TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/RHO-DUAL-COEFFICIENT-HOM-REVIEW.txt
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000001 | THE ACTUAL INVERSE CUBIC CHARACTER AND THE FULL q/p HOM PAIRING
000002 | Independent bounded original-source review, 2026-10-30.
000003 | No new reader, Fourier selection, state, metric, Atlas write or pinned edit.
000004 | 
000005 | VERDICT
000006 | 
000007 | The actual Pc already contains the inverse cubic character omega^2.
000008 | It must not be identified with the scalar native family center. More
000009 | substantively, the existing odd-spin products are already GLOBAL
000010 | K_C-valued Hom(Fq,Fp) sections. The original normalized two-torsion
000011 | pairings give this identification explicitly. Composition with the dual
000012 | Hom and trace cancels all actual q/p deck signs, with no requirement
000013 | that one naked coefficient eigenline be a parallel subsystem.
000014 | 
000015 | This is a positive typed bundle connection to an original dual coefficient
000016 | space. It does not by itself choose a rho Fourier line or its output
000017 | contraction. The checked original consumers make specific existing
000018 | choices: the positive structural Gram kernel and its matrix elements,
000019 | and separately the unweighted closed holonomy trace. v231/v233 establish
000020 | the stated phase arithmetic; they do not add a dual-line contraction
000021 | to those formulas. These precise choices, rather than a general absence
000022 | claim or an added symmetry requirement, are the bounded source result.
000023 | 
000024 | 1. THE ACTUAL Pc CHARACTER, NOT THE NATIVE CENTER
000025 | 
000026 | Retain GEOMETRIC-R-SPIN-COLLAR-REVIEW.txt, section6:
000027 | 
000028 |  G(cy)=G(y)Pc, Pc=[[0,0,1],[1,0,0],[0,1,0]],
000029 |  delta_c=Pc c*, delta_c W=W.
000030 | 
000031 | Pc has trace zero and spectrum {1,omega,omega^2}. The line
000032 | 
000033 |  E_(omega^2)=C.(1,omega,omega^2),
000034 |  p_(omega^2)=(I+omega Pc+omega^2 Pc^2)/3
000035 | 
000036 | is its genuine rank-one spectral line. A geometric c eigenvalue omega
000037 | therefore has an available inverse coefficient character. At the cubic
000038 | restriction, their tensor product has total character one. The operator
000039 | omega^2 I3 instead has trace 3omega^2 and a single eigenvalue; it is not
000040 | conjugate to Pc. v233's family/triality scalar is not an identification
000041 | of these two representations.
000042 | 
000043 | The actual Hom coefficient space also has inverse-character components:
000044 | with Q=diag(1,omega,omega^2), Ad(Pc)Q=omega^2 Q. Its omega^2 eigenspace
000045 | has dimension three, for example Q, QPc, QPc^2. This exhibits actual
000046 | coefficient representation content; none of these matrices is inserted
000047 | as a newly selected physical reader here.
000048 | 
000049 | The actual V4 sign D=diag(-1,-1,+1) does not preserve E_(omega^2).
000050 | This refutes only its identification with a global parallel scalar
000051 | family-center line. It is NOT an exclusion of a variable geometric
000052 | insertion, a full tensor product, or a physical source. The necessary
000053 | twist factors are already present, as follows.
000054 | 
000055 | 2. ORIGINAL TWO-TORSION PAIRINGS AND THEIR NORMALIZATION
000056 | 
000057 | The old normalized q+p Kummer construction has
000058 | 
000059 |  (h_a^q)^2=f_a^q=(w+omega^a)/(w+omega^(a+1)),
000060 |  (h_b^p)^2=f_b^p=(w-omega^b)/(w-omega^(b+1)),
000061 |  product h_a^q=product h_b^p=1.
000062 | 
000063 | For L_a^q=O(q_a-q_(a+1)), let e_a^q be its canonical meromorphic divisor
000064 | section. The equality div(f_a^q)=2q_a-2q_(a+1) gives the fixed isomorphism
000065 | 
000066 |  beta_a^q:(L_a^q)^2 -> O, beta_a^q((e_a^q)^2)=f_a^q.
000067 | 
000068 | The same construction gives beta_b^p. On the EXISTING joint cover the
000069 | flat nonvanishing frames t_a^q=e_a^q/h_a^q and t_b^p=e_b^p/h_b^p obey
000070 | beta(t^2)=1. They transform by precisely their V4 signs. Thus beta also
000071 | identifies each L with L^vee, isometrically for the retained sign-unitary
000072 | flat metric. The product-one normalizations retain the original triple
000073 | compatibilities; no new character or relative phase is appended.
000074 | 
000075 | These are the original square trivializations of the sign lines. They
000076 | are not the ordinary spin point norm Nq or an evaluation determinant Bq.
000077 | Those existing objects keep their separate phases and types.
000078 | 
000079 | 3. THE ODD-SPIN PRODUCTS ARE ALREADY GLOBAL HOM SECTIONS
000080 | 
000081 | The actual selected spin square S0^2=K_C and the old a+2 identities give
000082 | 
000083 |  Aqa=H^0(S0 tensor L_a^q)=H^0(O(q_(a+2))),
000084 |  Apb=H^0(S0 tensor L_b^p)=H^0(O(p_(b+2))).
000085 | 
000086 | Both spaces have dimension one. Their original multiplication is the
000087 | nonzero global map
000088 | 
000089 |  mu_ab:Aqa tensor Apb -> H^0(K_C tensor L_a^q tensor L_b^p).
000090 | 
000091 | Using beta_a^q to identify L_a^q with its dual gives EXACTLY
000092 | 
000093 |  K_C tensor L_a^q tensor L_b^p
000094 |        = K_C tensor Hom(L_a^q,L_b^p).
000095 | 
000096 | Therefore each mu_ab is already a holomorphic one-form-valued bundle
000097 | map L_a^q -> L_b^p. Its zero divisor is q_(a+2)+p_(b+2). It is neither
000098 | an ordinary naked K_C section nor an arbitrary extra coefficient leg.
000099 | All nine maps embed in H^0(K_C tensor Hom(Fq,Fp)); the six offdiagonal
000100 | ones are the previously proved geometric U6 product sector.
000101 | 
000102 | This identification uses the actual spin square and beta, so it carries
000103 | the original half-form and Kummer signs. The two spin-central minus signs
000104 | cancel in the product. Pullback, multiplication and beta are compatible
000105 | with the already chosen normalized c,s,R lifts. No separate parallel
000106 | one-dimensional coefficient subsystem is needed.
000107 | 
000108 | 4. FULL DECK COMPENSATION AND THE CANONICAL CONTRACTION
000109 | 
000110 | In flat frames on the joint cover write a component as
000111 | 
000112 |  mu_ab=alpha_ab (t_b^p tensor (t_a^q)^vee),
000113 | 
000114 | where alpha_ab is a scalar holomorphic differential there. For a deck
000115 | element d with signs epsilon_a^q,epsilon_b^p, invariance of the global
000116 | bundle map means alpha_ab carries their product. The displayed Hom
000117 | frame carries the SAME product. Hence the two signs cancel exactly.
000118 | This is the full tensor test that a naked E_(omega^2) test would omit.
000119 | 
000120 | For the actual reverse coefficient space Hom(Fp,Fq), ordinary composition
000121 | and trace give the canonical bundle morphism
000122 | 
000123 |  (K_C tensor Hom(Fq,Fp)) tensor Hom(Fp,Fq) -> K_C,
000124 |  (mu,B) -> tr_Fp(mu composed with B).
000125 | 
000126 | On components the reverse Hom frame is
000127 | t_a^q tensor (t_b^p)^vee. Its composition with the forward frame is the
000128 | identity on L_b^p, trace one. Thus all q/p deck signs disappear in the
000129 | trace. If B is a two-point kernel, its endpoints must first be composed
000130 | in their actual matching order; the same fiberwise evaluation applies.
000131 | This formula does not invent B or assert a new state. It identifies the
000132 | correct existing coefficient type for any actual supplied source kernel.
000133 | 
000134 | The retained flat unitary metrics give the adjoint pairing as well.
000135 | mu^dagger has the reverse Hom type and conjugate differential factor;
000136 | tr(mu mu^dagger) is the corresponding nonnegative K_C tensor bar(K_C)
000137 | density. The old positive Hodge norm on the degree-sixteen compact joint
000138 | cover is sixteen times its base integral; the factor is fixed by pullback.
000139 | No additional metric or free normalization is required.
000140 | This positive norm contraction cancels a common phase; it is not itself
000141 | a newly defined CP-odd reader.
000142 | 
000143 | For cubic descent, the full Hom action includes BOTH coefficient ends:
000144 | 
000145 |  delta_c^Hom(B)=Pc,p (c*B) Pc,q^-1.
000146 | 
000147 | Composition/trace intertwines these actions; the intermediate Pc factors
000148 | cancel. One must apply the total action to the actual sections/kernel.
000149 | The statement does not assert that every individual mu_ab or every rho
000150 | eigenvector is fixed by delta_c. Nor does it replace that calculation
000151 | by the scalar family-center phase. It proves the correct global bundle
000152 | contraction and leaves no naked-line-flatness requirement.
000153 | 
000154 | 5. WHAT THE CHECKED ORIGINAL CONSUMERS ACTUALLY SELECT
000155 | 
000156 | v117_monodromy_weyl_a3.py:14-38 and53-56 define the actual cubic monodromy
000157 | and quarter representation; the later exact G proof supplies the common
000158 | Pc chart. The three-dimensional Weyl monodromy is not a scalar triality
000159 | representation. v231_cp_mu6_phases.py:68-82 constructs the six-cycle and
000160 | reads its spectrum and a declared scalar orientation. v233_cp_triality_phase.py:
000161 | 66-89 checks the common omega^2 family arithmetic; it contains no Pc
000162 | spectral projector, Hom(Fq,Fp) field insertion or contraction with mu_ab.
000163 | 
000164 | The actual source consumer is written in
000165 | documents/archive--tfpt-45--source_extracts--03_em_flavor_source/sections/
000166 | 003-subsection-discrete-phase-operator-and-transport-grammar.tex:
000167 | 91-117 specify P_str, P_sing=(I+Z_c+Z_c^2)/3 and the positive kernel
000168 | (P_str D_f^dagger D_f P_str)^(-1/2). The actual amplitude uses its matrix
000169 | elements with family holonomy. The separate phase reader is the
000170 | UNWEIGHTED tr(Pi_triangle), explicitly in the same tree's
000171 | 005-source.tex:252-298. None of these formulas chooses p_(omega^2), Q,
000172 | or the above mu/B contraction as an additional oriented insertion.
000173 | Z_c is not silently identified with Pc by this review.
000174 | 
000175 | The retained original_insertions/PROOF.txt:25-49 follows an actual finite
000176 | one-form a_F[D_F,b_F] to qM. It retains the supplied full Yukawa/Majorana
000177 | inputs and does not select a fixed geometric cubic Fourier component.
000178 | A given coefficient can of course have such components; their existence
000179 | is not the missing abstract character and is not excluded here.
000180 | 
000181 | DECISION: the inverse character and the complete global Hom pairing are
000182 | already mathematically present in the original coefficient/sign bundles.
000183 | The specifically checked output choices remain the structural positive
000184 | kernel and the unweighted holonomy trace. No additional reader or Fourier
000185 | selection has been supplied or inferred by this review. The next actual
000186 | source-kernel composition can use the canonical Hom pairing above; it
000187 | does not need a new common state or a freely added matching character.