TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-octic-seam-origin-20261025/LOCAL-STRESS-HISTORY.txt Original file SHA-256: 4c73f82829a1678912c9a30505dceedd495d7b9173b97485860fb36c5d87e34c Inclusive original lines: 1-155 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 | THE ACTUAL FOLDED STRESS: LOCAL HISTORIES AND THE FIRST NONZERO JET 000002 | ================================================================= 000003 | 000004 | Claim: MATH.NATIVE.FOLDED-OCTIC-WARD.01 000005 | Status: PROVED within the supplied original quarter CAR/fold current 000006 | realization. This is a nonzero transition amplitude, not pure-state 000007 | preparation or an identification of this net with the primitive P1 seed. 000008 | 000009 | 1. The reference state is part of the operator dictionary 000010 | --------------------------------------------------------- 000011 | Write Omega_d for the ground vector of the actual shifted oscillator 000012 | module, not for a shift vector. In its underlying oscillator coordinates, 000013 | 000014 | L_n^d=L_n+d.h_n+||d||^2 delta_n0/2, 000015 | L_0^d Omega_d=h_d Omega_d, h_d=||d||^2/2=85/256. 000016 | 000017 | The fixed O/M/U+ dictionary gives b=U_+^T O^T d. The normalized native 000018 | octic state in this module is 000019 | 000020 | |F8;d> = F8hat(a_+,-1) Omega_d, 000021 | F8hat(x)=I8(x)/(96 sqrt(140)). 000022 | 000023 | It has relative oscillator weight eight and polarization degree plus 000024 | eight. The Hilbert product is linear in the second slot. The exact 000025 | checker obtains F8hat(b) != 0 in BOTH original quarters alpha=1/4,3/4. 000026 | It also proves ||F8;d||=1. No variable d is adjusted to obtain a hit. 000027 | 000028 | The unshifted statement remains valid: F8hat is primary for L_n and is 000029 | orthogonal to all stress descendants of the unshifted vacuum. In the 000030 | actual shifted representation it is not primary for L_n^d. Already 000031 | L_1^d F8hat has nonzero x0^7 coefficient b_0/(12 sqrt(140)). 000032 | 000033 | 2. All oscillator words in the translation orbit 000034 | ------------------------------------------------ 000035 | Put y_n=d.h_-n. On these creation polynomials the actual L_-1^d is 000036 | 000037 | L_-1^d = y_1 + sum_(n>=1) n y_(n+1) partial_(y_n). 000038 | 000039 | This follows from [L_-1,h_-n]=n h_(-n-1), with L_-1 Omega_d=0 for the 000040 | unshifted oscillator part. Consequently its full descendants are 000041 | 000042 | V_k=(L_-1^d)^k Omega_d 000043 | =k! [z^k] exp(sum_(n>=1) y_n z^n/n) Omega_d. (1) 000044 | 000045 | At k=8, equation (1) has all p(8)=22 oscillator partitions, including 000046 | the modes -2 through -8. None is replaced by a single-mode truncation. 000047 | The exact checker independently constructs V_k by the displayed 000048 | derivation and by the partition formula for every k=0,...,8. It then 000049 | computes its full norm using 000050 | 000051 | ||product_n y_n^(m_n) Omega_d||^2 000052 | =product_n m_n! (n ||d||^2)^(m_n), 000053 | 000054 | and obtains k!(||d||^2)_k at every level. At d=0 all positive-level 000055 | descendants vanish, recovering the unshifted vacuum control. 000056 | 000057 | Only the y_1^8 term can pair with |F8;d>; its coefficient in V_8 is one. 000058 | Thus the exact all-word identity is 000059 | 000060 | = 8! conjugate(F8hat(b)) != 0. (2) 000061 | 000062 | The complete unitary disk-Mobius orbit is 000063 | 000064 | U(w) Omega_d=(1-|w|^2)^h_d exp(w L_-1^d) Omega_d, 000065 | 000066 | =(1-|w|^2)^h_d w^8 conjugate(F8hat(b)), |w|<1. (3) 000067 | 000068 | Its norm is one by sum_n (2h_d)_n |w|^(2n)/n! 000069 | =(1-|w|^2)^(-2h_d). Higher levels in (3) are retained. A matrix element 000070 | extracts a component; it does not perform a physical projection. 000071 | 000072 | 3. Smooth local stress histories 000073 | --------------------------------- 000074 | For a real smooth fold-circle profile g, fix the Fourier convention by 000075 | 000076 | A_d(g)=sum_n g_(-n) L_n^d, 000077 | c_-1=coefficient of L_-1^d=g_1. 000078 | 000079 | The unshifted Sugawara operator L_n has polarization degree zero. The 000080 | linear shift d.h_n has degrees plus or minus one. A product of k copies 000081 | of A_d(g) cannot reach degree plus eight when k<8, so 000082 | 000083 | =0, k<8. (4) 000084 | 000085 | At k=8 every factor must contribute the positive linear shift. Their 000086 | commutators have zero polarization degree and cannot contribute. After 000087 | commuting away annihilators, all eight surviving creation modes have 000088 | positive oscillator weight. Total weight eight forces all to be -1. 000089 | The complete eighth coefficient therefore is 000090 | 000091 | 000092 | =8! conjugate(F8hat(b)) c_-1^8. (5) 000093 | 000094 | These statements apply to the whole smooth profile. To pass from a finite 000095 | Fourier sum to g, use smooth Fourier convergence and the current/stress 000096 | energy bounds on the common smooth-energy domain. The shifted finite 000097 | energy vector Omega_d lies in every domain required for (4)-(5). The 000098 | self-adjoint smeared stress generates its unitary flow, and the finite 000099 | order spectral theorem Taylor expansion gives 000100 | 000101 | 000102 | =t^8 conjugate(F8hat(b)) c_-1^8+o(t^8). (6) 000103 | 000104 | No analyticity of a general smooth vector is assumed. Differentiability 000105 | through order eight suffices. A projective scalar phase leaves the first 000106 | nonzero coefficient unchanged. If c_-1 != 0 the exact amplitude is nonzero 000107 | for all sufficiently small nonzero real t. The transition probability 000108 | begins at t^16 |F8hat(b)|^2 |c_-1|^16+o(t^16). 000109 | 000110 | For every proper fold interval choose a smaller subinterval of width 000111 | less than pi and a nonzero, nonnegative smooth bump there. Rotating to 000112 | its centre makes the real part of its first Fourier coefficient strictly 000113 | positive. Hence c_-1 != 0 is available with genuine local support. The 000114 | argument keeps the infinite Fourier tail of the bump. 000115 | 000116 | 4. Original localization and the scope of the origin result 000117 | ------------------------------------------------------------ 000118 | The original operator relation is 000119 | 000120 | Lhat_n=Lcal_(8n)^alpha/8+21/64 delta_n0. 000121 | 000122 | Thus the fold profile lifts to v(theta)=g(8theta)/8 on the original CAR 000123 | circle. A fold-local operation is supported on eight preimage intervals 000124 | there. This is the actual locality dictionary; it does not turn one 000125 | fold interval into one original interval. 000126 | 000127 | GEOMETRIC-SOURCE-JET.txt derives its one-particle jet by half-density 000128 | diffeomorphism transport of the supplied quarter Dirac operator. The 000129 | linear source generator is stress. Its DIRECT F8 matrix element is zero, 000130 | while the eighth ordered response (5) is nonzero. Demanding a nonzero 000131 | first-order F8 coupling therefore excludes an actual supplied mechanism 000132 | without justification. 000133 | 000134 | The profile g and pulse strength t specify an intervention. The remaining 000135 | P1 origin question is the exact embedding of this supplied source net, 000136 | ground state, reflection, clock and collar into the primitive seed. It is 000137 | not a demand that P1 choose every laboratory pulse. Full transport of all 000138 | data is a passive equivalence; an active unitary at a fixed detector is a 000139 | different comparison. Neither equation (5) nor (6) proves exclusivity of 000140 | the F8 ray, a six-outcome measurement instrument, or a 3+1-dimensional 000141 | physical reconstruction. 000142 | 000143 | Original source and standard analytic input 000144 | ------------------------------------------- 000145 | joint-native-algebra-selection-20261016/ 000146 | FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt:109-164 000147 | ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt:188-221 000148 | FULL-SOURCE-HULL-AND-QUARTIC-OPE.txt:81-91 000149 | v16-operator-pullback-20261014/FOLD-AUDIT.md:36-86 000150 | Carpi, Kawahigashi, Longo, Weiner, From vertex operator algebras to 000151 | conformal nets and back, arXiv:1503.01260, stress-energy covariance 000152 | and energy-bounded smeared fields. 000153 | https://arxiv.org/abs/1503.01260 000154 | 000155 | All relative source paths above are under experiments/theory-contracts/.