TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-paper-source-reconciliation-20261102/shorttime_20261105K/fullprofile/DIRECT-WHOLE-PROFILE-IR-AND-REMOVAL-TEST-20261105.txt Original file SHA-256: 333393f5433dd4cc5ce19485b804b2259ad2dde613c4a26ce43d18345d10c184 Inclusive original lines: 1-73 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 | Direct whole profile integration with the original IR regulator 000002 | 5 November2026. Final short original source report, no numerical model/run. 000003 | 000004 | Outcome 000005 | The original finite IR insertion gives a direct route to the ENTIRE spectral profile weight; no a4/local quartic replacement is necessary. Its positive r integral is constructively integrable in the stated finite ordinary volume branch. The decisive removal test also follows exactly: bounded positive whole spectral weights on an unbounded ordinary linear field domain give a mean squared radius of order1/r as r tends0. Thus performing these mass regulated integrals is not itself a complete primitive source solution. This is a scoped analytical test of this route and its actual limit, not a no-go for the entire theory or its original omega/pushforward. 000006 | 000007 | 1. Original operation, not a new physical prior 000008 | QFT007:3-29 defines 000009 | Z_k[J]=int_Hadm Dphi exp[-S_adm[phi]-/2+], 000010 | with an admissible R_k commuting with structural selection. The same whole source, reference, eta, state/time and maps are retained. No independent ambient measure is requested. 000011 | On a genuinely supplied finite bosonic Hilbert coordinate sector the operator rI, r>0, is an allowed identity insertion in this class, with r tending0 for regulator removal. In actual orthonormal coordinates its form is r|h|^2. In nonorthonormal ORIGINAL coordinates the form is r h^T G h with the original Gram G; a free coordinate Euclidean metric must not replace G. 000012 | The actual reflection split regulator has A=rI,C=0. This lies in the already proved R>=0,C<=0 preservation cone, FULL-SOURCE-REGULATOR-PROOF section4. It preserves the RP of an already given actual RP source. It does not create RP from an arbitrary pointwise spectral weight or prove that a divergent unregulated linear volume defines an RP probability. At the continuum/infinite rank limit, the original family and renormalization conditions remain necessary; a finite mass insertion is not an unproved universal continuum regulator. 000013 | 000014 | 2. Entire finite spectral action is bounded, without local truncation 000015 | For actual fixed finite compatible spectral ranks n_A,n_R and a bounded real profile f, 000016 | S_B(h)=Tr f(A(h))-Tr f(B(h)), 000017 | |S_B(h)|<=C=(n_A+n_R)||f||infty. 000018 | A and B here are the ACTUAL dimensionless whole numerator and bound reference arguments, including their scale, Higgs, Majorana, connection and all actual coefficients. They are not substituted mass only operators. R055 already proves this whole class statement. Relative-only infinite traces must stay paired; fixed rank bounds do not authorize splitting two divergent traces. 000019 | For a polynomial original Grassmann/Pfaffian coefficient P and an actual selfadjoint eta phase, 000020 | |P(h)e^(-S_B(h)-i eta(h))|<=e^C |P(h)|. 000021 | Consequently its whole finite positive r integral has Gaussian polynomial tail control. Completing an original real linear source square retains its actual scalar normalization. Splitting the original quadratic Gaussian in half, if useful for a bounded insertion envelope, is a compensated integral identity. It changes no physical density. An original restricted domain, nonconstant Jacobian, constraints, nonpolynomial reference factor or remaining S_extra is retained and needs its actual bound; it is not silently replaced by ordinary volume. 000022 | 000023 | 3. A whole profile derivative bound not requiring a nu half moment 000024 | On a fixed rank smooth chart let P(h) be an ACTUAL reducing spectral projector for the SAME actual selfadjoint dimensionless A(h), with retained |A|>=epsilon>0. Let its rank be n and suppose nu0=int dnu=f(0)0,|lambda|>=epsilon) s|lambda|e^(-s lambda^2)=1/(e epsilon) 000028 | gives 000029 | |partial_j Tr[P f(A)]|<=2n nu0 ||partial_j A||/(e epsilon). 000030 | Apply separately to the actual numerator and bound reference, with their actual gaps and derivatives; differentiate A=D/chi completely when chi varies. A summed gradient bound follows by the Euclidean norm of these actual coordinate derivative bounds. 000031 | No half moment int sqrt(s)dnu is needed. If nu0 is infinite but g_nu(epsilon^2/2) is finite, the alternative inequality 000032 | s|lambda|e^(-s lambda^2)<=2 exp(-s epsilon^2/2)/(e epsilon) 000033 | gives the valid replacement bound4n g_nu(epsilon^2/2)||partial_j A||/(e epsilon). 000034 | The gap of the original D_prim selection alone is NOT automatically a gap of the whole A or of its different reference B. Nor may a projector derived from a different operator be assumed reducing. Such cases require the actual complete projector/Duhamel derivatives. Threshold crossings, eta crossings and moving APS domains retain their real domain/cell controls. A modulus tail bound is not a smooth global Lipschitz proof across these crossings. 000035 | Within the exact bounded/gapped smooth chart, these bounds give the spectral component of the existing deterministic cell remainder directly for the whole profile. They do not choose new nu or smooth away eta/selector jumps. 000036 | 000037 | 4. Decisive r removal comparison, no invented model 000038 | Take exactly the conditional branch already tested by R055: ordinary unbounded linear volume on R^d, actual radius R=||h|| in the original metric, and a positive weight w=e^(-S_B) with e^(-C)<=w<=e^C. At zero real source define 000039 | Z_r=int w(h)e^(-rR^2/2) dh. 000040 | For every r>0, exactly 000041 | e^(-C)(2pi/r)^(d/2)<=Z_r<=e^C(2pi/r)^(d/2), 000042 | e^(-2C)d/r<=_r<=e^(2C)d/r. 000043 | Both inequalities are direct numerator/denominator comparisons with the ORIGINAL regulated Gaussian. More generally the Gaussian moment is 000044 | E_gamma_r R^(2q)=r^(-q)2^q Gamma(d/2+q)/Gamma(d/2), 000045 | and comparison multiplies it by e^(plus or minus2C). Thus a finite radial moment cannot be obtained by merely removing this regulator within that positive bounded weight/ordinary volume realization. The normalized mass of each fixed field ball also tends0. A scalar outer reference normalization can change Z_rel but cancels out of this normalized radius comparison. 000046 | 000047 | The same conclusion includes a nonzero ORIGINAL NONNEGATIVE polynomial factor p of degree m. Rescaling h=y/sqrt(r) gives the leading homogeneous polynomial p_m; its Gaussian integral is strictly positive. Euler homogeneity/Gaussian integration by parts give 000048 | int |y|^2 p_m(y)e^(-|y|^2/2)dy=(d+m)int p_m(y)e^(-|y|^2/2)dy. 000049 | Hence the Gaussian p weighted radius is asymptotic to(d+m)/r, and bounded positive w compares it by e^(plus or minus2C), giving order1/r again. This does NOT assert that an actual Pfaffian is nonnegative on every H. Signed/complex Pfaffians, eta phases, zero denominator fibers and oscillatory prescriptions cannot inherit the positive quotient inequalities. Their unnormalized source coefficients and genuine limiting prescription remain intact. 000050 | The constant C may depend on spectral rank. A joint rank/r limit is not decided by a fixed rank inequality; R056 supplies uniformity only in its own whole circle/constant Higgs product class. No blanket failure of every observable, algebraic state limit or full nonconstant source is asserted. 000051 | 000052 | 5. Why neither admissibility nor local SH removes this comparison by definition 000053 | The original lower gap projector is a lower spectral cutoff, not a field amplitude cap. H_adm=Ran(P_adm) is a Hilbert state subspace, not a compact configuration ball. The compact normal sphere has H^0(O(1))=C^2; its coefficients remain a vector space. Bounded coefficients on each compact spacetime realization do not give a common amplitude ceiling. The bound248 c3^2 is explicitly for V_metric, not all Higgs/gauge fields. Imposing a hard Higgs norm cutoff from it would add a new physical condition. 000054 | The exact relative profile remainder versus a positive local quartic cannot globally be a harmless bounded reweighting in the R055 class: along an allowed unrestricted ray S_local=B r_field^4+O(r_field^2), whereas S_B=O(1), so S_B-S_local=-B r_field^4+O(r_field^2). The quartic confinement is cancelled by the exact remainder there. This reuses the old R055/R056 result; it is not a new global exclusion. Other actual full S_adm terms or the actual source pushforward can change these premises and must be evaluated rather than erased. 000055 | 000056 | 6. Precise remaining value operation 000057 | Use the actual finite original matrices/profile/reference/eta and the actual configuration domain/density to compute the unchanged regulated whole source and its polynomial marked coefficients. Then control its actual physical limiting prescription and mapped OS time kernels. Positive r makes the stated finite bounded-profile/linear-volume subintegration executable; its radius theorem shows exactly why that step alone cannot be called the unregulated primitive solution. The local four component quartic method remains a valid different actual local caller. No new physical profile, Gram, prior, field cutoff, whole source value, primitive spectrum or ambient gate is introduced. 000058 | 000059 | Exact original provenance 000060 | - Archive31 primitive-relative-spectral-dynamics:7-28,80-135: actual nu, whole profile action and domain. 000061 | - Source45 Boundary001:194-271,303-380: actual D_prim selection, H_adm and bounded coefficient realization. 000062 | - Source45 Carrier007:100-166: compact base index and C^2 Higgs sections. 000063 | - Source45 QFT007:3-29: actual regulator/source insertion. 000064 | - FULL-SOURCE-REGULATOR-PROOF:section2,4: whole Z0 retaining reweighting and the RP preservation cone. 000065 | - R055 EXACT-CUTOFF-HIGGS-MEASURE-PROOF:section2-5 and R056 WHOLE-CIRCLE-HIGGS-MEASURE-PROOF:section2-5: exact old bounded class and explicit limits. 000066 | - Current tfpt2 section010:30-61 and v36.run: actual existing closed finite transfer measure, not an evaluated Higgs density. 000067 | Full current SHA256 and absolute paths are recorded in original-source-evidence.json. The primary profile expansion crosscheck is Chamseddine/Connes arXiv:0812.0165v1 section2.5 Lemma6/Theorem7 with its explicit field range, https://arxiv.org/html/0812.0165v1 . 000068 | 000069 | Minimal necessary marked response and limit target 000070 | For the actual original source word W, retain its actual marked coefficient P_W(phi) and define the unnormalized regulated value 000071 | M_(r,Lambda)[W]=int_original-domain P_W(phi) exp[-S_full(phi)-r||phi||_original^2/2] dmu_original(phi). 000072 | Use polynomial integration/tail rules only when the actual coefficient is polynomial; otherwise retain its actual whole kernel. The physical marked response uses this value and the SAME whole vacuum factor, with reference and eta intact. For example the already defined K_U(epsilon) is the same caller's reflected U, time-translated U coefficient divided by its whole vacuum factor. The desired first time derivative additionally takes epsilon down to0; a UY or ordered CAR word takes its actual mixed source derivative. 000073 | A finite r bounded integral or scalar gap is not a bound allowing these marked derivatives and regulator limits to commute. The required analytic target is whole caller uniform control of the corresponding marked moments/time difference quotients (or an equivalent direct controlled limit), along the actual original limiting prescription. In a positive field realization, norm tightness is needed for a limit on the SAME unbounded field coordinate domain; stronger marked moment control is needed for unbounded U. The explicit positive bounded weight/ordinary volume class above fails norm tightness and its squared radius diverges, even when every positive r integral can be computed. Conditional uniqueness of a finite transfer Perron state does not alter this comparison. Other actual source supports, densities, nonpolynomial factors, complex prescriptions or algebraic observable limits change its premises and are not excluded. No full primitive solution is inferred from restoration of a pointwise bounded integrand alone.