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Original ledger classification: Exact [E] spectrum/diagonalization/signature identities + [C] the bridge reading (exact sympy)
Recorded statement: The polarization frame round (one module, 9 checks, ~2 s; discovery probe polarization_frame_probe.py 9/9, verdict POLARIZATION-FRAMED). THE ROTATION SPECTRUM [E]: the braid rotation r has three SIMPLE eigenvalues = zeta_12 x {1, -1, -i}…
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Paper context: TFPT — Introduction and reading guide
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QGEO.POLFRAME.01
Original ledger classification: Exact [E] spectrum/diagonalization/signature identities + [C] the bridge reading (exact sympy)
Original claim text
The polarization frame round (one module, 9 checks, ~2 s; discovery probe polarization_frame_probe.py 9/9, verdict POLARIZATION-FRAMED). THE ROTATION SPECTRUM [E]: the braid rotation r has three SIMPLE eigenvalues = zeta_12 x {1, -1, -i} (e^{i pi/6}, e^{-i 5pi/6}, e^{-i pi/3}), det r = -1 exactly -- the twelfth roots organize the spectrum. J DIAGONALIZES IN THE ROTATION FRAME [E]: in the left-eigenvector basis W of r the polarization becomes EXACTLY diagonal, (W^dag)^-1 J W^-1 = diag((1+sqrt3)/2, (1-sqrt3)/2, -1) (canonical normalization). Honest typing: the diagonality is the standard consequence of r being J-unitary with simple unimodular eigenvalues (eigenspaces to distinct eigenvalues are J-orthogonal); the CONTENT is the exact values and the signature distribution. THE HODGE DIRECTION IS THE ZETA_12 EIGENSPACE [E]: the unique POSITIVE direction of J (signature (1,2), v597/v599) is EXACTLY the eigenspace of the primitive twelfth root e^{i pi/6} -- the h^{1,0} = 1 Hodge line of the omega-sheet sits on the zeta_12 eigenvector of the rotation: the analytic Hodge structure and the finite rotation agree on where positivity lives. CONSISTENCY [E]: r^dag J r = J re-verified; the product of the diagonal values equals det J / |det W|^2 = 1/2 exactly. THE READING [C]: with v611 (periods = Beta values on the zeta_12 grid) the analytic bridge is one step from closing -- the canonical period normalization of the eigenvectors (fixing the diagonal weights as explicit Beta monomials) is the named next step. GATE.QGEO does not move; no marker changes. [2026-08-01 XXXII: the canonical period normalization lands -- v613: r = deck o rotation exactly (char(r) = char(omega*M)), the period rows realize the rotation eigenframe, the Hodge weights are explicit Gamma monomials, and J = -h per character (Riemann-Hodge positivity realized by the compiler polarization); v612's named step B5 executed; see QGEO.PERNORM.01.]
Original classification detail
POLARIZATION-FRAMED: r eigenvalues zeta_12 x {1,-1,-i} simple; J exactly diagonal in the rotation frame (values (1+sqrt3)/2, (1-sqrt3)/2, -1); the positive Hodge direction = the primitive zeta_12 eigenspace; product = det J/|det W|^2 = 1/2; period normalization named; 9/9 checks
Dependencies as recorded
QGEO.PERIODS.01; QGEO.LANDSCAPE.01; QGEO.REAL.01; v597; v599; v610; v611
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- Architecture, E8 and the electromagnetic fixed point
- Standard Model and flavor structure
- E8 audit and bootstrap
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- =150 2026-08-01 (XXX) (the canonical period normalization lands ( v613 promoted) — THE ANALYTIC BRIDGE CLOSES: the quarter rotation acts DECK-FREE on the four branch segments (all [formula in source] wrap factors exactly [formula in source] ), and the exact dictionary is [formula in source] — [formula in source] EXACTLY, explaining [formula in source] structurally, with the deck power unique; the period rows (hol [formula in source] , hol [formula in source] , antihol [formula in source] ) realize the rotation eigenframe; the Hodge weights are explicit Gamma monomials ( [formula in source] , complex Beta formula); and [formula in source] PER CHARACTER: the compiler polarization is MINUS the analytic Hodge form — Riemann–Hodge positivity realized, v612's named step B5 executed)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportE8 audit and bootstrap
- =150 2026-08-01 (XXIX) (the polarization frame lands ( v612 promoted): the braid rotation has three simple eigenvalues [formula in source] , the polarization [formula in source] diagonalizes EXACTLY in the rotation eigenbasis (values [formula in source] , [formula in source] , [formula in source] ), and THE UNIQUE POSITIVE (HODGE) DIRECTION IS THE PRIMITIVE [formula in source] EIGENSPACE — the analytic Hodge structure and the finite rotation agree on where positivity lives; the canonical period normalization (Beta weights) is the named next step)Research changelog; TFPT — Introduction and reading guide
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
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- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Registered verification modules
- v612_polarization_frame.pyRegistered verification module · core
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QGEO.POLFRAME.01: die Polarisierungs-Rahmen-Runde — J DIAGONALISIERT EXAKT IN DER ROTATIONS-EIGENBASIS: die Braid-Rotation r hat drei EINFACHE Eigenwerte ζ₁₂·{1,−1,−i} (det r = −1); in der Links-Eigenbasis wird J exakt diagonal, diag((1+√3)/2, (1−√3)/2, −1) (ehrlich: Diagonalität = Standard-Folge der J-Unitarität; der GEHALT sind Werte und Signatur-Verteilung); **DIE HODGE-RICHTUNG IST DER ζ₁₂-EIGENRAUM**: die eindeutige positive Richtung von J (Signatur (1,2)) ist exakt der Eigenraum der PRIMITIVEN 12. Einheitswurzel e^{iπ/6} — analytische Hodge-Struktur und endliche Rotation stimmen überein, wo Positivität wohnt; Produkt der Diagonalwerte = det J/|det W|² = 1/2 exakt; benannter nächster Schritt: die kanonische Perioden-Normierung (Beta-Gewichte) (9 checks, ~2 s)
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