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Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/GEOMETRIC-CONE.txt
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000001 | I: THE ACTUAL SHARED GEOMETRIC CONE AND THE QUANTUM SOURCE WARD STEP
000002 | 2 November2026. relational_logic independent review. Canonical unchanged.
000003 | Entry plan88, Atlas351a3f2e6587f443125feb65. G/H retained.
000004 | 
000005 | RESULT
000006 | 
000007 | The Original already supplies a common tree-level bulk kinetic metric.
000008 | Its SAME Dirac anchor fixes g through its Clifford principal symbol; its
000009 | printed low-curvature Lorentz branch uses this SAME g for matter, Higgs
000010 | and gauge curvature. Different scalar field normalizations and internal
000011 | couplings do not require additional species-speed matching. On that
000012 | actual branch the physical principal characteristics share g^{mu nu}
000013 | p_mu p_nu=0, on the ACTUALLY retained nondegenerate propagating blocks.
000014 | The earlier R044/R045 reference-cancellation results remain in force.
000015 | A block cancelled by the actual relative trace is not assigned a
000016 | propagating field or a new cone by this theorem.
000017 | 
000018 | There is also a precise positive quantum continuation. If the WHOLE
000019 | original source/renormalized action and its flat vacuum/background obey
000020 | the actual Lorentz covariance identity, physical inverse-propagator pole
000021 | factors depend on the common q_g=g^{mu nu}p_mu p_nu, with their original
000022 | causal branch. Their internal mixing and species-dependent scalar form
000023 | factors cannot generate an independent spatial/time speed ratio.
000024 | 
000025 | The first overstrong inference to avoid is 'one Clifford principal symbol
000026 | alone implies the whole quantum source/state covariance'. QFT008 explicitly
000027 | states that covariance as a source lemma; its tailored OS theorem explicitly
000028 | uses it as a hypothesis. The finite-beta H source has a thermal rest frame
000029 | and is not thereby boost invariant. Unequal medium/group velocities are
000030 | not a counterexample to fundamental Lorentz covariance or a shared causal
000031 | front. No new Lorentz or continuum requirement is imposed on every optional
000032 | UV model.
000033 | 
000034 | 1. THE ORIGINAL METRIC IS FIXED BEFORE INTERNAL KINETIC READOUTS
000035 | 
000036 | Metrology001:49–63 assumes Z(A)=C-infinity(M) and a Dirac-type D_geo with
000037 | 
000038 |  sigma_1(D_geo)(x,p)^2=g^{mu nu}(x)p_mu p_nu I.             (1)
000039 | 
000040 | It says inner fluctuations of this same anchored datum generate the gauge
000041 | sector. It expressly distinguishes D_rel as comparison bookkeeping,
000042 | and:85–89 denies that Padm D_rel Padm supplies the local principal symbol.
000043 | Thus the actual metric source here is D_geo, not an equality of finite
000044 | dimensions, not c3 alone, and not the LR velocity of another caller.
000045 | 
000046 | The given local-scale realization is
000047 |  D_sigma,geo=exp(-sigma/2) D_geo exp(-sigma/2),
000048 |  K_sigma=D_sigma,geo+B_rel,
000049 | with its actual reference and domain, Metrology001:100–126 and G's
000050 | CURRENT-ARGUMENT-MAP. The anchor part has principal symbol exp(-sigma)c_g(p),
000051 | whose square is exp(-2sigma)q_g I. A common smooth finite real sigma changes
000052 | the conformal representative and not its null cone on the Lorentz branch.
000053 | 
000054 | No blanket bounded/orderzero claim is made about the package A_Sigma or
000055 | every operator named B_rel. The present bulk characteristic conclusion is
000056 | read from the ACTUAL printed physical low-curvature action below. It is
000057 | not inferred from the order or dimension of another compact normal-S2
000058 | operator, and it does not assert that an unspecified first-order addition
000059 | to D_geo must preserve(1).
000060 | 
000061 | 2. THE PRINTED PHYSICAL ACTION ALREADY HAS ONE KINETIC QUADRATIC FORM
000062 | 
000063 | Metrology002:145–170 gives, from the same relative residue, the terms
000064 |  |D Phi|^2/(16pi^2), Tr F^2/(192pi^2), R Phi*Phi/(96pi^2),...
000065 | using that common geometric contraction. Metrology004:35–83 writes the
000066 | Lorentz action with sqrt(-g), |D Phi|^2 and the gauge terms
000067 |  -SUM_a Tr F_(a,mu nu)F_a^(mu nu)/(4g_a^2).
000068 | The same package carries the original fermion kinetic/Yukawa sector.
000069 | QFT005:208–224 explicitly invokes the Wick-rotated quadratic operators:
000070 | scalar/bosonic normally hyperbolic, fermions Dirac hyperbolic, gauge fields
000071 | hyperbolic after covariant gauge fixing and BRST reduction.
000072 | 
000073 | The smallest principal-symbol test of those actual terms is:
000074 |  fermions: c_g(p) on every retained internal species, hence c_g(p)^2=q_g I;
000075 |  Higgs: Z_H q_g times its internal identity/mixing block;
000076 |  gauge: before gauge reduction the q_g g^(mu nu)-p^mu p^nu symbol,
000077 |         and in a covariant hyperbolic gauge q_g times the bundle metric.
000078 | 
000079 | Nonzero scalar Z_H, 1/g_a^2 or internal wavefunction matrices alter
000080 | normalization, not the zero set q_g=0. Gauge-Higgs mass mixing, Yukawa
000081 | matrices and Majorana masses are lower order for this characteristic
000082 | test. The physical gauge quotient removes the gauge-null directions;
000083 | it does not select a second spacetime metric.
000084 | 
000085 | This requires a nonzero/invertible retained kinetic block or the actual
000086 | physical pole. The displayed unsubtracted operator content alone must not
000087 | resurrect a gauge kinetic coefficient cancelled by its bound reference.
000088 | Such a cancelled block, or a nonpropagating clock, has no physical
000089 | propagation cone in this test; it is not a second-cone counterexample.
000090 | This explicitly retains the scoped R044/R045 cancellations.
000091 | 
000092 | This closes a supposed need to fit independent tree-level limiting
000093 | velocities INSIDE this stated geometric branch. The common g is already
000094 | the original input reconstructed from its full anchor. We have chosen no
000095 | additional metric, anisotropy, coupling or clock conversion. A Euclidean
000096 | elliptic symbol by itself is not a Lorentz cone: the conclusion uses the
000097 | Original's explicit Wick-rotated low-curvature Lorentz branch and its
000098 | reconstruction hypotheses, not a new continuation selected here.
000099 | 
000100 | 3. THE ACTUAL QUANTUM WARD IMPLICATION
000101 | 
000102 | QFT007:3–26 defines the complete original W_k and its Legendre Gamma_k.
000103 | QFT007:149–172 and:330–347 put the dressed inverse propagators and their
000104 | physical poles in the SAME Gamma_ren=lim_(k->0)Gamma_k package.
000105 | Thus the correct next quantum object is the whole source's Gamma^(2),
000106 | not an additional fitted speed attached to the tree action.
000107 | 
000108 | Suppose, precisely on an ORIGINAL flat Lorentz-invariant vacuum/background
000109 | branch, that the whole source and reference give
000110 | 
000111 |  Gamma_ren[rho(L) phi]=Gamma_ren[phi],
000112 |  rho(L)phi0=phi0.                                        (2)
000113 | 
000114 | All actual tensor backgrounds, domains, reference and eta are included.
000115 | Twice differentiating(2) at phi0 gives the exact momentum covariance law
000116 | 
000117 |  K(Lp)=D(L)^(-T) K(p) D(L)^(-1), K=Gamma_ren^(2).          (3)
000118 | 
000119 | For scalar multiplets its internal matrix coefficients are functions of
000120 | q_g. For chiral fermions the available Lorentz structures are slash-p
000121 | and scalar/chiral mass matrices, with internal form factors depending on
000122 | q_g. For the physical gauge polarizations, Lorentz covariance together
000123 | with the actual gauge/Slavnov identities reduces the inverse pole factors
000124 | to scalar functions of the same q_g. Keep the original positive/negative
000125 | energy or retarded/Feynman boundary prescription; it is not a new metric.
000126 | 
000127 | Consequently zeros have common invariant mass-shell kinematics, for
000128 | example p0^2=|p|^2+m_i^2 in common local orthonormal units. Different m_i,
000129 | residues, internal mixing, thresholds or complex resonance positions do
000130 | not introduce independently adjustable species speeds. Massive group
000131 | velocities at finite momentum need not equal massless velocity even here.
000132 | Equation(3) is a symmetry argument for all corrections satisfying(2),
000133 | not a numerical one-loop coincidence or a new RG attractor calculation.
000134 | 
000135 | If an intermediate regulator is not Lorentz covariant, keep its actual
000136 | modified Ward identity and the restoration at the stated endpoint.
000137 | [R_k,Padm]=0 alone is not Lorentz invariance. Scheme covariance in the
000138 | later comparison groupoid is also not spacetime Lorentz covariance.
000139 | 
000140 | 4. PRECISEST LOW-MOMENTUM ANISOTROPY TEST
000141 | 
000142 | For a local scalar physical two-derivative kernel K_i^(mu nu), its
000143 | anisotropic part is
000144 | 
000145 |  A_i^(mu nu)=K_i^(mu nu)
000146 |         -(1/4)g^(mu nu)g_(rho sigma)K_i^(rho sigma).       (4)
000147 | 
000148 | Under(2), A_i=0: the only invariant symmetric rank2 tensor is a scalar
000149 | multiple of g. Checking this tensor on the actual complete source is
000150 | sharper than adjusting separate apparent velocities. Analogously the
000151 | Lorentz-invariant gauge constitutive tensor has the metric wedge form
000152 | plus a possible invariant epsilon term, not an independently selected
000153 | preferred-direction tensor. Fermion kinetic symbols must obey the same
000154 | Lorentz representation law(3).
000155 | 
000156 | In an actual matrix low-momentum pole expansion, retain BOTH temporal
000157 | and spatial coefficients from the SAME caller and the SAME state:
000158 |  K(omega,k)=K_t omega^2+2 SUM_a K_ts^a omega k_a
000159 |                          -SUM_ab K_s^(ab)k_a k_b-M+... .
000160 | The mixed omega k terms have the SAME derivative order and must not be
000161 | hidden in the remainder. The general common-metric factorization test is
000162 |  K^(mu nu)=g^(mu nu) Z
000163 | on the actually retained physical block, with its common coordinate and
000164 | momentum conventions and an invertible internal Z. It keeps g^(0a)Z
000165 | whenever the supplied frame has a nonzero mixed metric component.
000166 | In the actual shared orthogonal frame g^(0a)=0, the corresponding test
000167 | also requires K_ts^a=0. On a positive, nondegenerate physical block, one
000168 | common isotropic two-derivative cone in that frame further means
000169 |  K_s^(ab)=v^2 delta_ab K_t;
000170 | equivalently every eigenvalue of K_t^(-1/2)K_s K_t^(-1/2) is the same
000171 | v^2 for each spatial direction. A static K_x=K_y test does not supply
000172 | K_t or this ratio. One may not set K_t to its target value afterwards.
000173 | This criterion tests the quadratic cone; even its success alone is not
000174 | sufficient for the Original's controlled IR universality of the full
000175 | interacting source.
000176 | 
000177 | Gauge Ward identities ALONE do not force(4). Gauge-invariant kinetic
000178 | terms can carry a fixed Lorentz-breaking rank2/rank4 tensor. The primary
000179 | Colladay-Kostelecky construction explicitly separates gauge invariance
000180 | from particle Lorentz invariance:
000181 |  https://arxiv.org/abs/hep-ph/9809521 .
000182 | No such deformation is introduced here. It identifies what would actually
000183 | be a new structure: an independent spacetime tensor or state/background
000184 | direction, rather than a scalar field normalization.
000185 | 
000186 | 5. WHERE THE ORIGINAL QUANTUM HYPOTHESES ENTER
000187 | 
000188 | QFT008:199–213 states graded Euclidean covariance of the full admissible
000189 | Schwinger source on U_scat. Its proof appeals to the geometric principal
000190 | symbol and low-curvature reduction. QFT008:219–254 then states its tailored
000191 | OS/CAR theorem CONDITIONALLY on RP, temperedness, graded Euclidean
000192 | covariance, graded symmetry and clustering. QFT005:163–184 invokes that
000193 | whole package to claim the reconstructed Wightman theory.
000194 | 
000195 | The useful positive composition is: where those ORIGINAL full-source
000196 | covariance/vacuum hypotheses hold, the common quantum pole kinematics
000197 | follows by(3), so a further independent species-speed equality is redundant.
000198 | It is not justified to infer the full covariance only from(1): lower-order
000199 | backgrounds, the state, a fixed defect direction or an untransported
000200 | reference can break boosts while preserving a Clifford principal symbol.