TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/WHOLE-CIRCLE-HIGGS-MEASURE-PROOF.txt Original file SHA-256: fa5c3001563f6989d6a8b7bd1bbf9e7847831f617c232c4d5af0f7901f272d27 Inclusive original lines: 1-119 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 | R056. WHOLE ORIGINAL CIRCLE DOES NOT REPAIR THE TESTED HIGGS VOLUME MEASURE 000002 | 31 October 2026. Uniform infinite-mode result in the existing constant source. 000003 | 000004 | 1. New question and retained original data 000005 | R055 left joint rank/IR limits explicitly undecided. Test that possibility 000006 | on the ACTUAL given constant quarter-circle Higgs source, not a new space. 000007 | Original R028 ORIGINAL-HIGGS-FIBER-PROOF:83-90 already gives 000008 | h_H=(n+alpha)gamma_F-i gamma_F Phi(H), alpha=1/4 or3/4, 000009 | h_H^2=(n+alpha)^2 I96+Phi(H)^2. 000010 | Phi(H)=L(H)+DM is the whole original Hermitian96 coefficient, all four 000011 | given Yukawas and symmetric Majorana intact. It is constant along the 000012 | original circle here. No nonzero/small/invertible Yukawa premise is needed. 000013 | Use the same H1 quarter domain, same adjoint and bounded mass perturbation. 000014 | All H dependence of Phi and of its bound reference coefficient is allowed. 000015 | 000016 | This is a conditional original SOURCE chart, not a derivation of the 000017 | primitive C_Sigma or its Higgs field measure. The whole geometric circle 000018 | operator is unbounded. It is NOT the bounded mass-only D_rel whose ordinary 000019 | infinite trace was already excluded. Do not replace the original primitive 000020 | action contract by a different geometric action without its actual map. 000021 | 000022 | 2. Uniform bound on the entire infinite original circle 000023 | Let A_alpha=(-i partial_theta+alpha)^2 on the same periodic trivialization; 000024 | its eigenvalues are (n+alpha)^2, n in Z. The square above is 000025 | L_H=A_alpha tensor I96+I tensor Phi(H)^2. 000026 | Take a positive Laplace profile F(x)=int exp(-t x)dnu(t), nu>=0, at a fixed 000027 | positive chi, with the EXPLICIT baseline condition 000028 | B_alpha=Tr_circle F(A_alpha/chi^2)I. 000046 | It reduces L_H and the first-order h_H. On its ACTUAL range, 000047 | 0<=T_N(H)=Tr_(Ran(P_N tensorI96)) F(L_H/chi^2)<=96 B_alpha, 000048 | and T_N(H) increases to T_H. The bound is independent of N and H. 000049 | Do not apply F to a zero extension and count its artificial F(0) complement. 000050 | Positive bounded full-trace insertions0<=W_H<=I also satisfy 000051 | Tr(W_H F(L_H/chi^2))<=96 B_alpha, without W_H commuting with L_H. 000052 | Compression BEFORE functional calculus is a different operation unless its 000053 | actual reducing/domain identity is established. This does not choose Padm. 000054 | 000055 | For a reference of this same genuine product type with selfadjoint 000056 | Phi_ref(H), fixed original geometric baseline with finite B_ref, the SAME 000057 | argument gives0<=T_ref,N(H)<=96 B_ref. Thus the real relative action obeys 000058 | -96 B_ref<=S_N(H)=T_N(H)-T_ref,N(H)<=96 B_alpha. (2) 000059 | Reference H dependence does not invalidate (2). Keep all absolute scalar 000060 | factors and the original eta phase; no reference term is silently frozen. 000061 | 000062 | 3. Joint geometric-mode and IR limit: the loophole is closed here 000063 | For the explicitly tested ordinary coefficient volume d^4H, (2) gives 000064 | c_-<=exp(-S_N(H))<=c_+ for every N,H, 000065 | c_-=exp(-96B_alpha), c_+=exp(96B_ref). 000066 | Consequently the unregulated bosonic volume integral diverges for every N 000067 | AND for the full infinite operator. Apply R055's exact C2 volume proof with 000068 | these constants, which now do not depend on N. For epsilon_N>0, 000069 | Z_N,epsilon=int exp(-S_N(H)-epsilon_N|H|^2)d^4H, 000070 | mu_N,epsilon(B_R)<=exp(96(B_alpha+B_ref))R^4epsilon_N^2/2. (3) 000071 | For ANY joint prescription N->infinity,epsilon_N->0, every fixed ball 000072 | has mass tending0. There is no tight probability family on the same C2. 000073 | No interchange of an uncontrolled limit and an integral is needed: the 000074 | uniform inequality (3) holds first. More modes cannot supply a missing 000075 | confining tail within this source/profile/volume class. 000076 | 000077 | At each fixed N any actual nonzero polynomial/Pfaffian source coefficient 000078 | retains R055's absolute-integral divergence. Its degree and coefficients 000079 | can depend on N. No uniform fermion-weight bound across N is asserted. 000080 | In particular an infinite renormalized determinant may introduce new H 000081 | dependence not covered by (2); derive its ACTUAL source prescription before 000082 | using it. Eta of a selfadjoint source is a pure phase and does not change 000083 | absolute damping. Signed/complex functionals are not positive probabilities. 000084 | Some normalized bounded observables can converge despite failure of 000085 | tightness. No claim rules out all algebraic state limits or a compactified 000086 | field space, which would need its original physical identification. 000087 | 000088 | 4. Existing compact spacetime product: same proof, no new geometry 000089 | The ALREADY GIVEN R031 conditional even spin product has 000090 | K_H=D_g tensor I96+gamma5 tensor Phi(H), constant H, 000091 | {D_g,gamma5}=0 and K_H^2=D_g^2 tensorI+I tensorPhi^2. 000092 | If its actual compact elliptic domain supports this grading and 000093 | B_g=Tr_g F(D_g^2/chi^2)