TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/MIXED-SPIN-WORD-PROOF.txt
Original file SHA-256: cb20452e22d19160eb0095b2f6092ac03b75ea084442547851fceed1441baf99
Inclusive original lines: 1-200
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | R071. Execute the actual mixed color-spin/CAR source word and its sewing.
000002 | 1 November2026. Continuation after R070. PHYSICAL F1 OPEN.
000003 | 
000004 | 1. Original question and given data
000005 | R070 fixes four original color roots requiring NS/R sector change. Now
000006 | execute their actual mixed four-field word with each original odd source
000007 | channel, retaining the original state, cocycle, entire sea, source time and
000008 | endpoint restrictions. No new field lattice, metric, Yukawa, profile or
000009 | physical measure is selected. Reuse the existing four-D8-module AIA and
000010 | the already constructed smooth half-kink/DHR lines; their existence is not
000011 | new. AIA intertwiners are not all mutually ordinary local fields.
000012 | 
000013 | Use exactly the fixed M from fold-results.json. In source coordinates
000014 |  q=M^T(e2-e3)=(1/2,-1/2,-1/2,-1/2,-1/2,-1/2,1/2,-1/2).
000015 | Also use its negative and +/-M^T(e1-e3). For each v=e_j, q.v=rho=+/-1/2.
000016 | q^2=2, v^2=1. These are source charges, not particle counts or spacetime
000017 | spin. q is a bosonic color-current field in the supplied E8 extension;
000018 | v is an odd original CAR field. No q=quark identification is made.
000019 | 
000020 | The given free circle has frequencies r+alpha, r in Z, alpha=1/4 or3/4,
000021 | beta=4pi. Its ACTUAL eightfold r=8n+j gives
000022 |  H=8 K_delta, K_delta=L0+delta.p,
000023 |  delta_j=(j+alpha)/8-1/2, j=0,...,7.
000024 | Indeed 8(n+1/2+delta_j)=8n+j+alpha. This is one original CAR source,
000025 | not eight newly inserted copies. On its given AIA parent charges lie in
000026 |  D8*=Z8 union(Z8+s), s=(1/2)^8.
000027 | The four modules are0,v,s,c; NS=0+v, R=s+c. The same given K_delta and
000028 | oscillators act on all four modules. Let T=8beta and Q=exp(-T).
000029 | 
000030 | 2. Whole original NS source and absolute normalization
000031 | For a coset Lambda=Z8 or Z8+s define
000032 |  Theta_Lambda(X)=sum_(k in Lambda) exp[-T(k^2/2+delta.k)+k.X].
000033 | The oscillator partition is P_osc=product_(n>=1)(1-Q^n)^(-8).
000034 | With the already chosen sea-energy subtraction,
000035 |  Z_NS=P_osc Theta_NS(0)
000036 |  =product_(r>=0)(1+exp[-beta(r+alpha)])
000037 |                  (1+exp[-beta(r+1-alpha)]).
000038 | This is the Jacobi triple product channel by channel. Thus normalization
000039 | is the entire original scalar CAR partition, not an independently fitted
000040 | theta constant. Absolute source normalization is retained. This positive
000041 | thermal Z is NOT the entire TFPT Ref/Eta/Pfaffian functional; those data
000042 | and all prior vertices remain separate, unchanged inputs.
000043 | 
000044 | 3. Exact ordered mixed word
000045 | Let Y_p(z) be the supplied charge vertex with the ORIGINAL cocycle. Write
000046 | the original-H cylinder field
000047 |  A_p(x)=exp[(h_p+delta.p)x]Y_p(exp x), h_p=p^2/2.
000048 | Time convention: x=8tau for evolution by H; x is K time. No physical time
000049 | calibration is inferred. An adjoint charge field includes its original
000050 | cocycle scalar; its actual Hilbert adjoint also reflects the cylinder
000051 | coordinate. Fix x1>x2>x3>x4 real with x1-x4<T, and charges(q,v,-v,-q).
000052 | On this analytic branch the paired charge-adjoint scalar is exactly1:
000053 |  E_q E_v E_v^* E_q^*=I with the actual associative charge cocycle.
000054 | This is checked by importing the old eps_ext and add_labels, not by
000055 | replacing the half-monodromy with freely chosen Klein signs.
000056 | 
000057 | Define
000058 |  E_T(x)=2sinh(x/2) product_(m>=1)
000059 |   [(1-Q^m exp x)(1-Q^m exp(-x))/(1-Q^m)^2],
000060 |  X=q(x1-x4)+v(x2-x3).
000061 | Then the normalized original NS trace of the full ordered word is
000062 |  F_NS=exp(delta.X) Theta_NS(X)/Theta_NS(0)
000063 |        * E_T(x1-x4)^(-2) E_T(x2-x3)^(-1)
000064 |        * [E_T(x1-x2)E_T(x3-x4)
000065 |           /(E_T(x1-x3)E_T(x2-x4))]^rho.                 (1)
000066 | Every oscillator mode and all integer source charges are included.
000067 | The cross-ratio and theta factors carry a joint response beyond separated
000068 | q and v two-point factors. This is a kinematic given-source correlation,
000069 | not a newly derived Yukawa or physical scattering interaction.
000070 | 
000071 | Derivation: In ordered multiplicative coordinates, commute all annihilation
000072 | exponentials to the right. Vacuum contractions give
000073 |  product_(i<j)(zi-zj)^(pi.pj). The normalized thermal oscillator trace adds
000074 |  exp[-sum_(n>=1) Q^n/(n(1-Q^n))
000075 |                    (sum_i pi zi^n).(sum_j pj zj^(-n))].
000076 | Neutrality sum pi=0 turns it into the displayed prime-form product. The
000077 | charge trace gives Theta(X); the original-H coordinate factor supplies
000078 | exp(delta.X). Cylinder h factors cancel multiplicative coordinate powers.
000079 | The original charge product fixes the remaining scalar. This derivation
000080 | works on the specified AIA branch; no even-lattice locality theorem is
000081 | silently transferred to fractional intertwiners. The pairwise exponent
000082 | matrix for(q,v,-v,-q) has rho,-rho,-2,-1,-rho,rho, giving(1).
000083 | Standard comparison: Mason/Tuite,math/0204323,Proposition3.13/Theorem3.14.
000084 | 
000085 | Independent original-kernel control: for p=e_j the two-point result equals
000086 |  sum_(r in Z+1/2) exp[x(r+delta_j)]/(1+exp[T(r+delta_j)]).
000087 | The entire filled negative sea sums to
000088 |  exp[(delta_j-1/2)x]/(1-exp(-x)); exponentially small thermal corrections
000089 | are summed separately. It agrees with the theta reader for every original
000090 | j and both quarters. Thus(1) is tied to the actual existing CAR source.
000091 | 
000092 | 4. Endpoint sector selection keeps intermediate fields
000093 | Right-to-left sector paths of the mixed word, starting at0, are
000094 |  0 -> s -> c -> s -> 0.
000095 | Starting at v,s,c gives the corresponding translated paths. The whole
000096 | word is center-neutral, but its intermediate spin/odd fields must remain.
000097 | Full monodromy of a q insertion around a v insertion is -1. It remains
000098 | visible in(1); neither endpoint charge neutrality nor cocycle choices
000099 | remove that analytic continuation.
000100 | 
000101 | A complete positive state extension is available CONDITIONAL on the given
000102 | parent representation: omega_NS(B)=Tr_NS(exp(-beta H) P_NS B P_NS)/Z_NS.
000103 | This compresses a WHOLE word, never each factor. For bounded B it is a
000104 | positive normalized functional on B(H_parent), and it exactly preserves
000105 | the given NS state. This is ordinary state extension/GNS, not a new
000106 | primitive selection theorem. Its intermediate R amplitudes are nonzero.
000107 | For bounded A,B the given midpoint form is also positive:
000108 |  Tr(P_NS exp(-beta H/2) A* exp(-beta H/2) B)/Z_NS
000109 |  =<exp(-beta H/4) A exp(-beta H/4)P_NS,
000110 |     exp(-beta H/4) B exp(-beta H/4)P_NS>_HS/Z_NS.
000111 | This bounded heat form is not by itself a full physical OS/RP or positive
000112 | vacuum-time theorem. Domains, reflection and geometry remain original
000113 | requirements; no infinite-local density matrix is asserted.
000114 | 
000115 | 5. Thermal sewing changes the block, not merely a scalar phase
000116 | Cycling a spinor insertion through a thermal trace changes its charge
000117 | coset from NS to R. Thus the scalar NS trace cannot be called a KMS state
000118 | on the ENTIRE sector-changing algebra. NS/R theta blocks have opposite
000119 | signs under X_j->X_j+2pi i, as their actual charge supports require.
000120 | They cannot be equated as complete source functions.
000121 | 
000122 | The full cyclic check is ALSO executed, with the original cocycle. Move
000123 | the first q field to x1-T at the right end; the new ordered charges are
000124 | (v,-v,-q,q). Both paired charge-adjoint cocycle products are exactly1.
000125 | For the unnormalized trace blocks,
000126 |  G_NS(q,x1;v,x2;-v,x3;-q,x4)
000127 |  =G_R(v,x2;-v,x3;-q,x4;q,x1-T).                       (3)
000128 | This follows by rectangular trace cyclicity and can be checked directly:
000129 |  Y=X-Tq,
000130 |  Theta_R(Y)=exp[T(q^2/2+delta.q)-q.X] Theta_NS(X)
000131 |  by k_R=n_NS-q. The prime-form product changes by
000132 |  exp[q.X-Tq^2/2], using E_T(T-d)=exp(T/2-d)E_T(d).
000133 | The remaining original-time factor changes by exp(-T delta.q).
000134 | All factors cancel. This uses the real ordered strips, not a guessed
000135 | fractional branch sign. With original center projection q has triality0,
000136 | so the same change of variable proves(3) for the center-restricted blocks.
000137 | NORMALIZED answers retain Z_NS,0 F_NS,0=Z_R,0 F_R,0(cyclic).
000138 | Dropping these absolute block normalizers would fail. The checker executes
000139 | this for all four actual color roots/eight odd channels/both quarters.
000140 | Individual q/v monodromy remains -1; cyclic consistency does not cure
000141 | their mutual ordinary-locality obstruction.
000142 | 
000143 | If one ADDITIONALLY demands a NORMAL beta-KMS state on the entire B(H_parent),
000144 | with this SAME given H and traceclass heat, it is uniquely Gibbs. Matrix
000145 | units between two finite-energy states give weights wi/wj=exp[-beta(Ei-Ej)].
000146 | All four connected modules then enter; zero R weight is impossible. This
000147 | is a conditional full normal typeI statement, not a required physical P1 axiom,
000148 | local typeIII theorem or derivation of the physical integration measure.
000149 | The NS-only functional above is still a valid positive state, with a
000150 | distinguished spin seam. Positivity does not force the extra KMS demand.
000151 | 
000152 | 6. Exact same spectrum, different marked source responses
000153 | Let P exchange the first and last four source coordinates and
000154 |  r=(1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2).
000155 | For BOTH original quarters,
000156 |  P^T(r+delta)=delta, r^2/2+delta.r=0.
000157 | Thus k->Pk+r bijects NS and R and preserves the actual given energy and
000158 | oscillator multiplicities. Consequently Z_R=Z_NS EXACTLY, not by numerical
000159 | coincidence. More generally Theta_R(X)=exp(r.X)Theta_NS(P^T X).
000160 | This heat identity does not prove an unchanged field/state/mark dictionary.
000161 | Under the additional whole-parent Gibbs condition, the unprojected NS/R
000162 | weights are1/2. This is not a new physical probability prediction.
000163 | 
000164 | The ORIGINAL fixed center has wF=(1,-1,0,-1,-1,-1,0,-1). Its full filter is
000165 |  Theta_Lambda,0(X)=sum_(l=0..2)Theta_Lambda(X+2pi i l wF/3)/3.       (2)
000166 | It is applied at the whole-word endpoints. For real X this is a sum of
000167 | positive actual charge weights. Replace Theta by(2) in(1) to obtain the
000168 | center-restricted mixed response; it is strictly nonzero. q has center0,
000169 | v may be charged, but the whole word and adjoint preserve this sector.
000170 | 
000171 | The affine NS/R heat identification DOES NOT preserve the fixed center:
000172 | Mr=e1+e4 has triality1. The original R zero-energy line is therefore removed
000173 | by this center, as previously proved; do not count that fact anew.
000174 | The NS vacuum survives. Hence Z_NS,0>=1 and Z_R,0<=Z_R-1=Z_NS-1<0.06.
000175 | For the last strict bound put x=exp(-pi)<1/20 and use
000176 |  log Z_NS<=(x+x^3)/(1-x^4)=a,
000177 |  exp(a)<=1/(1-a)<53/50 at that rational upper bound.
000178 | Thus identical full heat sums do NOT imply identical marked restrictions.
000179 | The same conditional full-center Gibbs rule would give NS weight greater
000180 | than1/1.06, not1/2. No such rule is imposed as the physical TFPT state.
000181 | 
000182 | 7. What is gained, and the actual remaining origin map
000183 | The missing original mixed field word has an explicit all-mode source
000184 | answer and actual sector sewing, tied to the unchanged original kernel.
000185 | This closes that conditional mathematical calculation. It does not select
000186 | the parent, the primitive source,96 matter embedding or H integration law.
000187 | The common-source candidate cannot be certified by its spectrum/Z0 alone.
000188 | Its actual C_Sigma/P_F or Z_adm map must reproduce(1),(2),the original
000189 | charged field/time/cover data and the full mixed words. Keep any given
000190 | original NS spin seam; do not silently replace it by a full-sector Gibbs
000191 | sum or impose a new global KMS requirement. Keep the old smooth DHR
000192 | construction as available support data, not as the physical insertion law.
000193 | No arbitrary new spin weights, boson Gaussianization or geometric model.
000194 | R054/R063/R067/R068, Ref/Eta/H0/Z0 and all Y/Flavor/Alpha,E8/clocks,memory
000195 | and State/Time/RP remain. Physical3+1 and universal spin2 remain F1 goals.
000196 | 
000197 | Evidence: check_mixed_spin_word.py,MIXED-SPIN-WORD.json. 598 controls in
000198 | the final run include actual old cocycle import, original entire-sea
000199 | comparison, partition identity, all paired mixed words, full cyclic
000200 | NS/R sewing and actual center filter with absolute block weights.