TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt
Original file SHA-256: 2abf4b44f4cb4ca713d4e520676ef8db0849e5142962521a9592550f1c16ddbe
Inclusive original lines: 36-99
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.

000036 | 2. Complete original global time/winding menu
000037 | 
000038 | Write U(t)=exp(-it K_alpha), W_n=W_1^n, Q the actual relative source number.
000039 | On the common finite-energy core the existing source theorem gives
000040 | 
000041 |  [Q,W_n]=n W_n,
000042 |  [K_alpha,W_n]=W_n c_n(Q),
000043 |  c_n(q)=nq/8+n(n-1)/16+alpha n/8.
000044 | 
000045 | Let F_alpha(q)=q(q-1)/16+alpha q/8. Then
000046 | 
000047 |  c_n(q)=F_alpha(q+n)-F_alpha(q),
000048 |  I_alpha=K_alpha-F_alpha(Q), [I_alpha,W_n]=[I_alpha,Q]=0.
000049 | 
000050 | In original folded charges p in Z8 and oscillator level N_osc,
000051 | 
000052 |  K_alpha=N_osc+sum_k p_k^2/2+((k+alpha)/8-1/2)p_k.
000053 | 
000054 | Consequently I_alpha is independent of alpha. This is checked as a symbolic
000055 | polynomial, and under ALL n=8z+r, z in Z,0<=r<8:
000056 | 
000057 |  T_n(p)_k=p_((k-r)mod8)+z+1_(k<r).
000058 | 
000059 | There is no winding cutoff. In particular T_8(p)=p+(1)^8; no finite cyclic
000060 | register replaces the actual charge carry. Opposite quarter transport
000061 | alpha->1-alpha,q->-q,n->-n preserves c_n(q).
000062 | 
000063 | The map |n> tensor psi -> W_n psi from ell2(Z) tensor H_(Q=0) to the
000064 | original full charged CAR representation is unitary. Its ranges are the
000065 | orthogonal charge sectors, and W_-n is the inverse on each sector. Thus
000066 | 
000067 |  Q=N_charge tensor I, W_1=shift tensor I,
000068 |  K_alpha=F_alpha(N_charge) tensor I+I tensor K_(Q=0).
000069 | 
000070 | This is a representation derived from the SAME source, not a newly chosen
000071 | rotor, Hilbert space or coupling. Restricting to the even CAR sector uses
000072 | W_(2n),charge2n. W1 initially leaves it: Me0 has norm1 outside E8, whereas
000073 | M(e0+e1)=(1/2)^8 is an E8 root. Global W2 is still not a local root vertex.
000074 | Extending the even formula over the additional E8 half requires its lift;
000075 | no ordinary local odd-CAR/E8 inclusion is inferred.
000076 | 
000077 | For a chronology U(t_m)W_(n_m)...U(t_1)W_(n_1)U(t_0), put
000078 | N_j=sum_(r<=j)n_r, N_0=0, T=sum_j t_j, N=N_m. EXACTLY
000079 | 
000080 |  History=W_N exp(-iTK_alpha)
000081 |              exp(-iQ sum_j(t_j N_j)/8)
000082 |              exp(-i sum_j t_j F_alpha(N_j)).
000083 | 
000084 | It follows by induction from U(t)W_n=W_n U(t)exp(-it c_n(Q)). Global
000085 | Q phases can be similarly reordered. This is an exact reduction, not a
000086 | unique parametrization modulo all spectral periods.
000087 | 
000088 | The full field mode R_(b,k) f_i has charge0,K=k,Gram B_k from the previous
000089 | proof. Its entire time/winding cyclic space has the actual factorization
000090 | ell2(Z) tensor Ran(R_(b,k)), with identity on that input multiplicity.
000091 | This statement retains every intervening charge, oscillator and opposite
000092 | polarization. No C5-closure assumption or executed projection is used.
000093 | A charge-return history is scalar there. In raw field coordinates its
000094 | returned matrix is scalar times B_k, NOT scalar times an artificially reset
000095 | identity. On mode superpositions total elapsed time still acts; histories
000096 | with the same N,T differ only by a common phase on a fixed-charge input.
000097 | Tuning intermediate wait times therefore supplies no new relative event
000098 | geometry. This excludes only that given global menu, not local insertions.
000099 |