TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/QUARTER-TOWER-HIGGS-INVERSE-PROOF.txt Original file SHA-256: fc4fb8e221aed1f4802ecb00a2d93578bd629ab08783378fcd364d135d1d90cc Inclusive original lines: 1-200 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 | UR.UNIFIED.QUARTER-TOWER-HIGGS-INVERSE.01 000002 | 31 October2026. R063. Source tomography from the whole existing mode tower. 000003 | 000004 | 1. Original question and changed information 000005 | R062 reads the complete original fixed rightup color-triplet marginal. 000006 | Its first three occupation moments do not determine the radial H law. 000007 | NEW question: does the ENTIRE original positive quarter mode tower, 000008 | not more copies of the same Pauli mode, determine that common radial law? 000009 | Use the SAME given original quark Yukawa operator/fixed right-familyvector 000010 | with y²>0 and the SAME spatially constant original H in every quarter mode. 000011 | Original quarter momenta are p_n=n+alpha,n>=0,alpha=1/4 or3/4. 000012 | The original h_H(p)=p gamma-i gamma Phi(H) gives, exactly as inR062, 000013 | t_p(u)=1/2+p tanh(ell sqrt(p²+y²u))/(2 sqrt(p²+y²u)), u=H*H. (1) 000014 | No extra mode, variable clock width or new Yukawa is introduced. For the 000015 | canonical elliptic quarter source ell=2pi: t_p(0)=1-C_p with 000016 | C_p=(1+exp(4pi p))^-1. General fixed ell>0 is retained in the theorem. 000017 | 000018 | PRECISE common-source premise: ONE normalized positive full source law 000019 | mu on finite u>=0 gives bar_t(p_n)=int t_p(u)dmu for ALL these modes. 000020 | All bosonic/fermionic/reference/eta/det/Pf/Z_F/Z0 weights belong to this 000021 | SAME law. It is not a separately normalized p-dependent conditional law. 000022 | There is no finite-moment or bounded-support assumption on mu. Additional 000023 | conditional fermion vertices, signed weights and a nonconstant H(x) operator 000024 | that mixes momenta are outside this constant-H mixture theorem. 000025 | This premise is a field-dependent consumer, NOT an identification of this 000026 | H-integrated source with the already given primitive seam source C_Sigma. 000027 | Success: exact injectivity, constructive inverse and finite-data bound. 000028 | Failure/stop: if actual master violates this shared constant-H mixture 000029 | premise, use its real full source; do not impose the premise as a new law. 000030 | 000031 | 2. Positive-real analytic response, even without H moments 000032 | Push mu to nu on v=y²u. Define for Re z>0 000033 | F_nu(z)=int z tanh(ell sqrt(z²+v))/sqrt(z²+v) dnu(v). (2) 000034 | For real p>0, F(p)=2bar_t(p)-1 and 01 as p->+infinity, since all u 000036 | are finite. This is a necessary common probability normalization condition. 000037 | The apparent square-root branch cancels: tanh(ell E)/E is meromorphic 000038 | in E². Logarithmically differentiating the standard cosh product gives 000039 | tanh(ell E)/E=(2/ell)sum_(j>=0)1/(E²+omega_j²), 000040 | omega_j=pi(j+1/2)/ell. (3) 000041 | Thus the conditional integrand is a positive-real function on Re z>0: 000042 | Re[z/(z²+a)]=Re(z)(|z|²+a)/|z²+a|²>0, a=v+omega_j²>0. (4) 000043 | Normal convergence/integrability is not inferred from formal positivity. 000044 | Every finite partial sum has real value atz=1 bounded by tanh(ell)<=1. 000045 | A positive-real analytic G with G(1)>0 real obeys, by Schwarz's lemma 000046 | under w=(z-1)/(z+1), 000047 | |G(z)| <=G(1)(1+|w|)/(1-|w|). (5) 000048 | This bounds all partial sums on each compact RHP uniformly in v>=0. 000049 | Convergence on the positive axis and Vitali give locally uniform convergence 000050 | there. The same uniform compact bound permits integration against ANY 000051 | probability nu. Hence F is analytic positive-real, with no moment expansion 000052 | at infinity or analyticity of a generic Higgs Fourier function assumed. 000053 | 000054 | 3. Whole original discrete tower fixes F uniquely 000055 | Set s_nu(z)=(F_nu(z)-1)/(F_nu(z)+1). This is analytic with |s|<=1 inRHP. 000056 | If two laws agree on the firstN ACTUAL quarter momenta, s_mu-s_nu has those 000057 | zeros and modulus<=2. Removing the finite RHP Blaschke factors and applying 000058 | the maximum principle gives, for every real q>0, 000059 | |s_mu(q)-s_nu(q)|<=2 B_N(q), 000060 | B_N(q)=product_(n=0..N-1)|(q-p_n)/(q+p_n)|. (6) 000061 | This standard bounded analytic function estimate can equally be proved 000062 | by moving to the unit disk and repeatedly removing a zero via Schwarz. 000063 | For real q, s_mu,s_nu in(-1,0], so the inverse Cayley map yields 000064 | |F_mu(q)-F_nu(q)|<=4 B_N(q), 000065 | |bar_t_mu(q)-bar_t_nu(q)|<=2 B_N(q). (7) 000066 | Explicitly bar_t=1/(1-s) and s in(-1,0], so derivative1/(1-s)^2<=1. 000067 | The occupation map is nonexpansive on this ACTUAL real range; an 000068 | uncontrolled Cayley Lipschitz constant near s=+1 is irrelevant here. 000069 | For p_n=n+alpha, sum p_n/(1+p_n²) diverges; explicitly log B_N(q) 000070 | =-2q log N+O(1) unless q is one of the sampled nodes, where B_N=0 already. 000071 | Thus equality of the WHOLE single-quarter tower forces F_mu=F_nu onRHP. 000072 | No Carlson exponential-type hypothesis or finite polynomial fit is needed. 000073 | This is not a claim that an arbitrary analytic function is fixed by samples 000074 | accumulating only at infinity: its positive-real/Schur class is essential. 000075 | 000076 | The original two quarters give a particularly explicit bound atq=1/2: 000077 | alpha1/4: B_N=1/(4N-1); alpha3/4: B_N=1/(4N+1). 000078 | If the SAME law supplies both sets of N answers, remove all2N zeros: 000079 | B_2N(1/2)=1/(16N²-1), so occupation ambiguity<=2/(16N²-1). (8) 000080 | With1000 exact answers from each quarter this is <=1.250000079e-7. 000081 | The targetq=1/2 evaluates the same analytic response; it does not add a 000082 | physical half-mode or identify another native clock. 000083 | This is a conditional pairwise bound between compatible laws with EXACT 000084 | sample agreement. It is not a noisy-data guarantee, a measure total-variation 000085 | bound or an assertion that the physical source values were already computed. 000086 | 000087 | 4. F fixes the actual radial law; constructive triangular inverse 000088 | Let sigma=sum_(j>=0)translation_(omega_j²)nu. It is locally finite since 000089 | omega_j² grows quadratically, and int(1+x)^-1 dsigma(x)0, Tonelli and(3) give 000091 | F(sqrt(z))/sqrt(z)=(2/ell)int dsigma(x)/(z+x). (9) 000092 | The ordinary Stieltjes transform on this integrability class is injective. 000093 | One justification: write1/(z+x)=int_0^infinity exp(-(z+x)t)dt; Laplace 000094 | uniqueness, twice, gives sigma from all positive z. Exponential weighting 000095 | makes the locally finite measures finite before the second uniqueness use. 000096 | Equivalently Stieltjes inversion recovers sigma with its atoms retained. 000097 | Now shift rho=translation_(-omega0²)sigma. It has the exact measure identity 000098 | rho=nu+sum_(j>=1)translation_(delta_j)nu, 000099 | delta_j=omega_j²-omega0²=(pi²/ell²)j(j+1). (10) 000100 | Write R(x)=rho([0,x]),V(x)=nu([0,x]), withV(x)=0 forx<0. Then 000101 | V(x)=R(x)-sum_(j>=1,delta_j<=x)V(x-delta_j). (11) 000102 | Every shift is >=delta1=2pi²/ell²>0 and only finitely many shifts occur 000103 | below any fixed x. Starting below delta1 and proceeding through finite 000104 | successive intervals reconstructs V everywhere uniquely, including atoms. 000105 | This is a triangular recursion, not a candidate-distribution search. 000106 | An equivalent exact inverse is Lnu(t)=Lsigma(t)/Theta(t), 000107 | Theta(t)=sum_j exp(-omega_j²t)>0. (12) 000108 | Finally u=v/y², using the GIVEN y, recovers mu. Therefore the entire actual 000109 | single-quarter two-point tower determines the full radial law uniquely. 000110 | Only radial law is determined. Angular H law, other-field joint law, actual 000111 | primitive localization and physical spacetime are not supplied by this inverse. 000112 | Under the stated common conditional source it fixes all radial triplet 000113 | higher words as CONSEQUENCES, including R062's third occupation moment. 000114 | It does not make the integrated fermion state quasifree or justify Wick 000115 | replacing those higher words by products of averaged covariances. 000116 | 000117 | 4b. Connection to the existing full invariant Higgs marginal 000118 | R057 already proves, under its EXPLICIT constant-H weak-invariant positive 000119 | marginal premise, H=sqrt(u)xi with Haar xi onS3 independent ofu. 000120 | This is NOT a new Haar measure choice and is NOT inferred from the tower. 000121 | R057 covariance/field-identification assumptions are retained when invoked; 000122 | the moment-free radial theorem above does not depend on those assumptions. 000123 | Adding precisely that existing premise, the uniquely recovered radial mu 000124 | therefore determines the ENTIRE constant-H marginal. Given the original 000125 | conditional whole fermionic word functional W_A(H), including its actual 000126 | Majorana/Pfaffian/charge/grading conventions, every finite original word is 000127 | omega(A)=int dmu(u) int_(S3) W_A(sqrt(u)xi) dHaar(xi). (12b) 000128 | Thus one given nonzero original quark Yukawa channel's whole tower fixes 000129 | all these conditional matter responses in this same class. It does not 000130 | supply other boson/geometry joint conditionals, field-local H(x) or an E8 000131 | physical carrier map. Angle-dependent conditional words are NOT fixed if 000132 | R057's invariance premise is omitted. No new96 independent physical CAR 000133 | fields or alternative Majorana prescription is introduced by this corollary. 000134 | This is why two-point information across the WHOLE constrained family can 000135 | identify a NONquasifree mixture, while Wick of averaged two-point entries 000136 | still gives incorrect higher words. Fixed-triplet and whole-tower inputs 000137 | are different information, so R062's finite ambiguity is preserved. 000138 | The unchanged original charged-lepton Yukawa channel is used as a direct 000139 | cross-channel diagnostic after reconstructing one shared radial law. 000140 | 000141 | 5. Finite automatic reader, not an arbitrary physical completion 000142 | A real multipoint Schur recursion provides target intervals from finite 000143 | tower values. Start S0(z)=(F(z)-1)/(F(z)+1). At a sample p_k let 000144 | c_k=S_k(p_k), b_k(z)=(z-p_k)/(z+p_k), and 000145 | S_(k+1)(z)=(S_k(z)-c_k)/[(1-c_k S_k(z))b_k(z)]. (13) 000146 | Schwarz-Pick ensures |S_(k+1)|<=1 for consistent analytic source data. 000147 | For real targetq use terminal S_N(q) in[-1,1], then recurse backwards 000148 | S_k(q)=(c_k+b_k(q)S_(k+1)(q))/(1+c_k b_k(q)S_(k+1)(q)). (14) 000149 | Endpoints, with ordering after each real Mobius map, give a certified 000150 | analytic-class interval. Intersect the resulting F interval with 000151 | 0<=F(q)<=tanh(ell q) for the original mixture. Zero pivots and |c_k|=1 000152 | are handled by the terminating constant-Schur case; the diagnostic code 000153 | uses only strict interior data. Numerical reconstruction needs sufficient 000154 | precision; an uncertified floating interval is not a rigorous enclosure. 000155 | This reader takes ACTUAL source answers as input. It does not generate them. 000156 | A positive-real finite Pick matrix is a necessary consistency test: 000157 | K_ij=(F(p_i)+F(p_j))/(p_i+p_j)>=0 as a MATRIX. (15) 000158 | From(3) it is a Gram sum with two features p/(p²+a),sqrt(a)/(p²+a). 000159 | This test ALONE is not sufficient for the special Higgs mixture class. 000160 | The recovered Stieltjes measure must also admit(10) with nonnegative nu 000161 | of total mass1 and original support; these are real additional conditions. 000162 | 000163 | 6. First original-source assignment test 000164 | For any p>0, t_p(u) is strictly decreasing in finiteu wheny>0. 000165 | Thus bar_t(p)=t_p(0) under a positive common law forces mu=delta0. 000166 | In particular the original zero-H elliptic right tower cannot AT THE SAME 000167 | TIME be the integrated source of a nontrivial positive H mixture in this 000168 | unchanged marked-field/ell/Y dictionary. This is a precise obstruction to 000169 | that SAME-source assignment, not a TFPT no-go or an imposed master equation. 000170 | The primitive free seam source before H integration remains allowed. 000171 | The actually given free covariance is replayed at2/4/8 quarter samples 000172 | under this EXPLICIT marked-field identity candidate; that identity is not 000173 | a derived physical carrier map. Nontrivial original carrier/field/source 000174 | transport can change the relevant 000175 | integrated observable; it must be derived, not silently set to the identity. 000176 | R054 already excluded a Gaussian replacement on all words; the present 000177 | tower route provides an actual inverse and a sharper two-point assignment test. 000178 | 000179 | 7. Checks and scope 000180 | Direct unchanged original96 h_H(p) at both original quarter offsets and 000181 | several real/complex H verifies(1),with Majorana retained. Fixed family 000182 | vector is independent ofp andH. Finite partial fractions, complexRHP 000183 | positive-real identities, exact telescoping bounds, positive Pick kernels, 000184 | finite Schur intervals and exact triangular measure deconvolution are 000185 | checked separately. Distinct positive laws can agree on a finite sample 000186 | prefix; the reader must not turn finite agreement into exact measure identity. 000187 | Old R062 same2/4,different6 examples are separated by additional actual 000188 | quarter responses. A normalized radial law with density1/(1+u)^2 and 000189 | infinite first moment has a well-defined response, checked without moment 000190 | expansion. No arbitrary radial law is chosen as the physical one. 000191 | No numerical diagnostic proves the infinite uniqueness theorem; sections2-4 000192 | give that proof with explicit hypotheses. No newLean/ledger/paper promotion. 000193 | 000194 | [OMITTED: non-scientific conversation or execution metadata] 000195 | [OMITTED: non-scientific conversation or execution metadata] 000196 | [OMITTED: non-scientific conversation or execution metadata] 000197 | [OMITTED: non-scientific conversation or execution metadata] 000198 | [OMITTED: non-scientific conversation or execution metadata] 000199 | [OMITTED: non-scientific conversation or execution metadata] 000200 | [OMITTED: non-scientific conversation or execution metadata]