TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-O1-HIGGS-LAW-PROOF.txt
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000001 | R057. THE ORIGINAL O(1) MODE KERNEL DOES NOT SELECT THE HIGGS LAW
000002 | 31 October 2026. Scoped classification and exact original-source readout.
000003 | 
000004 | 1. Original origin question and first additional identification
000005 | Original carrier007:95-144,185-204 supplies V=H0(CP1,O1)=C2 as the normal
000006 | Higgs mode space. R035 supplies an actual normalized mode embedding T only
000007 | WHEN its normal operator,domain,metric and given basis are supplied. R026
000008 | already supplies the wedge/whole96 Yukawa consumer. Retain all these results.
000009 | A geometric mode projector P=T T* is not a probability measure on V.
000010 | Test even the STRONGER,additional assumption that an actual positive Higgs
000011 | law has covariance cP, c>0. Neither this covariance identification nor c is
000012 | derived from the index. The original primitive source must supply them.
000013 | 
000014 | In normalized coordinates H in C2, grant an actual unitary WEAK SU2 action
000015 | and an equivariant isometric identification with the retained modes. This
000016 | is a stated premise, not an identification of normal spin2 with weak2 or
000017 | spacetime spin4. Grant a positive normalized invariant H MARGINAL, not an
000018 | arbitrary conditional law with other charged backgrounds held fixed.
000019 | Write c explicitly: E(H H*)=c I2, no fitted physical normalization.
000020 | 
000021 | 2. Complete invariant constant-fiber class
000022 | SU2 acts freely and transitively on each nonzero radius sphere in C2.
000023 | Its unique invariant angular probability is uniform on S3. Disintegration
000024 | therefore gives ALL positive invariant laws as
000025 |  H=sqrt(u) xi, u>=0, u~nu, xi uniform S3 conditional on u>0,
000026 |  with an arbitrary radial probability nu. At u=0 the angle is immaterial.
000027 | The uniform S3 law also has U2 invariance; this follows from uniqueness
000028 | or its Euclidean surface measure, without postulating a new physical U1.
000029 | For xi=(sqrt(t)exp(i theta1),sqrt(1-t)exp(i theta2)), t is uniform[0,1]
000030 | and both phases are uniform. Its exact balanced monomial moment is
000031 |  E |xi1|^(2a)|xi2|^(2b)=a!b!/(a+b+1)!.
000032 | Unbalanced phase monomials vanish. Consequently
000033 |  E H_i barH_j=(E_nu u)/2 delta_ij,
000034 |  E H_i H_j=0,
000035 |  E H_i barH_j H_k barH_l=(E_nu u^2)/6
000036 |                      (delta_ij delta_kl+delta_il delta_kj).       (1)
000037 | The covariance condition fixes only E_nu u=2c. A fourth moment is needed
000038 | only for the last identity and is not silently assumed for all laws.
000039 | Every such law gives the SAME whole mode covariance E(TH)(TH)*=cP.
000040 | No preferred frame or normal metric has been chosen in this classification.
000041 | 
000042 | 3. Source-specific separating witnesses, not physical candidates
000043 | Two well-defined invariant diagnostic laws have identical cP:
000044 |  A) fixed u=2c; E u^2=4c^2,
000045 |  B) complex Gaussian coefficients of covariance cI; radial density
000046 |     u exp(-u/c)/c^2 du; E u^2=6c^2.
000047 | These are witnesses against uniqueness from mode/kernel/covariance data.
000048 | They are NOT two derived TFPT completions of the same full master action,
000049 | and neither is selected as a physical prior. Gaussianity is extra input.
000050 | Jensen gives E u^2>=4c^2. Every finite value m2>=4c^2 is attainable:
000051 | put u=m2/(2c) with probability4c^2/m2, otherwise u=0. The infinite case
000052 | is also allowed by finite covariance when the radial tail has infinite
000053 | second moment. Thus the stated covariance/symmetry data leave an entire
000054 | radial law, not just a finite list of coefficients, undetermined.
000055 | 
000056 | Apply the ALREADY ESTABLISHED whole original96 identities with unchanged Y/MR:
000057 |  Tr Phi(H)^2=A u+M2,
000058 |  Tr Phi(H)^4=B u^2+4C u+M4,
000059 |  A=Tr DY^2, B=Tr DY^4, C=Tr DY^2 DM^2>=0,
000060 |  M2=Tr DM^2, M4=Tr DM^4.
000061 | Both witnesses give E TrPhi^2=2cA+M2. They differ by2c^2B in E TrPhi^4,
000062 | with identical Majorana remainder8cC+M4. If any supplied Y is nonzero,
000063 | B>0. The entire finite-quartic expectation range is
000064 |  [4c^2B+8cC+M4,infinity), under these data alone.
000065 | No coupling coefficient or physical Higgs mass is varied by this argument.
000066 | 
000067 | R054 supplies a second diagnostic on the original quark source: the fixed
000068 | sphere has24Var(m(u))=0; a nonconstant Gaussian radial law gives>0 if a
000069 | quarkY is nonzero. Zero of this particular quark witness does NOT establish
000070 | quasifreeness of the mixed quark source. The classical fixed-radius H law
000071 | itself is non-Gaussian; those two statements concern different source types.
000072 | 
000073 | 4. Exact readout of the actual radial law from the original source
000074 | Use real pairing j.H=Re(j*H) on R4 and a Fourier source, not an assumed
000075 | convergent real Laplace source. The characteristic function always exists
000076 | for a probability law. Haar angular integration gives the exact identity
000077 |  chi_H(j)=int [2 J1(sqrt(u)|j|)/(sqrt(u)|j|)] dnu(u),          (2)
000078 | with continuous value1 at u=0 or j=0. The full characteristic function
000079 | uniquely determines H's law and hence nu. A finite list of moments does
000080 | not suffice; even all moments need a determinacy condition if used instead.
000081 | When the fourth moment is finite, near j=0,
000082 |  chi_H(j)=1-c|j|^2/4+(E u^2)|j|^4/192+o(|j|^4),
000083 |  log chi_H(j)=-c|j|^2/4+(E u^2-6c^2)|j|^4/192+o(|j|^4).   (3)
000084 | For the two diagnostics, chi_G=exp(-c|j|^2/4) while
000085 |  chi_sphere=2J1(sqrt(2c)|j|)/(sqrt(2c)|j|).
000086 | 
000087 | Reuse the original source-pullback rule, now naming the required original
000088 | input exactly. If an ACTUAL Euclidean normal field varphi_N and actual
000089 | mode isometry T are supplied in the original positive source, H=T*varphi_N
000090 | has
000091 |  chi_H(j)=Z_orig[iTj,0,0]/Z_orig[0,0,0].                   (4)
000092 | The real adjoint and source pairing define the iTj convention. This is the
000093 | original full integral with all complement fields still integrated; do not
000094 | set them to zero or replace their effect by a bare Gaussian/Hessian.
000095 | The fixed relative reference cancels in this normalized ratio at fixed
000096 | background, but absolute Z0/ref/eta and their variations remain in the full
000097 | source. Mixed fermion sources must likewise be retained for mixed words.
000098 | Equation(4) is a readout, not an independently produced Z_orig, field map or
000099 | equivariant T. It does not infer classical joint probabilities from arbitrary
000100 | noncommuting quantum observables. The actual positive Euclidean field law
000101 | or a proved commuting field image is an explicit premise.
000102 | 
000103 | 5. What is and is not solved
000104 | The full O1 projector, even granted as actual field covariance, plus genuine
000105 | weak symmetry does not select the Higgs weight. The first additional choice
000106 | would be the field-source/covariance identification and then the radial nu.
000107 | Within the stated invariant marginal class, (2)-(4) give a precise test:
000108 | evaluate the ORIGINAL pulled-back Fourier source and compare its whole
000109 | response, not just its quadratic jet. This is the next original-source
000110 | computation, not a search over chosen radial distributions.
000111 | If other fields are held fixed, joint gauge symmetry need not leave H's
000112 | conditional law invariant; rotate/transport the whole source or use its
000113 | actual marginal. No global radial reduction of H(x), fixed gauge backgrounds,
000114 | spacetime source or interacting3+1D theory is asserted.
000115 | R055/R056 remain in their own trace/flat-volume scope. Neither their negative
000116 | result nor a canonical-looking Gaussian ensemble supplies this missing law.
000117 | Keep original Y/Flavor/Alpha,E8/clocks,whole96,R035G/T,R037P/Q,R043GWG,
000118 | state/time/RP,Z0/ref/eta/H0 and word memory. No new Lean or ledger promotion.
000119 | 
000120 | Primary comparison,not a TFPT origin proof: Shiffman/Zelditch/Zhong,
000121 | arXiv1005.4166,section2.2.1,eq17 explicitly STARTS with Gaussian coefficient
000122 | measure and then obtains the projector covariance. It does not derive
000123 | Gaussianity from the projector. Its preceding sphere/Gaussian comparison
000124 | for ZERO statistics must not be transferred to amplitude observables here.