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Original ledger classification: Identity [E] (block-diagonality = image of the mu4 marks) + premise [O] (mark-local DtN residual)

Recorded statement: THE v199 BEDROCK RESIDUAL REDUCED ONE GENUINE STEP FURTHER (2026-06-15): block-diagonality of the sub-principal symbol is NOT an independent postulate. (1) [E] the seam DtN is Lambda=|k|+M_f, M_f = multiplication by a curvature function…

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QGEO.SUBPRIN.01

Original ledger classification: Identity [E] (block-diagonality = image of the mu4 marks) + premise [O] (mark-local DtN residual)

Original claim text

THE v199 BEDROCK RESIDUAL REDUCED ONE GENUINE STEP FURTHER (2026-06-15): block-diagonality of the sub-principal symbol is NOT an independent postulate. (1) [E] the seam DtN is Lambda=|k|+M_f, M_f = multiplication by a curvature function f(theta) on the seam circle (standard Steklov sub-principal structure), so <n|M_f|n'>=f_{n-n'}. (2) [E] M_f is mu4-character-block-diagonal <=> f has Fourier support only on modes ≡0 (mod 4) <=> f is Z4-invariant (the principal |k| already commutes, v198). (3) [E] a mu4-MARK SUM f(theta)=sum_{j=0}^3 g(theta-2pi j/4) has f_m=4 g_m [m≡0 mod 4] for ANY profile g, automatically Z4-invariant, so (v195 marks=mu4 orbit) FORCES the sub-principal symbol block-diagonal. (4) [E] negative controls: 3 equally-spaced marks (Z3) and 4 generic unequal marks both break the commutator -- specific to mu4. (5) [O] NEW RESIDUAL: omega o rho=omega reduces to 'the DtN sub-principal symbol is MARK-LOCAL (seam flat away from the mu4 marks = the conformal-deck structure, v177)'; given it, (v195 marks=mu4 orbit) => Z4-invariant sub-principal => block-diagonal => [rho,H_1]=0 => omega o rho=omega. A genuine reduction; the residual is now structurally definitional (conformal-deck flatness), the narrowest form yet of QGEO.SYM.01. Python-only (exact mark-sum Fourier identity + block-diagonal criterion; Wolfram-mirrored)

Original classification detail

Identity/premise [E]/[O]: the v199 sub-principal block-diagonality is FORCED by the v195 mu4-mark orbit under a mark-local DtN (a mark-sum over mu4 points is Z4-invariant => Fourier support on modes ≡0 mod 4 => block-diagonal); residual reduces to DtN mark-locality (conformal-deck flatness), the narrowest form of QGEO.SYM.01; not a closure

Dependencies as recorded

QGEO.STATE.01; QGEO.MARKS.02; QGEO.SYM.01; v199_seam_state_invariance.py; v195_marks_lefschetz_character.py; v198_modular_commutator_reduction.py

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Registered verification modules

  • v201_seam_subprincipal_marks.pyRegistered verification module · registry
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    QGEO.SUBPRIN.01 — the v199 bedrock residual reduced one genuine step further: block-diagonality of the sub-principal symbol is NOT an independent postulate. [E] the seam DtN is Λ=|k|+M_f, M_f = multiplication by a curvature function f(θ) (standard Steklov sub-principal), so ⟨n|M_f|n'⟩=f_{n-n'}. [E] M_f is μ4-character-block-diagonal ⟺ f has Fourier support only on modes ≡0 (mod 4) ⟺ f is Z4-invariant (|k| already commutes, v198). [E] a μ4-MARK SUM f(θ)=Σ_{j=0}^3 g(θ−2πj/4) has f_m=4 g_m[m≡0 mod 4] for ANY profile g — automatically Z4-invariant, so (v195 marks=μ4 orbit) FORCES the sub-principal symbol block-diagonal; negative controls: 3 marks (Z3) and 4 generic marks break it. [O] new residual: ω∘ρ=ω reduces to 'the DtN sub-principal symbol is MARK-LOCAL (seam flat away from the μ4 marks = conformal deck v177)', the narrowest, structurally-definitional form of QGEO.SYM.01.

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