TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/SPIN-PRODUCT-BOUNDARY-WEIGHT-REVIEW.txt Original file SHA-256: 530b76ce72037cfb83ecb9bd04bb4472cfd47cfc494ef5ef29f3f05ed67e2fcc Inclusive original lines: 1-192 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 SPIN PRODUCT: BOUNDARY WEIGHT AND KERNEL INSERTION 000002 | Independent bounded divisor, type and domain review, 2026-10-30. 000003 | No new tangent, Clifford reader, source, metric or pinned-source mutation. 000004 | 000005 | VERDICT 000006 | 000007 | The proposed sections and their product formula are EXACT. The product 000008 | is regular on C and is a genuine K_C-valued Hom(Lqa,Lpb) section. It 000009 | does not automatically give a bounded zeroth-order operator in the 000010 | original base-circle L2 space: at each ramification mark its nonzero 000011 | one-form coefficient has base weight |theta-theta_j|^(-2/3). The two 000012 | W half-density factors cancel in a multiplier, and retaining Gq,Gp with 000013 | their flat metrics does not remove that weight. 000014 | 000015 | The ORIGINAL Clifford reader has a specific rank-one normal factor 000016 | c(dz)=z(N+S), not an arbitrary tangent matrix. On its surviving channel, 000017 | one-sided multiplication after a smoothing kernel requires an actual 000018 | vertex null condition. Smoothing alone does not imply that condition. 000019 | 000020 | There is nevertheless a canonical well-defined INTEGRABLE FORM/DIAGONAL 000021 | KERNEL pairing with the given smoothing cross kernel. The coefficient 000022 | is L1, the actual two-sided smoothing form is trace class, and its 000023 | diagonal integral converges. This is not the same assertion as declaring 000024 | M_mu c(h)j* a bounded L2 operator. It uses the original one-form and 000025 | kernel pairing, with no new state or Fourier selection. 000026 | 000027 | 1. EXACT OLD SPIN DIVISORS AND SECTION FORMULAS 000028 | 000029 | Write Kinf for the two points above w=infinity, Dq=sum q_a and 000030 | Dp=sum p_b. The retained old identities are 000031 | 000032 | div(Phi)=Dq-Kinf, div(Chi)=Dp-Kinf, Phi Chi=dw, 000033 | div(eLaq)=q_a-q_(a+1), div(eLbp)=p_b-p_(b+1), 000034 | div(w+omega^a)=2q_a-Kinf, 000035 | div(w-omega^b)=2p_b-Kinf. 000036 | 000037 | Hence the actual nonzero sections can be written 000038 | 000039 | psiqa=Phi eLaq/(w+omega^a), div(psiqa)=q_(a+2), 000040 | psipb=Chi eLbp/(w-omega^b), div(psipb)=p_(b+2). 000041 | 000042 | Their existing section normalizations only multiply these formulas by 000043 | their already specified nonzero constants. With tqa=eLaq/hqa and 000044 | tpb=eLbp/hpb, and using the original beta_q:Lqa^2->O to dualize the q 000045 | leg, their product is EXACTLY 000046 | 000047 | mu_ab= hqa hpb dw/[(w+omega^a)(w-omega^b)] 000048 | tensor (tpb tensor tqa^vee). 000049 | 000050 | Its divisor as a twisted canonical section is q_(a+2)+p_(b+2). 000051 | The apparent poles in this meromorphic expression at q/p points cancel 000052 | in the actual bundle and local branch coordinate. No global exclusion 000053 | can be inferred from those naked denominators. 000054 | 000055 | 2. THE FOUR ACTUAL RAMIFICATION WEIGHTS 000056 | 000057 | At either point w=0, hqa and hpb are finite nonzero and both denominators 000058 | are nonzero. Thus in its actual flat coefficient frame 000059 | 000060 | mu_ab=(A0+O(u))du tensor E_ba, A0 nonzero, u=w. 000061 | 000062 | At either infinity point put u=1/w. Then hqa,hpb tend to nonzero roots 000063 | of one, the denominator has leading term w^2, and dw=-u^-2 du. Again 000064 | mu_ab has a finite NONZERO leading coefficient times du. These are the 000065 | four ramification points over z=1,-1,i,-i, with z-z_j=constant*u^3+... . 000066 | 000067 | Along the actual lifted boundary theta-theta_j is of order u^3 and 000068 | theta'(u) of order u^2. In the original boundary one-form chart 000069 | 000070 | mu_ab=m_ab(theta)dtheta, |m_ab| is of order |u|^-2 000071 | =|theta-theta_j|^(-2/3). 000072 | 000073 | This coefficient is locally L1, but not L2 and not bounded. The regularity 000074 | of mu as a section upstairs therefore does not make its base-circle 000075 | coefficient a bounded multiplier. 000076 | 000077 | 3. THE ACTUAL G/W METRIC AND BOTH HALFDENSITIES 000078 | 000079 | The original charts are Wq=3^-1/2 sqrt(pi') Gq^-1 and the corresponding 000080 | Wp, with Hq=Gq^-dagger Gq^-1 and Hp similarly. On regular arcs the 000081 | flat-frame multiplier m E_ba has raw matrix 000082 | 000083 | M_raw=Gp (m E_ba) Gq^-1. 000084 | 000085 | Its norm BETWEEN the actual Hq,Hp fibers is the flat norm |m|, since 000086 | Gq,Gp are the retained unitary identifications for those metrics. Raw 000087 | Fuchs powers in a row cannot be used as extra suppression after their 000088 | metric or G^-1 deweighting has been discarded. 000089 | 000090 | Conjugating a zeroth-order map by Wp and Wq^-1 cancels the source and 000091 | target sqrt(pi') factors. It leaves m, not m sqrt(pi') and not m pi'. 000092 | For an actual deweighted source component f(theta), the upstairs field 000093 | has the mandatory factor sqrt(|theta'|), of order |u|. Multiplication by 000094 | the above m gives order |u|^-1 f(theta(u)), reproducing exactly the 000095 | same original base L2 test. No half-form factor is missing from it. 000096 | 000097 | All coefficient matrices retain BOTH Pc ends. Under delta_c their 000098 | Hom action is Pc,p (c* M) Pc,q^-1. The product and the matching kernel 000099 | contraction use that total action, not a selected scalar Fourier line. 000100 | An individual mu_ab is not asserted to be delta_c fixed. A base operator 000101 | uses the actual equivariant full insertion/consumer, or the full cover 000102 | pairing with its matching coefficient kernel. The estimates below apply 000103 | to its nonzero leading terms; a cancellation in that actual full sum must 000104 | be retained. No independent invariants/Fourier projector is chosen here. 000105 | 000106 | 4. THE ORIGINAL ONE-FORM CLIFFORD READER AND THE SHARP FIRST CONDITION 000107 | 000108 | From the actual principal symbol D=N partial_r-i S partial_x+..., 000109 | 000110 | c(dr)=N, c(dx)=-i S, dz=z(dr+i dx), 000111 | c(dz)=z(N+S), N+S=[[0,0],[2,0]]. 000112 | 000113 | This is the original Clifford multiplication appearing in central 000114 | one-forms [D_M,g]. It fixes the normal type of this holomorphic K-valued 000115 | insertion: one normal channel is killed and the other is mapped to its 000116 | opposite. There is no freely chosen tangent or internal reader here. 000117 | The surviving channel still carries the |theta-theta_j|^-2/3 factor. 000118 | Any charged/dual chart must retain that actual normal and Hom endpoint 000119 | map; a K-valued section is not multiplied into a scalar kernel by name. 000120 | 000121 | For a local one-form coefficient of order mu_u~u^m and an ACTUAL 000122 | deweighted kernel output in its surviving input channel of order 000123 | |theta-theta_j|^beta, the L2 criterion is 000124 | 000125 | beta > (1-2m)/6. 000126 | 000127 | For the present nonzero leading term m=0 it is beta>1/6. In particular 000128 | a nonzero vertex value fails. For a smoothing base output, annihilation 000129 | of the leading input jet by the leading normal/Hom matrix is sufficient; 000130 | the remaining u^-1 term in m is already locally L2 in the base measure. 000131 | The condition must be evaluated AFTER the retained G/W deweighting. 000132 | An apparent raw Frobenius zero does not count if the same chart removes it. 000133 | 000134 | This is the first exact test for M_mu c(h)j* as an L2 operator. Smoothing 000135 | does not itself force a null value at a marked point. In the given 000136 | constant-Phi circle realization, each Fourier block c(h(n)) is positive 000137 | invertible, so one can already obtain a nonzero surviving input component 000138 | at the mark from a smooth mode. Thus no general bounded-L2 composition 000139 | theorem follows just from the word 'smoothing'. Any cancellation in the 000140 | actual weighted corner/source kernel must be checked in that channel. 000141 | 000142 | 5. THE GIVEN SMOOTHING KERNEL DOES ADMIT AN INTEGRABLE FORM PAIRING 000143 | 000144 | Let Cq=c(hq), Cp=c(hp) be the already given positive cross operators, 000145 | c(t)=(2cosh(ell t))^-1, ell>0. The existing source map j intertwines them 000146 | because c is even. The actual cross-source kernel Cq j* has the reverse 000147 | Hom type. In matching frames its smoothing diagonal/cross trace is 000148 | bounded on the regular base charts, with the actual endpoint continuation. 000149 | The Clifford-contracted insertion therefore has the absolutely convergent 000150 | pairing 000151 | 000152 | integral tr[c(mu)(theta) (Cq j*)(theta,theta)] dtheta. 000153 | 000154 | Locally absolute convergence follows from integral |t|^-2/3 dt