS AND COMMON H/GRAVITY RESPONSE
000002 | 31 October 2026. R044/R045. Continues F1 after R042/R043.
000003 | Scoped ordinary local coefficient audit; no replacement physical profile or ledger promotion.
000004 | 
000005 | 1. Original consumer actually inspected
000006 | Current tfpt2 section010:273-315 states the relative master
000007 |  S=Tr f((Drel+A_Sigma)/chi)-Tr f(Dref/chi)+i pi Deltaeta/2,       (1)
000008 | and cites v255 and v259 for its expansion and fixed moments. This is NOT
000009 | the archived -6chi² Res_1 zeta+FP_0 zeta of R029/R030. Keep both source
000010 | contracts distinct. The current gravity overview section007:1-50 also
000011 | distinguishes its entanglement/Einstein route from the spectral route.
000012 | 
000013 | v255:1-20 prints 2 f4 chi^4 a0+2 f2 chi² a2+f0 a4, but defines
000014 |  f4=int u f(u)du, f2=int f(u)du, f0=f(0).
000015 | Its run():63-120 does not instantiate the full original operator or an
000016 | argument map. The expansion-structure check tests N=96 and three labels;
000017 | several gravitational/quartic checks have literal True inputs. This audit
000018 | does not reinterpret those declarations as an executed source derivation.
000019 | v259:1-51 and55-106 evaluate f(u)=exp(-u) and exactly obtain1,1,1.
000020 | Those scalar integrals are correct. The code does not establish whether u
000021 | is a signed modular eigenvalue, |K|/chi or K²/chi² in the full action(1).
000022 | The seam-fixed assertion is recorded with True, not an operator comparison.
000023 | The cited v258 run():79-157 checks inverse KMS functional calculus on
000024 | supplied random/Yukawa/mass blocks. Its physical C_F=P_F C_Sigma P_F
000025 | identification is explicitly conditional and recorded with True. Reuse the
000026 | existing inversion theorem; this code is not an independent original
000027 | normal-reduction/spacetime/spur construction. Do not discard solved flavor.
000028 | 
000029 | Already reviewed R016/R017/R019 are INPUT, not new discoveries here:
000030 | signed exp(-K) on a two-sided unbounded spectrum is not an even positive
000031 | heat cutoff; the original positive boundary factor is Q=exp(-beta|h|).
000032 | Q does not by itself identify the full original bosonic K, trace, reference
000033 | or action with exp(-|K|/chi), nor with a fermionic logdet. No new profile
000034 | search, tensor-product spacetime reconstruction or Yukawa fit is performed.
000035 | 
000036 | 2. Same ordinary coefficients fix the argument dictionary
000037 | State the extra realization explicitly: an ordinary4D Dirac-type consumer,
000038 | with heat coefficients defined by
000039 |  Tr exp(-tK²) ~ sum_j a_j(K²) t^((j-4)/2).                   (2)
000040 | Use justified local/interior asymptotics and its actual elliptic domain;
000041 | boundary/log terms, null modes and reference jets are not silently removed.
000042 | For a squared-argument cutoff F with the requisite moments, the first
000043 | three bulk weights in Tr F(K²/chi²) are
000044 |  (weight a0,weight a2,weight a4)=(m2,m1,m0),
000045 |  m0=F(0), m1=int_0^infty F(u)du, m2=int_0^infty u F(u)du.   (3)
000046 | This follows from Mellin substitution, or integrating (2) against the
000047 | Laplace measure when it is positive with the requisite inverse moments.
000048 | For an even linear-argument f(r)=F(r²), r>=0, the substitution u=r² gives
000049 |  m1=2 int r f(r)dr, m2=2 int r³ f(r)dr.                    (4)
000050 | Thus CC's f2=int r f(r)dr and f4=int r³ f(r)dr give factors2.
000051 | They are not int f and int u f for that SAME linear argument.
000052 | Primary check: Chamseddine/Connes/Marcolli hep-th/0610241, eq(1.3),
000053 | Theorem3.13 and Remark6.5 explicitly distinguish the two conventions:
000054 | https://arxiv.org/pdf/hep-th/0610241 . Their curvature sign is translated
000055 | to R031 below; no TFPT profile or physical source identity is imported.
000056 | 
000057 | Consequently the SAME ordinary coefficients have the following weights:
000058 |  actual expression                   a0     a2     a4
000059 |  exp(-K²/chi²)                        1      1      1
000060 |  exp(-|K|/chi)                       12      2      1
000061 |  literal v255+v259 combination         2      2      1.       (5)
000062 | For Poisson, CC's f0=1,f2=1,f4=6. For heat, CC's f2=f4=1/2.
000063 | The printed (a2,a4) weights match the Poisson candidate. The printed a0
000064 | does not. The leading factor6 in this comparison is not R030's factor6
000065 | between a2_H and a4_RH; their operators and tested ratios are different.
000066 | 
000067 | 3. Exclude a whole class, while preserving the actual relative reference
000068 | Reuse R017's positive Laplace class F(u)=int exp(-su)dnu(s), dnu>=0,
000069 | finite m0,m1,m2. Tonelli gives m0=int dnu,m1=int dnu/s,m2=int dnu/s².
000070 | Cauchy Schwarz then requires m0 m2>=m1². The NEW application to (5)'s
000071 | literal full coefficient triplet would require (m0,m1,m2)=(1,2,2), giving
000072 |  m0 m2-m1²= -2.                                          (6)
000073 | No profile in this ENTIRE class realizes that full triplet for the same
000074 | ordinary coefficients. Positive common amplitude and argument scaling
000075 | multiply this determinant by a positive factor and do not fix the sign.
000076 | This is not an exclusion of all positive functions outside this class.
000077 | 
000078 | Crucial relative exception: the master subtracts an actual reference.
000079 | If both operators have the same ordinary principal symbol, rank, geometry
000080 | and trace, their a0 terms cancel exactly. Then the discrepancy in a0 is
000081 | invisible to that relative action, and (6) does NOT exclude its surviving
000082 | (a2,a4)=(2,1). These two weights agree with Poisson but do not uniquely
000083 | select it. For example the previously known single exponential with
000084 | m0=1,m1=2 has m2=4; m2 is unobserved after exact a0 cancellation.
000085 | If geometry varies with an unscaled bound reference, a0 generally survives: