TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/QUARTER-FOURFIELD-ORDERED-HOLDOUT-20261102.txt Original file SHA-256: eebc8cf5837acb4a201d4d782a3f5ac708143bff3da369c26697a3609b0501f0 Inclusive original lines: 1-118 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | QUARTER / MATTER / FOUR-FIELD HOLDOUT ON THE ORIGINAL R062 TRIPLET 000002 | 2 November 2026. Conditional source calculation; no F1 closure. 000003 | 000004 | Purpose. P0 of the new synthesis requests a joint clock, matter and four-field 000005 | word. Use the ACTUAL fixed right-up colour triplet from R062, with the given 000006 | Yukawa singular vector and original p=1/4 block. Its one-mode quarter is 000007 | the charged quarter action of the existing free source. The tested bounded 000008 | triplet implementer is a SUBSYSTEM insertion, not yet an identification with 000009 | the entire original quarter clock. R062 already reconstructs its whole marginal 000010 | from four number-characteristic samples; the present contribution is the 000011 | explicit marked ordered clock/matter/four-field holdout and exact controls. 000012 | 000013 | Inputs and limits. For fixed constant Higgs H, the triplet marginal has 000014 | rho_H=t(H)^N(1-t(H))^(3-N), N=sum_c a_c^* a_c, with 000015 | t(H)=1/2+p*tanh(ell*sqrt(p^2+y^2 H*H))/(2*sqrt(p^2+y^2 H*H)); 000016 | y>0 is an actual given Yu singular value. See HIGGS-COLOR-MODULAR-SOURCE-PROOF, 000017 | sections 1,2 and HIGGS-MIXTURE-WICK-PROOF, sections 1,3. This uses a fixed 000018 | family vector independent of H, three real colour slots, vanishing anomalous 000019 | two-point functions and the conditional quasifree source. Arbitrary dynamical 000020 | H(x), extra conditional vertices, nonpositive H weights, and the physical 000021 | source-to-triplet map are not established by this computation. 000022 | 000023 | For N_c=a_c^*a_c, E_cd=a_c^* a_d and O=3N^2-sum_cd E_cd E_dc, 000024 | CAR gives O=4N(N-1). It is a four-field observable on this actual triplet. 000025 | For Q_z=z^N and |z|=1, Q_z a_c^* = z a_c^* Q_z. At z=i this has 000026 | the same creation phase as the charged particle branch of the old quarter 000027 | source, see seam-collar-multifield-20261016/PROOF.txt lines 157-170. Its global 000028 | filled-sea phase and the complement of this triplet have not been transported. 000029 | 000030 | For a conditional triplet state, 000031 | _H = 24 z^2 t^2 (1-t+zt), 000032 | _H 000033 | = 16 z^2 t(1-t)^2 + 24 z^3 t^2(1-t). (1) 000034 | The CAR sign, exact operator order and all eight Fock states are checked in 000035 | check_quarter_fourfield_holdout_20261102.py. This order is algebraic: without 000036 | the ORIGINAL time-evolution insertions it is NOT a computed multitime word. 000037 | An arbitrary charge-breaking pairing or additional fermion vertex requires 000038 | the original conditional response, not formula (1). 000039 | 000040 | COUPLED-PAIR QUARTER: KEEP THE ORIGINAL COMPLEMENT. 000041 | For one given singular value and each colour the constant-H one-particle 000042 | Hamiltonian has the actual paired L/R form h_H^2=E^2 I and tr h_H=0. The 000043 | conditional C=(1+exp(2 ell h_H))^-1 has tr C=1, C_RR=t and 000044 | d=det C=1/[4 cosh(ell E)^2]. 000045 | Use the full six particle modes, not only the three retained R modes. The 000046 | explicit fixed background component of the common charge action is 000047 | Q_pair,z=z^(N_R+N_L), O_R=4N_R(N_R-1). 000048 | The two unnormalized R occupation weights after insertion of z^N_L are 000049 | w0=d+z(1-t-d), w1=t-d+z*d. 000050 | These follow from the four diagonal probabilities of the conditional two-mode 000051 | quasifree density: P00=P11=d, P10=t-d, P01=1-t-d. Its off-diagonal L/R 000052 | coherence is not assumed zero; it enters d. Summing all three paired colours, 000053 | _H 000054 | =16 z^2 w0^2 w1 + 24 z^3 w0 w1^2. (3) 000055 | At z=i this is a concrete H-dependent complex answer with the paired matter 000056 | complement retained. For z=1 it reduces to the z=1 case of (1). Formula (3) 000057 | is an actual finite coupled-component answer under R062's conditional source; 000058 | it is not the entire source: all other singular pairs, particle/hole modes, 000059 | opposite quarter, their relative phases, original reflection/eta/reference, 000060 | additional vertices and the H measure must also be included. In particular, 000061 | the exact covariance and its complement must be transported under the actual 000062 | clock; (3) does not establish the original full physical clock identification 000063 | or provide separated physical-time insertions. The direct independent 64-state 000064 | CAR check uses C_RR=3/5, det C=1/5 and C_RL=1/5; it yields at z=i the exact 000065 | value 8/5-8i/25. This single rational control tests the identity, not the 000066 | original numeric choice ell=2pi. The general identity is derived above. 000067 | 000068 | Assume precisely R054's positive normalized joint H law mu including all 000069 | other fields, Pfaffian, reference, Eta and Z[0]. Write m_j=int t(H)^j dmu. 000070 | The uninserted triplet four-field word is =24m_2. The bounded marked word 000071 | at z=i reads 000072 | = -24(m_2+(i-1)m_3), 000073 | 000074 | = -16(m_1-2m_2+m_3)-24i(m_2-m_3). (2) 000075 | The third moment is the genuinely six-fermion part of this known triplet 000076 | marginal; it is not a new independent field, coupling or discovery of the 000077 | non-Gaussianity already established by R054/R062. For the unnormalized original 000078 | relative functional, multiply each expectation by Z_rel[0] only IF the 000079 | source-to-triplet insertion and the entire above conditional decomposition 000080 | are established on that same source. Z_rel[0]=Z_adm[0]/Z_ref[0], not 1 in 000081 | general. A global clock insertion can act on the traced complement and give 000082 | H-dependent complex weights, so its insertion cannot in general be replaced 000083 | by Q_z on the marginal or its positive mixture. 000084 | 000085 | EXACT TWO-MEASURE COUNTERCONTROL. Let four t values be 13/25, 27/50, 000086 | 14/25, 29/50, and choose respective probabilities (5,1,7,3)/16 and 000087 | (3,7,1,5)/16. Both have m1=11/20, m2=303/1000, but m3 equals 000088 | 167197/1000000 versus 167203/1000000. Hence they agree on the full triplet 000089 | two-point data and the uninserted colour-invariant four-field response, 000090 | yet the second marked word gives respectively 000091 | -111197/62500 - 407409 i/125000, 000092 | -111203/62500 - 407391 i/125000. 000093 | Their difference is 3/31250 - 9i/62500. At the original p=1/4, ell=2pi, 000094 | all four t belong to (1/2,(1+tanh(pi/2))/2), so strict monotonicity supplies 000095 | unique finite u for each actual nonzero given y. These distributions are 000096 | mathematical comparison witnesses within the R054 assumptions: they are not 000097 | asserted to be equal to the unknown original TFPT H measure, and do not match 000098 | the full original system's many other observables or prove two admissible F1 000099 | models. They show exactly what the labelled marginal holdout can distinguish. 000100 | 000101 | SOURCE CHAIN. The original source is the formal Z_adm[J,eta,bar-eta] with 000102 | Z_rel=Z_adm/Z_ref[0]; the original regulator action W_k and its higher S_adm 000103 | derivatives are stated in archive--tfpt-45--source_extracts--04_qft_source, 000104 | sections 002 lines 256-266 and 007 lines 3-24, 115-140. The previous R024/25 000105 | full-source regulator treatment already retains its unnormalized scalar, 000106 | complement source and higher vertices. Neither quoted definition provides the 000107 | actual H pushforward, complete time transport or the required identification 000108 | of the marked triplet quarter with the global original clock on the source. 000109 | Completing P0 requires those explicit maps and a positive-time word with the 000110 | source's own time transports, not merely (2). A global quarter's traced 000111 | complement is a twisted conditional trace; it is not generally a scalar 1. 000112 | Keep R054/R062, both quarters, full E8, Q memory, state/reflection, existing 000113 | Yukawa/Flavor/Alpha and reference/Eta/Z0 unchanged. 000114 | 000115 | Reproduce: python3 -B aps_source_review/check_quarter_fourfield_holdout_20261102.py 000116 | from experiments/theory-contracts/tfpt-unified-source-derivation-20261029. 000117 | Result: CAR, quarter action, both word identities and two-measure comparison 000118 | PASS exactly in SymPy. No new Lean or physical experiment is claimed.