TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C6-WHOLE-WORD-PRODUCER-PROOF.txt Original file SHA-256: b0809e2f642c6d67f2c0bd6afee3b3391a732c177a0178c59c6457c71f2956e8 Inclusive original lines: 1-142 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 | R076. WHOLE ORDERED-WORD EXPANSION OF THE ORIGINAL C6 SQUARE ROOT 000002 | 1 November2026. Original component calculation; physical F1 remains open. 000003 | 000004 | 1. Decisive question 000005 | 000006 | The original EM3/EM5 gives D_y=yI-delta U6, Q=D_y*D_y+epsilon^2I, 000007 | Green Y=Q^-1 and the structural inverse-square-root component K=Q^-1/2. 000008 | Can these given operators supply their whole ordered transport-word 000009 | coefficients, rather than new fitted diagonal weights? Decide this exact 000010 | component and stop; do not optimize a new stochastic physical model. 000011 | The old finite inverse/cubic formulas, polarity and first variations are 000012 | reused. The new calculation retains all word lengths and intermediate 000013 | labels and tests concatenation before collapsing to an endpoint matrix. 000014 | 000015 | Work in the actual bare six-cycle basis of original v17.U6_power: 000016 | U[i,(i+1) mod6]=1. Row paths use that convention; they are not a new 000017 | physical-time convention for operators acting on quantum state columns. 000018 | Put C=(U+U^-1)/2, a=y^2+delta^2+epsilon^2, b=y delta, 000019 | kappa=a-2b=(y-delta)^2+epsilon^2, q=2b/a. 000020 | Original delta=delta_ph in(1/3,2/3), y=1,1/3,2/3 gives kappa>0, 000021 | 00 the norm-convergent binomial functional calculus gives 000026 | Q^-nu=a^-nu sum_(n>=0) (nu)_n q^n C^n/n! . (1) 000027 | C1=1. Therefore Q^-nu1=kappa^-nu1; its common row factor is fixed. 000028 | R_nu=kappa^nu Q^-nu is symmetric stochastic in this six-cycle basis. 000029 | For EVERY literal ordered sign word w=(s1,...,sn), si=+ or-, starting 000030 | at a specified vertex, (1) gives the normalized bookkeeping weight 000031 | p_nu(w)=(1-q)^nu (nu)_n/n! (q/2)^n. (2) 000032 | The complete vertex path, backtracks, displacement and winding are 000033 | retained. Endpoint sums recover R_nu. The length distribution is 000034 | p_nu(N=n)=(1-q)^nu (nu)_n q^n/n!, 000035 | E[z^N]=[(1-q)/(1-qz)]^nu, 000036 | E[N]=nu q/(1-q), VarN=nu q/(1-q)^2. (3) 000037 | 000038 | For ORIGINAL K use nu=1/2: (1/2)_n/n!=binom(2n,n)/4^n. 000039 | For ORIGINAL Green use nu=1. The normalization is recorded explicitly: 000040 | K=kappa^-1/2 R_1/2, Y=kappa^-1 R_1. (4) 000041 | It cannot be silently removed from Hol K, the given coupling consumer, 000042 | from its derivatives, or from an absolute source normalization. 000043 | 000044 | (2) is the canonical positive expansion in the specified original 000045 | alphabet and analytic functional family. It is NOT a uniqueness theorem 000046 | for all path measures representing one endpoint matrix at fixed q. 000047 | U^6=I already makes many distinct words have the same endpoint action. 000048 | Nor is a canonical expansion a selection of the primitive physical 000049 | measure: ordinary Markov positivity is not the full OS/CAR source law. 000050 | 000051 | 3. Whole-word sewing, not just a squared endpoint matrix 000052 | 000053 | For every n, Vandermonde gives 000054 | sum_(k=0)^n (nu)_k(mu)_(n-k)/(k!(n-k)!)=(nu+mu)_n/n!. 000055 | Consequently concatenating independently distributed complete words of 000056 | parameters nu and mu, including every split position and its actual 000057 | intermediate vertex, gives EXACTLY p_(nu+mu) on each literal full word. 000058 | Thus R_nu R_mu=R_(nu+mu), and two original half-word expansions yield 000059 | the whole Green word expansion. This preserves every ordered path before 000060 | endpoint reduction. nu is a power parameter, not derived physical time. 000061 | An intermediate physical selector or a new readout is not inserted. 000062 | 000063 | Counts can be read with formal edge labels or noncommuting symbols: 000064 | replace C by C(z)=1/2 sum_directededges z_e E_e in (1). 000065 | Near z_e=1 convergence persists. Differentiation counts visits; 000066 | noncommuting markers, or the retained literal word, keep their order. 000067 | Commuting count variables alone do NOT preserve all word order. 000068 | Example ++-- and +-+- both return to0 at length4, while only the first 000069 | visits edge1->2. These are different words of equal bare coefficient. 000070 | Formal z labels are bookkeeping, not an original physical field J. 000071 | 000072 | 4. Actual original homogeneous delta derivative, with the amplitude retained 000073 | 000074 | No new tangent is required to vary the already given parameter delta. 000075 | For the ABSOLUTE coefficient in (1), for a word of length n, 000076 | d_delta log weight_abs = -2nu delta/a+n(1/delta-2delta/a). (5) 000077 | For the normalized word weight (2), 000078 | d_delta log p_nu(w)=q' [n/q-nu/(1-q)]. (6) 000079 | Its mean is0. Adding -nu kappa'/kappa to (6) recovers(5). 000080 | Thus the entire ordered-word derivative matches the original operator 000081 | derivative; normalization has not hidden a response term. Homogeneous 000082 | delta commutes with the bare C6 calculus. General noncommuting original 000083 | field variations still require ordered Frechet derivatives and actual 000084 | selector/frame/reference/domain transport; (5) is not their substitute. 000085 | 000086 | For fixed original delta and all epsilon>=0, q(epsilon)<=q(0)<1. 000087 | At nu=1/2 the omitted normalized mass above N is bounded by 000088 | q^(N+1)/sqrt(1-q). This also bounds endpoint row errors. All finite 000089 | degree count derivatives have convergent polynomial-times-geometric 000090 | tails on compact gapped parameter sets. Therefore the complete source 000091 | component can be evaluated without enumerating its 2^n individual words. 000092 | 000093 | 5. Closed loops, determinant and the first unproved physical identification 000094 | 000095 | The SAME bare matrix gives exactly 000096 | F_y=-logdetQ=-6log a+sum_(n>=1) q^n Tr(C^n)/n. (7) 000097 | Tr(C^n) counts based closed row paths with their1/2 step factors; 000098 | all windings and returns remain. The1/n counts cyclic repetitions, 000099 | not a universal weight for arbitrary open transfer words. 000100 | Equation(7) is an algebraic determinant expansion. F_tr is specified as 000101 | a stationarity functional in EM3, not as logZ_adm in that definition. 000102 | If F_y were an action term, e^-F_y=detQ, not detQ^-1. Positive loop 000103 | coefficients of -logdet alone therefore select neither boson nor fermion 000104 | statistics. No new Pfaffian, physical auxiliary field or measure is used. 000105 | 000106 | This computes the given C6 contribution completely in its word basis. 000107 | It does not supply QFT004's additional transport-count/carrier/selector 000108 | weights or identify formal z with the actual primitive field source. 000109 | Bare translation symmetry gives equal weights to translated paths; it 000110 | does not forbid a later original label-dependent coupling. Actual Hol 000111 | and the bound reference/Eta/H0/Z0 must be carried by that coupling. 000112 | 000113 | For a selector P, P Q^-nu P always means endpoint compression of(1). 000114 | (PQP|RanP)^-nu is generally a different consumer unless P reduces Q. 000115 | Stepwise P insertions can erase charged intermediates. The original 000116 | K_str already specifies its own compressed Gram on RanP_str; the bare 000117 | six-cycle result transfers there only with its actual reduction or full 000118 | compressed operator proof. No bare-cycle row measure is imposed on it. 000119 | Neither finite cycle vertices nor length n are physical space/time. 000120 | 000121 | The next original calculation is the actual field/coupling/selector 000122 | image of one marked word or the original S_adm->T_V[J] rule, compared 000123 | with this now exact entire component. Reuse R075; no further generic 000124 | walk-length, gap, profile, Y or scale fit. Whole R035/R037/R043 sources, 000125 | R054/R063/R067/R068/R069/R071, charged histories, memory, State/Time/RP, 000126 | E8/clocks and existing Yukawa/flavor/alpha remain unchanged in scope. 000127 | 000128 | 6. Evidence and primary formula 000129 | 000130 | The unchanged native v17 U6_power is called by the checker. Original 000131 | delta is the already derived EM3 algebraic branch, not a new fitted value. 000132 | Checks compare the positive all-length series with the direct spectrum, 000133 | original cubic Green and original four-point square-root interpolation, 000134 | including explicit remainder bounds. Small literal words test their 000135 | retained intermediate vertices and all split positions. Symbolic source 000136 | scores and native Gram derivatives are checked independently. 000137 | The infinite statements follow analytically from(1)-(7), not a cutoff. 000138 | NIST DLMF15.4.6, together with its defining hypergeometric series, gives 000139 | the standard binomial identity used in(1); no new probability theorem 000140 | is attributed to the finite numerical replay. 000141 | https://dlmf.nist.gov/15.4.E6 000142 | complete_TFPT_solution=false; ledger_changed=false.