TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/BOUNDARY-OPERATION-PROOF.txt Original file SHA-256: a55bc1d8bcda606e274092873dedf00709d40bd093d3ec0e08ce7ff25ecf5a47 Inclusive original lines: 24-137 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. 000024 | Prior results reused: this directory's PROOF.txt:34-128, and 000025 | seam-full-collar-reconstruction-20261015/PROOF.txt:65-97. 000026 | The actual retained product equation is (partial_r+B)psi=0, length ell>0, 000027 | with identified/labeled boundary fibers and supplied selfadjoint B. It gives 000028 | T=exp(-ell B), a closed positive injective possibly UNBOUNDED operator. 000029 | The closed endpoint solution space is L=graph(T), not all boundary pairs. 000030 | Its orthogonal projector and canonical normalized graph map are 000031 | 000032 | G=[[A,C],[C,I-A]], A=(I+T^2)^-1, C=T(I+T^2)^-1, 000033 | V=(A^(1/2),T A^(1/2))^T : H -> H direct_sum H. 000034 | 000035 | Both entries of V are bounded, V*V=I, VV*=G and Ran V=L. The bounded map 000036 | T A^(1/2) is understood spectrally; no multiplication of unbounded T on an 000037 | incorrect domain is used. These are consequences of the given source PDE, 000038 | not an identification of G with a physical quantum covariance. 000039 | 000040 | 2. First check: a raw boundary mask does not descend to the kernel quotient 000041 | 000042 | Let P0=diag(I,0) select the incoming boundary. This label is geometric. 000043 | The naturally available positive comparison is K_G(x,y)=. 000044 | Its null space is Ker G. A linear map F on boundary data can act on its 000045 | quotient only if F Ker G is contained in Ker G. A represented adjoint pair 000046 | also needs the corresponding adjoint compatibility; a raw Hilbert adjoint 000047 | cannot simply be assumed to be the K_G adjoint. 000048 | 000049 | At a scalar spectral mode T=t>0, let 000050 | v=(1,t)/sqrt(1+t^2), w=(-t,1)/sqrt(1+t^2). 000051 | Then G=v v*, G w=0, whereas 000052 | 000053 | K_G(P0 w,P0 w)=t^2/(1+t^2)^2 > 0. 000054 | 000055 | Thus P0 maps the ZERO kernel class to a NONZERO class. It is not an operation 000056 | on this minimal response-state quotient. This is an exact null-stability 000057 | failure in the actual collar graph, not a comparison of arbitrary toy models. 000058 | It forbids the inference 'a labeled boundary mask is automatically a physical 000059 | operator on the state reconstructed from this same K_G'. It does not forbid 000060 | physical boundary interventions governed by a DIFFERENT full history kernel. 000061 | 000062 | Adding an invisible ancillary space cannot repair this exact intertwining 000063 | while preserving K_G: any realization W with W*W=G satisfies W w=0; an 000064 | operator O with O W=W P0 would imply W P0 w=0, contradicting the value above. 000065 | More memory cannot turn the same inconsistent quotient action into a map. 000066 | 000067 | 3. A canonical noncommuting RESPONSE calculus really does follow 000068 | 000069 | For a smooth scalar boundary function f, take the incoming data map 000070 | F_f=diag(M_f,0). It uses only geometric scalar multiplication; no finite 000071 | matrix algebra or internal carrier choice. The kernel-derived compression is 000072 | 000073 | L_f=V* F_f V=A^(1/2) M_f A^(1/2). 000074 | 000075 | This is a bounded source-to-solution response operator, with 000076 | L_f*=L_(conj f), norm at most ||f||_infinity. It is now well-defined on the 000077 | graph space. Compression, however, changes the product: 000078 | 000079 | L_f L_g=A^(1/2) M_f A M_g A^(1/2), 000080 | L_f L_g-L_(fg)=-V* F_f (I-G) F_g V, 000081 | [L_f,L_g]=A^(1/2)(M_f A M_g-M_g A M_f)A^(1/2). 000082 | 000083 | The displayed product is derived ordinary composition of two geometric 000084 | solution-response steps. It is not asserted to equal the sewing of two 000085 | physical interventions: that identification requires an intervention law. 000086 | For f=1, L_1=A rather than I. Thus this compression of incoming insertions 000087 | is not even a unital representation of the original local function algebra. 000088 | Include the identity response separately when speaking of its generated 000089 | bounded star algebra; it is a nonlocal RESPONSE algebra, not a selected 000090 | physical local insertion net. No equality with all bounded operators is made. 000091 | 000092 | The coherent comparison of two such responses on specified boundary vectors 000093 | is fixed, without a fitted parameter: 000094 | 000095 | K((f,x),(g,y))= 000096 | =. 000097 | 000098 | This is a positive Gram kernel, with complex cross terms and all word 000099 | comparisons defined by composing the derived bounded L's. It is a geometric 000100 | Hilbert-space comparison, not yet a Born probability, a physical preparation, 000101 | or an original P1-selected field-state functional. The normalization and 000102 | available boundary vectors must not be silently promoted to those claims. 000103 | 000104 | 4. Evaluate on the retained actual quarter source, without shift truncation 000105 | 000106 | Use the existing conditional realization B e_n=(n+alpha)e_n, alpha=1/4, 000107 | n in Z, and U=M_exp(i theta). This realization is not selected by raw P1. 000108 | For every given ell>0 put 000109 | t_n=exp(-ell(n+alpha)), a_n=(1+t_n^2)^-1. 000110 | Let L=L_u. Then on the FULL bilateral basis 000111 | 000112 | L e_n=sqrt(a_n a_(n+1)) e_(n+1), 000113 | L*L e_n=a_n a_(n+1)e_n, 000114 | LL* e_n=a_n a_(n-1)e_n, 000115 | [L*,L]e_n=a_n(a_(n+1)-a_(n-1))e_n != 0. 000116 | 000117 | The sequence a_n is strictly increasing for ell>0, so the inequality holds 000118 | analytically for every n, not just a finite check. The selfadjoint real/imaginary 000119 | parts of L are also noncommuting bounded response operators. Internal indices, 000120 | five-dimensional carriers and Yukawa matrices have not entered. 000121 | 000122 | One explicit exact scale choice FOR THE CHECK, ell=4 log 2, yields 000123 | t_n=2^(-(4n+1)), a_0=4/5, a_1=1024/1025, a_-1=1/65. 000124 | For input e0, the two response orders have squared norms 000125 | ||L e0||^2=4096/5125, ||L* e0||^2=4/325, 000126 | =52428/66625. 000127 | These numbers are response norms, not empirical probabilities or a derived 000128 | physical scale. For x=(e0+e1)/sqrt2 the coherent overlap 000129 | =sqrt(a0 a1)/2=32 sqrt(205)/1025; 000130 | =i times this value. Selecting this x and phase is a diagnostic 000131 | boundary-vector comparison, not a source-derived physical preparation. 000132 | 000133 | Only the face masks without phase/position insertions give A and I-A, which 000134 | commute. The noncommutativity above uses the actual mixing of B modes by 000135 | geometric multiplication, subsequently projected to the full solution graph. 000136 | It does NOT prove a noncommutative internal fiber or physical gauge group. 000137 |