TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/SOURCE-RETURN-MEMORY.txt
Original file SHA-256: c22c2144e4986a0f9c2f591fdc1667d3ad8233d4ff39e433c22031cc366dbd11
Inclusive original lines: 115-186
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.

000115 | 4. Exact original-source matrices and sharp universal-in-code bounds
000116 | -------------------------------------------------------------------
000117 | The previous K is unchanged:
000118 | 
000119 |   K = [[188/3, -40sqrt3 i/3, 0,0,0],
000120 |        [40sqrt3 i/3,80,0,0,0],
000121 |        [0,0,48,0,0], [0,0,0,48,0], [0,0,0,0,32]].
000122 | 
000123 | The full same-source response is
000124 | 
000125 |   H = [[81152/3, -24064sqrt3 i/3, 0,0,0],
000126 |        [24064sqrt3 i/3,54272,0,0,0],
000127 |        [0,0,11264,0,0], [0,0,0,11264,0], [0,0,0,0,6144]].
000128 | 
000129 | Hence
000130 | 
000131 |   Delta = [[203312/9, -55072sqrt3 i/9, 0,0,0],
000132 |            [55072sqrt3 i/9,142016/3,0,0,0],
000133 |            [0,0,8960,0,0], [0,0,0,8960,0], [0,0,0,0,5120]].
000134 | 
000135 | There is a strict exact lower bound Delta >=5120 I5. In Delta-5120I
000136 | the nontrivial2block has first diagonal157232/9>0 and determinant
000137 | 5627150336/9>0; remaining entries are3840,3840,0. Thus Delta has rank5.
000138 | EVERY nonzero original code input has nonzero exterior response, not just f0.
000139 | The value5120 uses fixed source units and is not a probability.
000140 | 
000141 | There is also a sharp scale-independent geometric statement:
000142 | 
000143 |   K^2 <= (9/44) H,
000144 |   ||P L psi||^2 / ||L psi||^2 <= 9/44 for every0!=psi in code. (6)
000145 | 
000146 | For (9/44)H-K^2 the2block has first diagonal106192/99>0 and determinant
000147 | 516112384/121>0; remaining entries are0,0,2560/11. The equality space
000148 | is exactly span(original f2,f3). At least35/44 of the squared Fock norm
000149 | of L psi is exterior. This is NOT a physical instrument probability or a
000150 | statement about every other source channel/profile/history.
000151 | 
000152 | For f0 the two genuinely different next answers are
000153 | 
000154 |   <f0,L^2 f0>=81152/3,
000155 |   <f0,K^2 f0>=40144/9,
000156 |   difference=203312/9=||Q L f0||^2.                             (7)
000157 | 
000158 | The first TWO memory terms are also checked: with C1=B^dagger D B,
000159 | 
000160 |   P L^3 P=K^3+K Delta+Delta K+C1.                              (8)
000161 | 
000162 | The full matrices for C1 and PL^3P are in the exact JSON certificate.
000163 | Computing (8) checks actual exterior continuation, not merely a leakage norm.
000164 | 
000165 | 5. A necessary state size, not a new ten-state model
000166 | ---------------------------------------------------
000167 | The response Hankel/Gram block from M0=I5,M1=K,M2=H is
000168 | 
000169 |   H1 = [[I5,K],[K,H]].
000170 | 
000171 | Its Schur complement is Delta>0, so rank(H1)=10 exactly. It is the
000172 | Gram of the ten ORIGINAL vectors {f_i,Lf_i}, i=0..4. No new states,
000173 | geometry or fitted representation have been chosen.
000174 | 
000175 | For any fixed linear realization M_n=C A^n E, n>=0, with time-independent
000176 | input/output maps E,C and d-dimensional state space,
000177 | 
000178 |   H1 = [C; C A] [E, A E],   hence rank(H1)<=d.
000179 | 
000180 | Therefore d>=10: even changing those fixed input/output coordinates cannot
000181 | make a closed five-state linear evolution reproduce the first3 moments.
000182 | No Hermiticity or positivity is assumed for the hypothetical A.
000183 | An extra feedthrough term only at n=0, time-dependent maps, or nonlinear
000184 | realizations are outside this precise class. This is a lower bound only;
000185 | ten has not been proved sufficient for the full response sequence.
000186 | It is not a particle count, new internal algebra or physical10D geometry.