TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/SHARED-METRIC-SOURCE.txt Original file SHA-256: 329344c7ed896a68ae1960ad5e2c86afecbdeef065242e279209a201a1632281 Inclusive original lines: 1-187 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 | SHARED VACUUM KINETIC GEOMETRY FROM THE WHOLE ORIGINAL SOURCE 000002 | L20261102I. 2 November2026. Original G/H sources and their scopes retained. 000003 | 000004 | OUTCOME 000005 | 000006 | The original geometric branch already supplies a shared kinetic metric. 000007 | Under the original WHOLE source covariance and invariant-vacuum hypotheses, 000008 | this extends to the physical pole geometry of the renormalized action. 000009 | No independent species-speed fit is needed within that branch. A precise 000010 | complete tensor test and a quantitative matrix-error bound are given below 000011 | for a proposed application to another actual low-momentum source. 000012 | 000013 | This is not an independent construction of a Lorentz-invariant vacuum 000014 | from arbitrary Hamiltonians. The native IR scripts supply useful, explicitly 000015 | typed witnesses; their comparison of Bloch momentum, holonomy and deck phase 000016 | is neither a full common-cone test nor a physical Lorentz counterexample. 000017 | 000018 | 1. USE THE ACTUAL COMMON GEOMETRIC ANCHOR 000019 | 000020 | The original D_geo, not the finite internal coefficient or a compressed 000021 | comparison operator, has Clifford symbol c_g(p) with c_g(p)^2=q_g(p)I. 000022 | The Lorentz low-curvature action contracts its retained Higgs and gauge 000023 | kinetics with that same g. A common smooth conformal factor multiplies 000024 | q_g without changing its null set. Fermion mass/Yukawa/Majorana matrices 000025 | are lower order in this physical-spacetime characteristic test. 000026 | Internal gamma_F, family/color labels and the physical Clifford action 000027 | are different data; equal dimensions are not used as their dictionary. 000028 | 000029 | On the actually retained nondegenerate blocks, scalar normalizations and 000030 | internal wavefunction mixing change amplitudes, not the common metric. 000031 | Gauge-null directions are removed by the actual Ward/constraint quotient. 000032 | R044/R045 reference cancellations remain valid. No cancelled kinetic 000033 | coefficient or nonpropagating clock is assigned an invented propagation cone. 000034 | No bounded/orderzero claim is made about every B_rel or package A_Sigma; 000035 | the conclusion uses the printed full Lorentz branch and its actual terms. 000036 | 000037 | 2. WHOLE SOURCE COVARIANCE, THEN THE SAME GAMMA 000038 | 000039 | Let the actual source, state, admissible field/source maps, reference, 000040 | boundary data and renormalization prescription obey their original Lorentz 000041 | covariance. The source variables transform contragrediently to the actual 000042 | field representations, including separate fermion and conjugate legs. 000043 | Change of variables in this SAME full source preserves its Ward identity; 000044 | its Legendre procedure carries it to the SAME Gamma_ren. If only a formal 000045 | perturbation series is supplied, the conclusion is coefficientwise and 000046 | does not promote that series to an analytically existing full theory. 000047 | 000048 | For a real multiplet write its transformation as rho(L), including the 000049 | spacetime pullback, and suppose the actual flat vacuum phi0 is invariant. 000050 | Then 000051 | Gamma_ren[rho(L)phi]=Gamma_ren[phi], 000052 | K(Lp)=D(L)^(-T)K(p)D(L)^(-1), K=Gamma_ren^(2)|phi0. 000053 | For complex/fermion legs use their actual paired representations instead 000054 | of treating this real Hessian congruence as a new reality condition. 000055 | Invertible representation factors cannot change the physical zero set. 000056 | 000057 | Scalar internal matrices are functions of the common q_g. Dirac/Weyl 000058 | inverse propagators use the actual slash-p/chiral structures with internal 000059 | form factors. Lorentz covariance plus the actual gauge Ward identities 000060 | similarly constrains the physical gauge pole factors. The original causal 000061 | boundary prescription is retained. Physical mass shells, thresholds and 000062 | residues can differ while their spacetime invariant remains the same q_g. 000063 | Different massive group velocities do not constitute different metrics. 000064 | 000065 | This proves the positive all-correction implication under the original 000066 | covariance/vacuum conditions; it is not a claim that a principal symbol 000067 | alone proves those conditions. QFT008:199-213 states covariance as a 000068 | whole-source lemma; its OS theorem219-254 explicitly takes it as a 000069 | hypothesis. The existing OS repair uses FULL moments from Z/Z[0], not 000070 | connected cumulants as the reflected moment family. That repair is reused. 000071 | 000072 | An intermediate regulator commuting with Padm is not for that reason 000073 | Lorentz covariant. Retain the actual modified Ward identities and their 000074 | endpoint restoration. Likewise general covariance of Gamma[g,u,...] 000075 | does not mean invariance at a fixed nonzero vector u. Scalar source 000076 | normalization cannot remove a genuine independent spacetime tensor. 000077 | 000078 | 3. THE COMPLETE QUADRATIC METRIC TEST 000079 | 000080 | For an actual Hermitian physical second-order kinetic block, retain ALL 000081 | same-order terms: 000082 | K(omega,k)=K_t omega^2+2 SUM_a K_ts^a omega k_a 000083 | -SUM_ab K_s^(ab) k_a k_b-M+.... 000084 | The dots here are genuinely different-order terms, not omitted omega*k. 000085 | All matrices must be coefficients of the SAME field/source/state kernel. 000086 | 000087 | One common kinetic metric means the complete coefficient tensor factorizes 000088 | as K^(mu nu)=g^(mu nu)Z on the retained invertible block. In a common 000089 | orthogonal chart with g0a=0 this requires BOTH 000090 | K_ts^a=0 and K_s^(ab)=c^2 delta_ab K_t. 000091 | The positive K_t can contain internal mixing. Its removal by one internal 000092 | wavefunction change does not rescale spacetime separately for each field. 000093 | The c in these equations is the original metric conversion in one common 000094 | coordinate chart. No physical coupling is selected by a common unit choice. 000095 | 000096 | Equivalently, for every unit spatial n the whitened matrices 000097 | B(n)=K_t^(-1/2) SUM_a K_ts^a n_a K_t^(-1/2), 000098 | C(n)=K_t^(-1/2) SUM_ab K_s^(ab)n_a n_b K_t^(-1/2) 000099 | obey B(n)=0 and C(n)=c^2 I. This is necessary and sufficient for the 000100 | stated COEFFICIENT-TENSOR FACTORIZATION on that nondegenerate Hermitian 000101 | quadratic block. It is not a characterization of every dressed quantum 000102 | pole kernel or of full IR universality. 000103 | Spinor blocks must keep their actual Clifford/representation test rather 000104 | than being treated as arbitrary scalar coefficient matrices. 000105 | 000106 | 4. WHOLE-CLASS CONTROLLED ERROR BOUND, WITHOUT A NEW MODEL 000107 | 000108 | Suppose an ACTUAL source calculation in this same chart supplies, uniformly 000109 | over the stated spatial directions/IR region, 000110 | ||B(n)||<=eta*c, ||C(n)-c^2 I||<=epsilon*c^2, 000111 | c>0, eta>=0, 0<=epsilon<1. 000112 | No numerical values of eta or epsilon are inferred from the native static 000113 | responses. These are measurable/boundable full-kernel residuals. 000114 | 000115 | Every positive characteristic ratio v=omega/|k| of this principal block 000116 | satisfies 000117 | sqrt(eta^2+1-epsilon)-eta 000118 | <= v/c <= sqrt(eta^2+1+epsilon)+eta. (1) 000119 | 000120 | Proof: if (v^2 I+2vB-C)x=0, normalize x. Put b=/c, 000121 | s=/c^2 and r=v/c. Then |b|<=eta, 1-epsilon<=s<=1+epsilon, 000122 | r^2+2br-s=0. Its positive root is r=-b+sqrt(b^2+s). 000123 | It increases with s and decreases with b, giving (1). No simultaneous 000124 | diagonalization or commutation of B,C is assumed. Since C is positive, 000125 | the Hermitian companion [[-2B,sqrt(C)],[sqrt(C),0]] supplies real nonzero 000126 | characteristic roots and the same quadratic equation. 000127 | 000128 | For eta=0 the interval is [sqrt(1-epsilon),sqrt(1+epsilon)]. 000129 | For exact original Lorentz covariance epsilon=eta=0 and every such block 000130 | has v=c. This supplies a quantitative route from an actual symmetry 000131 | residual to common quadratic kinematics. It does not prove the required 000132 | residual bound for an arbitrary lattice Hamiltonian, or control all higher 000133 | vertices. It also does not identify low-energy characteristic velocities 000134 | with the UV causal front of an unspecified nonlocal effective action. 000135 | 000136 | 5. WHAT THE ACTUAL ORIGINAL IR EXECUTIONS ESTABLISH 000137 | 000138 | The unchanged v1011 B8 source evaluates the complete256D physical 000139 | logTr exp(-beta H(theta)) at its original g.8,beta4,t_hop.6,masscoef.3. 000140 | It gives Kx=.2921723188988497 and Ky=.2921723189292652. 000141 | This is a genuine coupled static isotropy response. Its later spinor 000142 | companion is a separate Wilson slab multiplied by a static link ratio, 000143 | as its original preregistration expressly states. It supplies neither 000144 | K_t nor the complete physical temporal/spatial pole kernel in section3. 000145 | 000146 | The unchanged v1012 IR comparison uses Bloch momentum for the QWZ edge, 000147 | global holonomy for a small gauge torus, and a deck phase for a0D clock. 000148 | Its native SPLIT is already source-typed. Those operations are not one 000149 | spatial momentum action on one coupled source. The replay preserves the 000150 | positive edge witness and does not refute the original shared metric. 000151 | No new speed fit, Hamiltonian, temperature or parameter scan is used. 000152 | 000153 | 6. STATE, TIME AND CONE TYPES REMAIN SEPARATE 000154 | 000155 | The H beta-KMS state is thermal for its GIVEN Hamiltonian time. A thermal 000156 | rest direction permits medium responses depending on both q_g and u.p. 000157 | Their group/sound velocities need not coincide. This does not disprove 000158 | the vacuum covariance used in section2. Nor does a thermal state alone 000159 | prove spacelike commutator support for an arbitrary lattice model. 000160 | 000161 | A P1 modular KMS flow identified with a geometric wedge boost by its 000162 | actual Bisognano-Wichmann map is another flow. A vacuum can have a KMS 000163 | restriction for that boost. Neither equal beta values nor the label KMS 000164 | identifies this modular flow with the ordinary Hamiltonian time in H. 000165 | The established clock/source dictionaries are retained, not replaced by 000166 | a new compulsory physical clock or temperature choice. 000167 | 000168 | The LR graph cone is a bound on commutator tails. Static stiffness, a 000169 | quadratic principal cone, Lorentz-invariant pole kinematics and retarded 000170 | causal support are distinct properties with their actual stated premises. 000171 | The original controlled IR condition is not replaced by any one of them. 000172 | 000173 | VERDICT AND EVIDENCE 000174 | 000175 | Use the original common metric and whole-source Ward route. On that 000176 | covariant vacuum branch common kinematics follows without another 000177 | species-speed parameter. For a proposed lattice application the complete 000178 | same-source tensor, including mixed time-space terms, is the concrete 000179 | test; (1) controls its deviations when the actual uniform bounds are given. 000180 | This does not supply the original source/vacuum covariance by assumption 000181 | or prove the complete physical TFPT. The optional a->0 decision is retained. 000182 | 000183 | Original passages and primary sources: GEOMETRIC-CONE.txt. 000184 | Actual native calls, defaults and outputs: NATIVE-IR.txt/json and 000185 | ORIGINAL-IR.txt with actual-IR-velocity.json. Their field scopes are retained. 000186 | G/H, whole moments, original Ref/Eta/Z0/H0 and earlier exclusions remain. 000187 | No original paper, ledger or physical completion marker is changed.