TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/automated_source_search_20261029/PROOF.txt
Original file SHA-256: 0c6874e4f294f5c11e23e4b453786b683271463820a324503d8ceb78c1bd4563
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.
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000001 | AUTOMATED SOURCE SEARCH: REFLECTION TYPE BEFORE SOURCE OPTIMIZATION
000002 | 29 October 2026
000003 | 
000004 | Question and scope
000005 | 
000006 | The requested end result remains an original P1 derivation of the common
000007 | physical source, preserving localization, state, charges, reflection and time.
000008 | This calculation addresses one necessary identification on that route. It
000009 | does not choose a new physical source or optimize weights to match a known
000010 | carrier marginal. A contradiction in one actual current mode is sufficient
000011 | to reject the proposed ordinary-Weyl intertwiner. Passing that test is not a
000012 | proof of a P1 source map.
000013 | 
000014 | The changed premise relative to the preceding reflection note is explicit:
000015 | test whether its CONDITIONAL formulas (4) can even extend to the ordinary
000016 | current Weyl product, while retaining both conjugate linearity and reversal
000017 | of product order of the earlier OS word prescription. The preceding note
000018 | correctly left that extension unproved. Under these joint premises its
000019 | single-chiral formula is now excluded, rather than merely awaiting proof.
000020 | 
000021 | Original inputs and already solved work
000022 | 
000023 | 1. The given original current reader has
000024 |    [J(phi),J(psi)] = sum_(m>0) m
000025 |        [conj(phi_m) psi_m - phi_m conj(psi_m)] I.
000026 |    W(phi)=exp(i J(phi)). The retained cover has level1 and the folded total
000027 |    current J_tot,n=J_cover,8n has level8. Both have nonzero central cocycle.
000028 |    Source: joint-native-algebra-selection-20261016/
000029 |    ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt, lines83-104.
000030 | 
000031 | 2. Original v519 theta_mono, lines769-784, reverses disjoint Majorana words
000032 |    with eta=i. Its tested OS rule conjugates scalar coefficients. It is NOT
000033 |    an established map of the completed CAR/current Weyl algebra. We do not
000034 |    infer that OS reconstruction must preserve both product types literally.
000035 |    Grassmann symbols, operator products and their reconstruction are distinct.
000036 | 
000037 | 3. The full geometric two-face graph is already known, not a new search
000038 |    target: seam-full-collar-reconstruction-20261015/PROOF.txt, lines63-123.
000039 |    Given B, collar length and identified boundary fibres determine it. The
000040 |    physical CAR cross covariance equals its cross block only after the actual
000041 |    physical field/state identification. Projector-state rigidity removes a
000042 |    separate Gaussian assumption; it does not supply that identification.
000043 | 
000044 | 4. P1's displayed primitive action is a relative spectral trace, not by
000045 |    itself an explicit Grassmann quadratic kinetic form. Source:
000046 |    documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000047 |    sections/001-section-primitive-core-and-carrier-interface-handoff.tex,
000048 |    lines302-316. Existing Berezin/Schur reduction and the special logarithmic
000049 |    determinant test are already in seam-collar-multifield-20261016/PROOF.txt,
000050 |    lines271-325. Do not replace the unspecified physical coupling by a fitted
000051 |    quadratic form and then call its inverse a P1 prediction.
000052 | 
000053 | Exact type theorem
000054 | 
000055 | Use the conventional real cocycle sigma defined by
000056 |    [J(f),J(g)] = i sigma(f,g) I,
000057 |    W(f)W(g) = exp(-i sigma(f,g)/2) W(f+g).
000058 | Changing the overall convention for sigma changes none of the conclusions.
000059 | 
000060 | Let Theta(W(f))=W(Lf), with real linear L. Let c=+1 for scalar linearity and
000061 | c=-1 for conjugate linearity. Let o=+1 for preservation of product order and
000062 | o=-1 for its reversal. Comparing the two sides of the Weyl product gives
000063 |    sigma(Lf,Lg) = c o sigma(f,g).                         (1)
000064 | This is an equality of forms, not a congruence modulo a phase: all real
000065 | scalings of f and g must work.
000066 | 
000067 | In particular, a conjugate-linear, order-reversing rule requires L to be
000068 | symplectic. For a chiral spatial reflection r f(x)=f(-x),
000069 |    sigma(rf,rg) = -sigma(f,g).
000070 | The extra field sign in L=-r cancels twice and changes nothing. Thus
000071 |    Theta(W(f))=W(-r f)
000072 | cannot be conjugate-linear AND order-reversing on this same ordinary
000073 | single-chiral Weyl algebra. With a commutator word normalized to -i, scalar
000074 | conjugation requires +i, while reversal of the four reflected factors gives
000075 | -i. This is a contradiction before state, dynamics or regulator choices.
000076 | 
000077 | The obstruction is local. On a proper interval choose a real nonconstant
000078 | smooth compactly supported f and g=f'. Then the chiral form is proportional
000079 | to integral f f'' = -integral (f')^2, which is nonzero. Reflecting their
000080 | supports reverses that form. It is unnecessary to approximate a localized
000081 | profile by a finite Fourier polynomial. A Fourier pair is an additional
000082 | exact global witness in the old current mode convention.
000083 | 
000084 | A constant sign f -> -f and additive scalar-current shifts, equivalently
000085 | Weyl characters, cannot repair the obstruction: their phases cancel in
000086 | the four-factor group commutator. A channel unitary preserving the positive
000087 | current inner product cannot convert a positive-level signature to its
000088 | negative. None of this excludes a different reconstructed product type.
000089 | 
000090 | Surviving algebraic routes
000091 | 
000092 | For the same chiral reflection r, a conjugate-linear MULTIPLICATIVE rule
000093 | passes (1). So does a LINEAR order-reversing rule. The original OS-to-operator
000094 | reconstruction must determine the correct type; the word rule alone does not.
000095 | 
000096 | For a target with two oppositely oriented current faces use
000097 |    sigma_pair(F,G)=sigma(f_L,g_L)-sigma(f_R,g_R),
000098 |    K=diag(1,-1), S=face exchange.
000099 | Then S^T K S=-K. Spatial reflection and face exchange together give two sign
000100 | reversals, hence L=-S r is symplectic and L^2=I. It passes the conjugate-linear
000101 | order-reversing algebraic test. This is a conditional target type suggested
000102 | by the existing two-face geometry. The actual P1 cross covariance, source
000103 | representation, opposite current orientations, positivity, locality and time
000104 | have NOT been derived by writing this signature down.
000105 | 
000106 | The result therefore directs the next original-source test: retain both
000107 | original faces and derive their current product and OS reconstruction map,
000108 | or prove that reconstruction changes the product type on the target. Do not
000109 | search for coefficients that repair the excluded literal single-face rule.
000110 | 
000111 | What the automatic search did
000112 | 
000113 | The checker enumerates a declared menu of64 variants: four channel/orientation
000114 | choices, scalar linear/conjugate-linear, preserved/reversed product order,
000115 | identity/reflected coordinate, and both overall field signs. This is complete
000116 | for that finite menu only. It is not a complete search over physical theories.
000117 | 
000118 | Every row is checked twice in different ways: an exact integer symplectic
000119 | matrix identity and direct multiplication of Weyl commutator words with
000120 | rationally scaled profiles. Rational scales prevent accidental phase
000121 | agreement modulo2pi at level8. Thirty-two rows preserve the algebraic type;
000122 | the candidate single-chiral anti/anti reflection fails for both field signs
000123 | and both existing level conventions. No fitness function or genetic fitting
000124 | is needed for these exact constraints.
000125 | 
000126 | [OMITTED: non-scientific conversation or execution metadata]
000127 | [OMITTED: non-scientific conversation or execution metadata]
000128 | [OMITTED: non-scientific conversation or execution metadata]
000129 | [OMITTED: non-scientific conversation or execution metadata]
000130 | [OMITTED: non-scientific conversation or execution metadata]
000131 | 
000132 | Verdict
000133 | 
000134 | Established: an exact local necessary reflection/type obstruction to the
000135 | specified single-chiral ordinary-Weyl identification, with two distinct
000136 | algebraic repair routes and a conditional opposite-face candidate.
000137 | 
000138 | Not established: physical P1 source identification, full OS intertwiner,
000139 | electromagnetic readout, physical spacetime dynamics or complete TFPT.
000140 | 
000141 | complete_TFPT_solution=false
000142 | ledger_changed=false