TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-paper-source-reconciliation-20261102/shorttime_20261105J/shorttime20261105J_gravity/FULL-FOUR-COMPONENT-LOCAL-HIGGS-TRANSFER.txt
Original file SHA-256: c861727fb396d9b0f3f16c558f2e549676c7a8ac05e0f746bb223ba89807d67b
Inclusive original lines: 1-44
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | Direct original four-component integration strategy, 5 November 2026
000002 | 
000003 | Decision question: can the bounded integration operate on the actually supplied local Higgs action without pretending that its local radial chart is a global gauge pushforward?
000004 | 
000005 | Original source COMMON-CURVATURE-RESPONSE-PROOF.txt:119-128 gives, conditional on the actual R031/R045 fixed-H reference branch, SH=int[|DHc|²-mu0²|Hc|²+lambda|Hc|^4+xiR|Hc|²], lambda>0. DYNAMICAL-HIGGS-SOURCE-PROOF.txt:34-78 retains the original real four-component field and further full-action vertices. This is an existing local action, not a newly selected model or a statement that its quartic truncation equals the entire untruncated spectral action.
000006 | 
000007 | At an actual finite cell-coordinate presentation of that local action retain all four real components h_i in R^4, all cells i=1,...,M, the actual real symmetric quadratic form K, linear source l and beta_i=lambda V_i>0:
000008 | 
000009 | S(h)=h^T K h/2-l^T h+sum_i beta_i |h_i|^4.
000010 | 
000011 | The entries, cell map, field normalization and fixed gauge/geometry arguments are the ORIGINAL ones. There is no radial gauge fixing here. A numerical reference Gaussian of precision tau I, tau>0, is only an exact change of integration weight: its density and determinant cancel algebraically in the same unnormalized original integral. It is not a new physical Gaussian source or a selection of a vacuum. The method still needs the original real-coordinate measure and any actual domain/rest factors.
000012 | 
000013 | Let R=||h||, c=min_i beta_i/M, alpha=max(0,tau-lambda_min(K))/2 and j=||l||. Cauchy Schwarz gives sum_i beta_i|h_i|^4>=c R^4. Young bounds give
000014 | 
000015 | alpha R²<=c R^4/4+alpha²/c,
000016 | j R<=c R^4/4+3j^(4/3)/(4c^(1/3)).
000017 | 
000018 | Set C=alpha²/c+3j^(4/3)/(4c^(1/3)). Then
000019 | 
000020 | W(h)=S(h)-tau R²/2+C>=c R^4/2>=0.
000021 | 
000022 | Thus f=exp(-W) is globally bounded by exp(-c R^4/2), and the SAME local four-component integral is
000023 | 
000024 | I[P]=(2pi/tau)^(2M) exp(C) E_(N(0,tau^-1I))[P(h) f(h)],
000025 | 
000026 | up to the unchanged eta-independent original baseline factors. This is an exact identity for every actual polynomial mark P. All four components and all angular directions remain integrated. Any original H-dependent reference or extra factor stays inside the original operation.
000027 | 
000028 | For k>0, T_k=sup_R R^k exp(-cR^4/2)=(k/(2ec))^(k/4), T_0=1. With A=K-tau I and beta_max=max_i beta_i,
000029 | 
000030 | ||grad W||<=||A||R+j+4 beta_max R^3.
000031 | 
000032 | For P(h)=sum_p d_p h^p, set
000033 | B_P=sum_p |d_p| T_(|p|),
000034 | D_P=sum_p |d_p|[|p|T_(|p|-1)+||A||T_(|p|+1)+jT_(|p|)+4beta_max T_(|p|+3)],
000035 | 
000036 | where the degree0 derivative term is omitted. The monomial derivative bound ||grad h^p||<=|p|R^(|p|-1) proves these formulas. On standard normal coordinates h=tau^-1/2 z, Lip(Pf)<=D_P/sqrt(tau). In dimension d=4M the existing normal-weighted midpoint and Gaussian-tail argument therefore gives
000037 | 
000038 | |E(Pf)-Q_P|<=D_P/sqrt(tau) h_grid sqrt(d)/2+B_P 2d exp(-T²/2).
000039 | 
000040 | This is a constructive convergent integral strategy on the originally specified local SH branch, without a small-coupling expansion and without discarding the angular field. It does not require K>0. Tau is an algorithmic reference parameter only; the exact value is independent of it. The bounded radial implementation has not been falsely presented as an implementation of these four-component formulas. No native physical data or integral value was computed in this extension.
000041 | 
000042 | Crucial scope: the literal radial 3^N recurrence does not carry over by replacing N with4M. The O(4) quartic gradient contains mixed cubic monomials; a finite isolated Jacobian quotient is not established. Polynomial Pfaffian/minor marks can instead be integrated directly with the displayed bounds. A gauge quotient, discontinuous domain restriction, ghosts, eta/ref factors or a genuinely nonpolynomial untruncated spectral action is not absorbed into this polynomial potential by notation. Its actual remaining factor and error/domain operation must stay inside. No assertion that an a4/local action equals the full primitive S_adm is made.
000043 | 
000044 | This eliminates the local radial-chart obstacle for the existing full-component LOCAL action. It does not yet evaluate the complete primitive joint physical source, UY, OS time kernel, continuum limit or common spectrum. The decisive next application is the actual original finite local-action coefficients and polynomial source marks, followed by the actual full rest or a proved original equality with the requested source. Existing same-caller maps are closed and preserved.