TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/RECURSIVE-WORD-SOURCE-PROOF.txt Original file SHA-256: 06d651e747e392a5a9c021c071fea3865687203aa58d75cee26fceb7035f5af5 Inclusive original lines: 1-110 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 | [OMITTED: non-scientific conversation or execution metadata] 000002 | [OMITTED: non-scientific conversation or execution metadata] 000003 | [OMITTED: non-scientific conversation or execution metadata] 000004 | 000005 | 1. Question and retained input 000006 | 000007 | Can one keep a stable reconstruction rule while source responses change, 000008 | without discarding exterior information at each coarse observation? 000009 | Use the actual existing source, not another fitted Hamiltonian. Preserve the 000010 | existing flavor/Yukawa/alpha results. Source-specific physical identifications 000011 | remain at their existing scope; this calculation does not reopen their values. 000012 | 000013 | The general mathematics below uses familiar block elimination. Its substantive 000014 | native application is the actual minimal reachable source process in section 4. 000015 | The previously established positive-word reconstruction and future quotient 000016 | F1.M1-M3 are reused, not counted as new results. 000017 | 000018 | 2. Complete ordered response and nested reconstruction 000019 | 000020 | Let D be a common invariant complex vector domain for an alphabet A_a. 000021 | For infinite alphabets use coefficientwise finite words and a well-defined 000022 | formal sum with finite contribution to each coefficient. Assume P^2=P, 000023 | R^2=R, PR=RP=P and P D,R D subset D. Finite-rank original finite-energy 000024 | readouts satisfy this. Hilbert positivity additionally requires orthogonal 000025 | P and the actual Hilbert adjoints, not names assigned to letters. 000026 | 000027 | Let x_a be formal free NONCOMMUTING variables commuting with operator 000028 | coefficients. Define T=sum x_a A_a, M=I-T and G_P=P M^-1 P. 000029 | All inverses in this section are formal in the word-length completion. 000030 | The constant term of M is I. Thus M^-1=sum T^n exists coefficientwise, 000031 | and the coefficient of x_a1 ... x_an is exactly P A_a1 ... A_an P. 000032 | This statement concerns every finite word, with no depth truncation. 000033 | 000034 | For Q=I-P, write D_Q=Q-QTQ on QD. It has constant identity Q. 000035 | The exact Schur expression on PD is 000036 | 000037 | F_P(M)=P-PTP-PTQ D_Q^-1 QTP, 000038 | G_P=F_P(M)^-1. 000039 | 000040 | All these formal inverses exist. Crucially the variables within the memory 000041 | term KEEP THEIR ORDER. E.g. for T=xL+yJ+zJdagger with JP=PJ=P, 000042 | and L=[[K,Bdagger],[B,D]], J=diag(I,J_Q), it is 000043 | 000044 | F_P= (1-y-z)I_P-xK 000045 | -x Bdagger (I_Q-xD-yJ_Q-zJ_Qdagger)^-1 x B. 000046 | 000047 | Replacing the last term by x^2 Bdagger (...)^-1 B is WRONG for free x,y,z. 000048 | It would change the word x y x into x x y. No such reordering is used here. 000049 | 000050 | Nested theorem: F_P(F_R(M))=F_P(M), with the inner complement R-P 000051 | and the SAME formal variables, or the same spectral parameter after a 000052 | legitimate common specialization. Proof: F_R(M)^-1=R M^-1 R. Its 000053 | P compression is P M^-1 P. The formal inverse on PD is unique, hence 000054 | both eliminations give the same F_P. This is not a statement that the 000055 | linear pencil I-sum x_a P A_a P closes: its nonlinear memory terms remain. 000056 | 000057 | Full legs are retained by the block inverse, with block ordering as written: 000058 | Q M^-1 P = D_Q^-1 QTP G_P, 000059 | P M^-1 Q = G_P PTQ D_Q^-1, 000060 | Q M^-1 Q = D_Q^-1 + D_Q^-1 QTP G_P PTQ D_Q^-1. 000061 | Thus a prescribed additional readout can be transported with its actual legs. 000062 | The single boundary response does not reconstruct an unobservable decoupled 000063 | sector. If an original finite Grassmann Gaussian functional is also retained, 000064 | its det(D_Q) factor must be kept at commutative specialization. We assert no 000065 | ordinary multiplicative determinant for free noncommuting variables and no 000066 | unregulated infinite determinant. Interacting original Z is not made Gaussian. 000067 | 000068 | For bounded operators the identities also hold analytically in a common 000069 | Neumann neighborhood. For unbounded fields the coefficientwise theorem is 000070 | on D only; it supplies neither convergence nor a physical selfadjoint time. 000071 | At a numerical point invertibility of M alone does not ensure invertibility 000072 | of D_Q: M=[[1,-1],[-1,0]] is invertible but its lower diagonal block is zero. 000073 | An actual analytic Schur application must control these inverse domains. 000074 | 000075 | For a unitary U the exact identity is 000076 | G_(UPUdagger)[U A Udagger]=U G_P[A] Udagger. 000077 | Keeping a fixed P that fails [P,U]=0 does not give a unitary internal clock 000078 | on PD. Transport P as well, or retain the full clock word legs and memory. 000079 | No identification among the different existing TFPT clocks is inferred. 000080 | 000081 | 3. Positivity, minimality and structural fixed point 000082 | 000083 | With actual input vectors e_i in PD define 000084 | K((u,i),(v,j))=. 000085 | For every finite c, c* K c=||sum c_(u,i) A_u e_i||^2>=0. 000086 | No completely positive channel assumption is needed for this Gram identity. 000087 | The kernel is not the unsymmetrized ordered response matrix itself. 000088 | 000089 | Let H_reach=span{A_w e_i}. If this space is finite dimensional and the 000090 | alphabet is closed under actual adjoints, it reduces every A_a. It is also 000091 | fully observable: if P A_w v=0 for every word and v in H_reach, then 000092 | =0 for every w,i; hence v=0. The transfer Hankel matrix 000093 | has rank dim(H_reach): it factors through H_reach and, after reversing and 000094 | adjoining row words and applying the invertible input Gram, contains a 000095 | full-rank Gram block. Thus no smaller linear realization with fixed input 000096 | and output maps can preserve all words, including the empty word. 000097 | 000098 | This is a statement about LINEAR realization, not arbitrary nonlinear 000099 | encodings or a claim about the number of physical spatial dimensions. 000100 | Reachability can be found automatically: add generator images of a basis 000101 | until none adds an independent vector. Once the space is invariant, an 000102 | induction covers every length. This replaces enumeration of infinitely 000103 | many words, rather than setting an arbitrary billion-word cutoff. 000104 | 000105 | The resulting representation is a structural fixed point: minimize it again 000106 | with the same source alphabet and readouts and it is unchanged up to an 000107 | intertwining isometry. Individual word operations need not fix a state. 000108 | Nested exact observation also leaves the answer unchanged. This resolves 000109 | the logical confusion between stable organization and a frozen trajectory. 000110 | It does not turn source-response words into a selected physical history.