A registered result in its source context
Original ledger classification: Lemma decomposition [E] (5/6 discharged) + [C] (Troyanov) + [O] (one open lemma)
Recorded statement: ROUTE (i) (RP-STATE UNIQUENESS) LEMMA DECOMPOSITION (2026-06-18): turns Route (i) from one hard target into a 6-lemma chain with 5 lemmas discharged [E]/[F] and exactly one open lemma isolated (does NOT prove the open premise). L1 [E] RP…
Statement, assumptions and sources
Paper context: tfpt research contracts
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QGEO.ROUTEI.01
Original ledger classification: Lemma decomposition [E] (5/6 discharged) + [C] (Troyanov) + [O] (one open lemma)
Original claim text
ROUTE (i) (RP-STATE UNIQUENESS) LEMMA DECOMPOSITION (2026-06-18): turns Route (i) from one hard target into a 6-lemma chain with 5 lemmas discharged [E]/[F] and exactly one open lemma isolated (does NOT prove the open premise). L1 [E] RP data define the raw DtN intrinsically (v194 QGEO.DTN.01, OS on the quasi-free state). L2 [E] the four branch marks force the pillowcase conformal class (v195 QGEO.MARKS.02 Lefschetz/character + v216 QGEO.MARKS.03 n=2chi=4 + v214 QGEO.PILLOW.01). L3 [E] the flat orbifold Steklov/DtN operator = sqrt(-Delta_flat): the DtN principal symbol is |xi|=sqrt(-Delta_bdry) (Lee-Uhlmann) with no curvature correction for a flat metric (flat-strip DtN(k)/k->1; v280). L4 [E] the quasi-free covariance is determined by the DtN, C=(1+e^H)^-1 with H=log-DtN (v258 PS.DIRAC.03). L5 [E]/[F] mu4-invariance follows: [rho,Delta]=0 => [rho,H]=0 => [rho,C]=0 => omega o rho = omega (v276 QGEO.SYM.03 + Lean FORM.QGEO.04 + v280). TROYANOV REDUCTION [C]: 4 cone points of angle pi => chi_orb = 2-4(1-1/2) = 0 => the unique constant-curvature representative is flat = the tau=i square (j=1728), so the residual reduces to 'the raw seam is the constant-curvature geometric state'. THE ONE OPEN LEMMA [O]: the raw RP seam state is the constant-curvature/geometric state (Lambda = sqrt(-Delta_g) for constant-curvature g) -- the state-identification theory-selection step; proving it closes Route (i).
Original classification detail
[E] 5/6 Route-(i) lemmas discharged + the L3 flat-DtN verification; [C] the Troyanov reduction (constant curvature -> flat tau=i); [O] the one constant-curvature/geometric-state lemma. Decomposes Route (i) and isolates its single open lemma; does NOT close it.
Dependencies as recorded
QGEO.SYM.01; QGEO.OBLIG.01; QGEO.DTN.01; QGEO.MARKS.03; PS.DIRAC.03; QGEO.SYM.03; QGEO.STEKLOV.01
Open the original ledger source, row 379Location recorded in the ledger
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- Research contracts
Sections that explicitly cite this claim
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- 2026-06-18 (the two proof routes decomposed: Route (i) and Route (ii) into lemma chains, v284/ v285)Research changelog; TFPT — Introduction and reading guide
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- 2026-06-22 (Route B carried out: rigidity [formula in source] the icosian-ring identity reduce the whole keystone to ONE question – does the raw seam carry the golden ratio? – the simplest form of L2: v348)Research changelog; TFPT — Introduction and reading guide
- Research Contract 2 — [formula in source]tfpt research contracts
- 2026-06-22 (can the one open arrow be fully solved? the closure-mode classification: the flat [formula in source] spherical base locus, four modes, no free theorem – only rigidity or axiom [formula in source] : v347)Research changelog; TFPT — Introduction and reading guide
- 2026-06-18 (closing sprint: the Seam Equivalence Theorem named + both routes attacked, v286– v288)Research changelog; TFPT — Introduction and reading guide
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- Research Contract 2 — [formula in source]Research contracts
Registered verification modules
- v284_route_i_rp_uniqueness.pyRegistered verification module · registry
Read the description recorded in the module registry
QGEO.ROUTEI.01 — Route (i) (RP-state uniqueness) decomposed into a 6-lemma chain with every dischargeable lemma discharged and the one irreducible lemma isolated (does NOT prove the open premise). [E] 5/6 lemmas are existing [E]/[F] results — L1 RP→DtN (v194), L2 the four marks force the pillowcase class (v195/v216/v214), L3 the flat orbifold Steklov operator = √(−Δ_flat) (verified: flat-strip DtN(k)/k→1, principal symbol |ξ| with no curvature term), L4 covariance C=(1+e^H)⁻¹ from the DtN (v258), L5 μ4-invariance (v276/Lean/v280); [C] Troyanov reduction: 4 cone points of angle π ⇒ χ_orb=0 ⇒ the unique constant-curvature representative is flat (τ=i), so L_open reduces to 'the raw seam is the constant-curvature geometric state'; [O] the ONE open lemma — the raw RP seam state is the constant-curvature/geometric state (the state-identification theory-selection step).
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