TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/P1-W8-LOCAL-MAP-AND-TWOPOINT-TEST.txt
Original file SHA-256: 072b6588666504ec4f04024e279030bbdf83b724102bac181ee3696133c1be28
Inclusive original lines: 27-119
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.

000027 | 2. The original source direction, not an inverse theorem
000028 | --------------------------------------------------------
000029 | The original operational seed is
000030 | 
000031 |  S=(A_loc,tau_t,Theta,omega,[u_Sigma],D_coll).
000032 | 
000033 | The original C_seed functor sends this full tuple to its admissible boundary
000034 | datum. It does not run in the reverse direction to supply the local quantum
000035 | algebra, all state moments or the actual current insertions. The later
000036 | relative reflection-positivity lemma explicitly ASSUMES an admissible
000037 | fermionic CAR kernel with reflection positivity. It does not construct that
000038 | kernel from the scalar c3 or the finite compiler closure.
000039 | 
000040 | Read original passages:
000041 |  documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000042 |  sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex:
000043 |  103-143 and245-248;
000044 |  sections/001-section-primitive-core-and-carrier-interface-handoff.tex:319-342;
000045 |  documents/archive--tfpt-45--source_extracts--02_carrier_source/sections/
000046 |  007-subsection-bosonic-relative-index-and-higgs-selection.tex:365-394.
000047 | 
000048 | The already established QWZ edge -> original CAR -> current -> W8 chain is
000049 | retained. No freely chosen model is added to fill the missing P1 arrow.
000050 | 
000051 | 3. A genuinely local necessary current test
000052 | --------------------------------------------
000053 | Let F be the EXISTING eightfold circle map. For a proper quotient interval I,
000054 | let J_F(I) be the current algebra inside A_CAR^even(F^-1 I). Its physical
000055 | source support remains the union of eight original arcs. A P1 localization
000056 | map must specify its actual target region; the quotient is not silently
000057 | identified with original c1 point localization.
000058 | 
000059 | The old marked profiles sin4theta and1-cos4theta have full spatial support.
000060 | They provide a correct global phase test but not a proper-interval test.
000061 | For the latter take real nonzero chi in C_c^infinity(I), with I avoiding the
000062 | specified marks if that is required, and put
000063 | 
000064 |  kappa_chi=(1/(2pi))integral(chi_prime)^2 >0,
000065 |  f=chi,  g=-2chi_prime/kappa_chi.                         (1)
000066 | 
000067 | This is a normalization of an intervention test profile, not a physical
000068 | coupling or a selected fundamental constant. With the original level-one
000069 | Fourier convention, exact Parseval gives
000070 | 
000071 |  c(f,g)=sum_(k>0) k[conj(f_k)g_k-f_k conj(g_k)]
000072 |        =i/(2pi) integral f_prime g = -2i.
000073 | 
000074 | Hence for W(f)=exp(iJ(f)),
000075 | 
000076 |  W(sf)W(tg)W(-sf)W(-tg)=exp(2ist)I.                     (2)
000077 | 
000078 | All four operators in (2) are in the SAME proper current region. The
000079 | original eight-arc support is retained under the fold. A unital current
000080 | embedding must preserve this operator identity. Ordinary commuting phase
000081 | multipliers, or the previously tested common two-face gauge action with
000082 | cocycle zero, fail this direct current identification. This is not a no-go
000083 | for all P1 constructions and is not a performed measurement protocol.
000084 | 
000085 | Passing (2) for one pair is only a necessary test. Constructing the local
000086 | current map requires the full generating Weylsystem, its state kernel,
000087 | time covariance, reflection and compatible target support regions.
000088 | 
000089 | 4. The conditional P1 operator is fixed once the local map is proved
000090 | --------------------------------------------------------------------
000091 | Suppose the ORIGINAL P1 net contains a normal unital star embedding iota_I
000092 | of the existing current algebras with the support map above, preserving
000093 | state, time and reflection. On its cyclic GNS image,
000094 | 
000095 |  V_iota A Omega = iota(A) Omega_P
000096 | 
000097 | is isometric, extends onto the closed cyclic image, and intertwines the
000098 | given time there. The real-time covariance makes that image reducing for
000099 | the time generator. The prescribed point-split current limits then fix
000100 | 
000101 |  O_P,8(rho) V_iota psi
000102 |        =V_iota Y_CAR(W8,rho) psi,  psi in D_fin.           (3)
000103 | 
000104 | The same coefficients, subtraction, adjoints and source shift are retained.
000105 | Closures on this image are transported from the common core. No claim of a
000106 | new self-adjoint extension outside that image is made.
000107 | 
000108 | Equation (3) fixes a candidate AFTER the actual iota has been constructed;
000109 | it does not define iota by assuming the desired answer. The original P1
000110 | passages give no concrete generating Weylsystem for the left side of (2).
000111 | This is the first missing object on this direct route.
000112 | 
000113 | A neutral current cyclic map also does not automatically transport the
000114 | charged seam implementer between charge sectors. The full marked source
000115 | must retain that additional transport. For a complete Seedop identification,
000116 | the same map must still handle Theta, D_coll, D_ref and their closed domains.
000117 | The earlier global unitary criterion remains sufficient but is stronger
000118 | than needed just for the current subfield image.
000119 |