Original claim records.

Original ledger status

SCALE.OVERDET.01

SCALE.OVERDET.01

THE SINGLE MASS ANCHOR IS OVER-DETERMINED; NO DERIVABLE ABSOLUTE SCALE IS A THEOREM (2026-06-18): sharpens the No-Unit theorem (v153). [E] one anchor: every dimensionful output is a pure number x M_bar^d (rho_Lambda/M_bar^4 = (3/4 pi^2) e^{-2 alpha^-1} = 7.13e-121 v60/v7, M_scal/M_bar = c3^{7/2} v7, the whole mass spectrum); the absolute neutrino scale (v272) and v_geo are the SAME anchor. [N] OVER-DETERMINATION: two independent routes to M_bar agree -- Route 1 (gravity) M_bar = (8 pi G)^{-1/2} = 2.4353e18 GeV; Route 2 (cosmology) invert the compiler prediction rho_Lambda/M_bar^4 with the measured rho_Lambda^{1/4} = 2.24 meV -> M_bar = 2.438e18 GeV, agreeing to 0.11%. [E] the anchor is the gravitational unit (G), the one measured dimensionful input of all physics (modern SI fixes hbar, c exactly). [E] THEOREM not gap: the seam is a conformal c=8 chiral CFT (v157/v158), scale-invariant by construction so provably no intrinsic scale; the scale is gravitational, entering only through the ambient measure (QG.AMB.01). [A] TFPT is scale-complete up to one over-determined anchor (the Planck mass) -- maximal predictivity for a dimensionless compiler, not a missing derivation.

Recorded status: Identity [E] + over-determination [N] + re-typing [A]

Original ledger status

QGAMB.ROADMAP.01

QGAMB.ROADMAP.01

OBLIGATION ROADMAP FOR QG.AMB.01 (the ambient nonperturbative QG measure), CONSOLIDATED INTO ONE MODULE (2026-06-18; QG.AMB.01 since reclassified as a DECOUPLED general QG problem, NOT a TFPT gate, v335): the (now decoupled, non-TFPT) general QG problem had no module of its own; this gives it one without claiming closure. [E] Tier A gap decoupling: Delta_eff = 6 log(3/2) - 2*248 c3^2 = 2.433 - 0.785 = 1.648 > 0 (v36/v76), so every testable low-energy readout (masses, mixings, alpha^-1, R+R^2) is independent of the un-built ambient measure -- QG.AMB.01 is gap-decoupled, not a blocker. [E]/[P] Tier B boundary-net reduction: the ambient bulk measure reduces to identifying the seam-Calderon boundary measure with the holomorphic (E8)_1 conformal net (c = 248/31 = 8 = rank E8; v77/GATE.METRIC.03), the gap giving clustering -> tightness; residual = the constructive seam-Calderon -> (E8)_1 map + bulk reconstruction (REDTEAM.A.01: reduced, not closed). [C] the perturbative layer is in hand (S_pert EG-constructible v269 + one-loop numbers v271/v273). [O] the nonperturbative interacting ambient measure (the bulk path integral / projective limit) is NOT constructed -- the one genuine structural frontier. [X] QG.AMB.01 closes IFF the seam net is identified with (E8)_1 AND the bulk reconstruction is proved; a proof that the seam net is NOT (E8)_1 or that the gap fails (Delta_eff<=0) kills the reduction.

Recorded status: Roadmap [E]/[C] + [O] open core + [X] promotion/kill

Later scope correction: the DtN form estimate alone does not prove the transfer gap.

Original ledger status

REDTEAM.F.01

REDTEAM.F.01

RED TEAM Target F -- adversarial audit of the perturbative 4D-QFT + scale round (v269-v275), published in release 5.2. TWO attacks land and force a narrowing (now folded into v269/v274); the rest survive as typed. (1) GRAVITY S_pert NON-UNITARY: power-counting renormalizability is not unitarity -- the spectral-action R^2/Weyl^2 sector gives a 4-derivative propagator 1/(p^2(p^2+M^2)) whose massive spin-2 mode has a NEGATIVE residue (the Stelle ghost), so S_pert is unitary for the matter+gauge sector only; the gravity perturbative S-matrix carries the ghost, which is exactly why the nonperturbative QG.AMB.01 (asymptotic safety v207 / boundary (E8)_1 net) is the real frontier, not a new gap. (2) OVER-DETERMINATION CONDITIONAL: v274 Route-2 inverts rho_Lambda/M_bar^4=(3/4pi^2)e^{-2 alpha^-1} = LAMBDA.BRANCH.01 [C], so the 0.11% gravity-vs-cosmology agreement is conditional on the Lambda-branch (the gravity route is unconditional). Already-honest: J_PMNS (v270) inherits delta's [C] provenance (assembled mu6/triality phase, not an M_nu output); the nu-scale (v272) is [O] with the NO floor Sigma m_nu=0.0586 eV as the one kill test. FIREWALL HOLDS: no positive-claim doc carries an unscoped overclaim (the red-team paper itself is excluded as it quotes forbidden phrases to forbid them). Verdict: SURVIVES (narrowed).

Recorded status: Red-team [P] (survives, narrowed)

Original ledger status

QGEO.SYM.03

QGEO.SYM.03

FLAT PILLOWCASE CLOSES THE MODULAR COMMUTATOR TO ALL ORDERS (2026-06-18): collapses the bedrock QGEO.SYM.01 to ONE geometric postulate (does NOT prove the premise). For a quasi-free seam state QGEO.SYM.01 = (omega o rho = omega) <=> [rho,C]=0 <=> [rho,H]=0 (C=(1+e^H)^-1, v258). [E] ISOMETRY=>COMMUTATOR: if H=f(Delta_flat) (the DtN of a flat metric, no curvature corrections) and the order-4 deck rho (z->iz) is a flat-tau=i isometry, then [rho,Delta]=0 => [rho,H]=[rho,f(Delta)]=0 EXACTLY to all orders -- verified numerically on the discretised square torus (rho^4=I, [rho,Delta]=0, [rho,sqrt(-Delta)]=0; a generic non-isometry permutation fails). [E] this UPGRADES the leading-order results v198 (principal symbol |k| commutes) and v201 (subprincipal block-diagonal) to the FULL operator H. [C] tau=i is forced by the order-4 requirement (cross-ratio 2 <=> j=1728, the unique order-4 CM by Z[i]; hexagonal tau=omega has j=0, order-6, excluded; v214/v267). [O] QGEO.SYM.01 therefore collapses to the single geometric statement 'the raw RP seam state is the flat tau=i pillowcase quasi-free state (H=sqrt(-Delta_flat))' -- everything downstream [E]; this one statement is the bedrock, needs a human constructive-QFT proof or stays an honest axiom. Sharpest non-circular form.

Recorded status: Reduction [E] (flat => commutator all orders) + [C] (tau=i forced) + [O] (one geometric postulate)

Original ledger status

QGAMB.TIERB.01

QGAMB.TIERB.01

SEAM-CALDERON -> (E8)_1 MATCHING CERTIFICATE (2026-06-18): the concrete Tier-B step of QG.AMB.01 (v275) -- computes the full (E8)_1 matching target and pins the residual to ONE bit (does NOT close it; the rt_A 'reduced, not closed' verdict stands). [E] c=8 (Sugawara, c((E8)_1)=248/31=8=g_car+N_fam), matching the seam (v77), but c=8 alone does not pin the net. [E] THE DISCRIMINATOR IS HOLOMORPHY: det Cartan(E8)=1 (one primary/anyon, holomorphic) vs the c=8 counterexample det Cartan(D8=SO(16))=4 (four primaries 1,v,s,c, non-holomorphic) -- same c, different det, so the single discriminator is |det K|=1. [E] THE (E8)_1 CHARACTER: the unique holomorphic c=8 character E4/eta^8=j^{1/3}=q^{-1/3}(1+248q+4124q^2+...) has a SINGLE primary and exactly 248=dim E8 weight-1 currents (the affine E8 current algebra; 240 roots from Theta_E8=E4). [E] EQUIVALENCE CHAIN (v234/v235): holomorphic <=> det Cartan 1 <=> E8 the unique even unimodular rank-8 lattice (v83) <=> seam link a homology 3-sphere (|Gamma^ab|=1) <=> |det K|=1 (Chern-Simons) -- all Tier-B faces are ONE condition. [C] SEAM-SIDE MATCH: the seam is pinned to the flat tau=i pillowcase (v276) and one Kummer/K3 carries seam+carrier-16+E8 (v260), so the boundary chiral data matches (E8)_1 on every computable invariant. [O] RESIDUAL = the single holomorphy bit (det K=1 / mu-index 1 / single sector); reduced to one bit, NOT closed.

Recorded status: Reduction [E] (full (E8)_1 match + discriminator) + [C] (seam-side) + [O] (one holomorphy bit)

Original ledger status

QFT4D.SPERT.04

QFT4D.SPERT.04

S_pert -> S_phys LSZ BRIDGE + ONE-LOOP UNITARITY (2026-06-18): connects the EG perturbative S-matrix (v269/v271/v273) to the physical asymptotic S-matrix (S_phys = LSZ on the OS-reconstructed Wightman functions, v240); the three-layer S-matrix is now S_top (v243) | S_pert (v269) --LSZ--> S_phys (v240). [E] LSZ BRIDGE: S_phys matrix elements = (amputate external legs)(on-shell residue sqrt(Z)) of the connected time-ordered correlators, and the EG time-ordered products of S_pert ARE those correlators (the OS Wick-rotation of the seam Wightman functions), so the perturbative amplitudes feed S_phys. [C] MASS GAP: Delta=6log(3/2)>0 (v64/v240) gives isolated one-particle states, so Haag-Ruelle scattering theory constructs the in/out asymptotic states and LSZ is valid. [E] ONE-LOOP UNITARITY (optical theorem): the s-channel one-loop bubble B(s)=int_0^1 dx ln(m^2-x(1-x)s-i eps) has discontinuity Im B(s)/pi = x_+ - x_- = sqrt(1-4m^2/s) for s>4m^2 EXACTLY -- the two-body phase-space factor -- so 2 Im M = sum_int dPi |M|^2 (cutting rule) holds at one loop: S_pert is unitary order by order for the matter+gauge sector. [E] THRESHOLD: Im B=0 for s<4m^2, turning on exactly at the physical two-particle threshold s=4m^2 (no spurious thresholds). [O] GRAVITY CAVEAT (rt_F): the R^2/Weyl^2 gravity sector has a negative-residue massive spin-2 mode (the Stelle ghost), so the gravity perturbative S-matrix is NOT unitary -- S_phys for gravity is the nonperturbative QG.AMB.01 frontier. [C] FIELD-STRENGTH Z: the EG scheme gives finite Z so the LSZ reduction is well-defined.

Recorded status: Bridge [E] (LSZ structure + one-loop optical theorem) + [C] (gap/Z) + [O] (gravity ghost)

Original ledger status

QGEO.OBLIG.01

QGEO.OBLIG.01

QGEO.SYM.01 AS ONE PRECISE CONSTRUCTIVE-QFT LEMMA + PROOF-TREE COMPLETENESS (2026-06-18): the formal write-up of the one human math obligation (does NOT close it). LEMMA (constructive-QFT form): let Sigma = P^1 minus mu_4 carry the free c=8 RP/OS quasi-free state omega_Sigma with DtN Lambda_Sigma; PREMISE (the one open input, QGEO.SYM.01): the raw seam carries the FLAT tau=i pillowcase metric g_flat (the Troyanov-unique constant-curvature representative, cone angles pi, of the conformal class fixed by the order-4 deck), so Lambda_Sigma = sqrt(-Delta_{g_flat}); CONCLUSION: omega_Sigma o rho = omega_Sigma (the carrier clock is a symmetry of the seam STATE) -> mark-locality -> metric-sector inclusion -> E8. [E] PROOF-TREE COMPLETENESS: the reduction DAG from 'free RP seam' (v155/v175 -> marks v195/v216 -> conformal class tau=i v214/v267 -> Troyanov flat -> commutator all orders v276/Lean -> state-invariance v198/v201/SeamDeckClosure -> mark-locality v210/v264 -> E8 v154/v277) has EXACTLY ONE un-discharged leaf: 'the raw seam state IS the flat g_flat state' (Lambda_Sigma = sqrt(-Delta_flat)). [E] the GEOMETRIC SIDE is fixed (flat DtN diagonal in Fourier modes, rho commutes with any spectral function to all orders, Lean FORM.QGEO.04), so the obligation is STATE-IDENTIFICATION not geometric ambiguity. [C] two proof routes: (i) RP-state uniqueness on the flat tau=i orbifold, (ii) Troyanov + OS reconstruction. [O] one constructive-QFT statement closes QGEO.SYM.01.

Recorded status: Obligation lemma [E] (proof-tree complete) + [C] (routes) + [O] (one residual)

Original ledger status

FORM.QGEO.04

FORM.QGEO.04

FLAT-METRIC ALL-ORDERS CLOSURE FORMALISED IN LEAN (2026-06-18): TfptCarrier/SeamDeckClosure.lean (theorems diagonal_commutesClock, flat_all_orders_clock) machine-proves the v276 all-orders step -- any SPECTRAL FUNCTION g of the Fourier-diagonal flat Laplacian (specFun g d) commutes with the carrier clock rho=diag(i^n) (CommutesClock), i.e. given the flat tau=i metric the DtN modular Hamiltonian H=g(Delta_flat) is clock-invariant to ALL orders, not merely the principal (v198) and sub-principal (v201) orders. Upgrades the conditional block-diagonality (markLocal_blockDiagonal / SeamDeckPremise.clock_invariant) to the full operator. Only the standard axioms {propext, Classical.choice, Quot.sound}, no sorry; full lake build green (3356 jobs), listed in AxiomCheck.lean. The premise (the seam IS flat) stays the [O] QGEO.SYM.01 input. QA NOTE (2026-08-08, round 36, v878_ledger_integrity.py): claim_id RENAMED FORM.QGEO.03 -> FORM.QGEO.04 -- the id FORM.QGEO.03 was double-booked on 2026-06-18 (it had been assigned to the CohomologyGrading formalisation on 2026-06-15): a primary-key violation found by the round-36 ledger audit; every flat-closure reference (papers, scripts, registry, website) updated to FORM.QGEO.04; the CohomologyGrading row keeps FORM.QGEO.03. The claim content is UNCHANGED.

Recorded status: Formal [F] (Lean theorem, all-orders closure)

Original ledger status

QGEO.STEKLOV.01

QGEO.STEKLOV.01

FLAT tau=i PILLOWCASE DtN EXPERIMENT (2026-06-18): a direct numerical experiment on the actual seam orbifold, realising the QGEO chain on the geometric side (does NOT prove the premise). [E] the pillowcase = the Z2-even sector of the flat torus T^2=C/(Z+iZ) modulo z->-z; the 4 cone points are the 2-torsion fixed points (deficit pi each, sum 4pi=2pi*2 Gauss-Bonnet). [E] the order-4 deck rho (z->iz) descends to the pillowcase ([rho,Z2]=[rho,P_even]=0) and permutes the 4 cone points as 2 fixed + 1 swapped pair (the order-4 mark structure). [E] the geometric chain is numerically EXACT: [rho,Delta]=0 => [rho,H]=0 (H=sqrt(-Delta)) => [rho,C]=0 (C=(1+e^H)^-1) => omega o rho = omega -- the v276/v279 all-orders closure realised on the real orbifold. [C] the Steklov/DtN is positive self-adjoint, fixed by the flat metric alone (numerical evidence for QGEO.OBLIG.01 route (i), RP-state uniqueness). [O] confirms the conclusion GIVEN the flat geometry; the premise (the raw seam state IS this flat pillowcase state) stays the one open QGEO.SYM.01 input.

Recorded status: Numerical experiment [E] (geometric chain) + [C] (RP-state) + [O] (premise)

Original ledger status

QGAMB.MODULAR.01

QGAMB.MODULAR.01

MODULAR-DATA / ANYON-CONDENSATION LAYER OF THE Tier-B HOLOMORPHY BIT (2026-06-18): complements v277 (det+character) with the anyon/modular-data view. [E] #anyons = |det Gram|: E8 -> 1 (holomorphic, single sector), D8=SO(16)_1 -> 4 (anyons 1,v,s,c), carrier D5(+)A3 -> det(D5)*det(A3)=4*4=16. [E] the anyon-condensation tower D5(+)A3 (det 16) -> D8 (det 4) -> E8 (det 1), each step condensing a Lagrangian Z2 boson (|det|/4=2^2 per step), the Kitaev E8 quantum-Hall endpoint (v92/v235). [E] Gauss-Milgram for the D8 anyons (topological spins 0,1/2,1,1): (1/sqrt|A|) sum_a e^{2pi i h_a} = e^{2pi i c/8} recovers c=8 mod 8 (the chiral central charge = the modular anomaly). [E] E8 is the unique holomorphic c=8 net (1 anyon <=> det 1 <=> even unimodular rank-8 = E8); SO(16)_1 has the SAME c=8 but 4 anyons, so holomorphy (det 1) is THE discriminator (matches v277). [O] the open Tier-B bit: does the seam-Calderon net condense to the det-1 holomorphic endpoint (|det K|=1)? Reduced to one bit, not closed.

Recorded status: Reduction [E] (modular data + condensation tower + Gauss-Milgram) + [O] (one bit)

Original ledger status

QGAMB.UNIFY.01

QGAMB.UNIFY.01

THE E8-AT-tau=i UNIFICATION: TWO OBLIGATIONS -> ONE OBJECT (2026-06-18): the candidate simplification. The (E8)_1 character as a modular function of the seam torus modulus tau is chi_E8(tau)=Theta_E8/eta^8=E4/eta^8=j(tau)^{1/3} (since j=E4^3/eta^24=(E4/eta^8)^3=chi_E8^3). [E] tau=i is the order-4 CM point (cross-ratio 2 <=> j=1728 <=> tau=i, unique order-4 CM by Z[i], v214/v267) -- the flat tau=i pillowcase geometry of QGEO.SYM.01. [E] chi_E8 = j^{1/3} (verified chi^3 = q*j as a q-series), so chi_E8(i)=1728^{1/3}=12 -- the (E8)_1 partition function at the SAME modulus tau=i, the holomorphy of QG.AMB.01. [E] ONE OBJECT, TWO FACES: tau=i is simultaneously the order-4 conformal CM point (QGEO.SYM.01, flat geometry) and the modulus at which the holomorphic (E8)_1 character is the seam partition function (QG.AMB.01, holomorphy). [C] OBLIGATION COUNT 2->1: prove the single statement 'the raw seam state is the (E8)_1 holomorphic CFT at tau=i' and BOTH QGEO.SYM.01 (flat tau=i, order-4 CM) AND QG.AMB.01 (E8 holomorphic, det K=1) follow -- the flat geometry IS the CM modulus, the holomorphy IS the E8 algebra. [O] the single unified premise stays open (it IS the seam-identification) but is now ONE object, not two separate premises -- a genuine reduction of the open count, not a closure.

Recorded status: Unification [E] (chi_E8=j^{1/3}, tau=i shared) + [C] (count 2->1) + [O] (one premise)

Original ledger status

PRED.LIVE.01

PRED.LIVE.01

LIVE KILL-TEST SCORECARD OF THE RECENT ROUND'S COMPUTABLE PREDICTIONS (2026-06-18): a data-confrontation consolidation (confronts, does not add physics), each row tagged with the decisive near-term experiment. [N] PMNS angles: sin^2 th12=1/3-phi0/2=0.30675 vs NuFIT 0.303(12) [+0.31 sigma]; sin^2 th13=phi0 e^{-5/6}=0.02311 vs 0.02225(59) [+1.45 sigma]; sin^2 th23=0.5 maximal (octant-ambiguous) -- decisive JUNO/Daya-Bay/DUNE. [C] leptonic CP: J_PMNS=-0.0297 (J_max 0.0342 vs NuFIT 0.0332 ~3%), delta=240 vs NuFIT ~232(30) -- decisive DUNE/Hyper-K. [X] Sigma m_nu NO floor=0.0586 eV vs Planck+BAO<0.12, DESI~<0.072 (consistent at the edge; Sigma<0.0586 excludes the NO m1~0 spectrum) -- decisive DESI/CMB-S4. [X] proton decay tau_p(p->e+pi0)=1.55e35 yr for the E8-allowed +(15,1,1) content vs Super-K>2.4e34 (safe); minimal 16-Higgs 4.4e33 EXCLUDED -- decisive Hyper-Kamiokande (reach ~1.4e35). [N] closed readouts alpha^-1=137.0359992 (1.9 sigma known offset), Omega_b=(1-1/4pi)phi0=0.04894, birefringence beta=phi0/(4pi)=0.2424 deg -- decisive CMB birefringence. [N] all confronted rows consistent < 2 sigma; 2 sharp near-term kill tests (Sigma m_nu floor, proton decay).

Recorded status: Data confrontation [N]/[C] + [X] (two near-term kill tests)

Original ledger status

QGEO.ROUTEI.01

QGEO.ROUTEI.01

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).

Recorded status: Lemma decomposition [E] (5/6 discharged) + [C] (Troyanov) + [O] (one open lemma)

Original ledger status

QGAMB.ROUTEII.01

QGAMB.ROUTEII.01

ROUTE (ii) (SEAM-NET CONDENSATION) LEMMA DECOMPOSITION + ROUTE-COINCIDENCE (2026-06-18): a 4-lemma chain with 3 discharged and the open lemma shown to COINCIDE with Route (i)'s (the v282 unification at the lemma level; does NOT prove the open premise). M1 [E] no nontrivial abelian sector <=> |det K|=1 <=> holomorphic (mu-index 1) (v234 GATE.HOLO.01: #(1-dim irreps)=|Gamma^ab|=|H1|=1 <=> E8; v235 GATE.HOLO.02 Chern-Simons det K=1). M2 [E] holomorphic c=8 <=> (E8)_1 the unique even unimodular rank-8 lattice; the c=8 counterexample SO(16)_1 has |det|=4 (v277 QGAMB.TIERB.01 + v281 QGAMB.MODULAR.01 #anyons=|det Gram|). M3 [E] the condensation tower D5(+)A3 (det 16) -> D8 (4) -> E8 (1) is the unique mu4-Lagrangian glue (v92 GATE.METRIC.07 + v281 + v235). THE DISCRIMINATOR [E]: det 1 (E8, holomorphic) vs det 4 (SO(16)) at the same c=8. ROUTE COINCIDENCE [E]: Route (i)'s L_open ('raw seam = constant-curvature flat tau=i state', v284) and Route (ii)'s M_open ('the free RP seam condenses the mu4-Lagrangian glue' = QGEO.SYM.01) are the SAME lemma -- both 'the raw seam IS the (E8)_1-at-tau=i object' (v282); proving EITHER route closes BOTH. THE UNIFIED OPEN LEMMA [O]: construct a canonical equivalence between the raw seam KMS/DtN state and the holomorphic (E8)_1 net at tau=i -- the state-identification step.

Recorded status: Lemma decomposition [E] (3/4 discharged + route coincidence) + [O] (unified lemma)

Original ledger status

SEAM.EQUIV.01

SEAM.EQUIV.01

THE SEAM EQUIVALENCE CONTRACT (2026-06-18): concentrates the whole open structure on ONE named arrow (does NOT prove it). THE NAMED THEOREM: the raw reflection-positive seam state (gap, chirality, carrier clock) reconstructs canonically the SAME state as the holomorphic (E8)_1 boundary net at the tau=i pillowcase, respecting the DtN, KMS modular flow, mu4 deck, Q-system extension, det K=1 and the (E8)_1 character (short: the raw seam is (E8)_1 at tau=i). [E] four objects (RawRPSeam, FlatTauIPillowcaseDtN, HolomorphicE8Net, SREKitaevE8Phase). [E] six arrows: 2 closed (Route i: FlatDtN->omega o rho=omega, v276/v280/v284; Route ii: SRE->HolomorphicE8Net, v235/v277/v281), 1 conditional (FlatDtN<->HolomorphicE8Net at tau=i, v282), 2 open premises (RawRPSeam->FlatDtN and RawRPSeam->SRE, the SAME lemma per v284/v285), 1 Gral (RawRPSeam->HolomorphicE8Net = SEAM.EQUIV.01). [E] IMPORT FIREWALL (anti-circularity): no Route-i module (v276/v280/v284) imports any Route-ii module (v277/v281/v285) or vice versa, verified by scanning the sources -- the routes converge ONLY through this contract, killing the 'E8 smuggled into the geometry and pulled back out' attack. [E] exact-label taxonomy {Formal Exact (Lean), Numerical Exact (reproducible experiment), Conditional Exact (standard theorem under a stated premise), Open Selection (theory-selection premise)}. [O] the one open arrow: free Gaussian RP bulk => holomorphic single-sector boundary net. Writes seam_equivalence.json. NOT closed by this: v_geo (metrology primitive), F_transfer (downstream), QG.AMB.01 (nonperturbative ambient measure). ROUTE SPLIT (2026-07-22; ID restructuring after the external review, NO status change in substance): the target statement now carries two named routes -- SEAM.EQUIV.MMST.01 (the conditional scaling-limit route: closed modulo cited existence theorems) and SEAM.EQUIV.TWISTOR.01 (the open Costello-Li construction). PARENT LOGIC (the claim text of this row): closed if MMST.01 OR TWISTOR.01 closes; currently: MMST conditional-closed modulo cited theorems, TWISTOR open => parent stays [O] as an unconditional claim. UPDATED 2026-08-27 (v973): the route narrowing -- the MMST route's cited-theorem coverage is carved into COVERED (bilinear half) vs NOT-COVERED (the named N1-N4 list, headed by the mu4/GSO net extension), the twistor route's declared measure is unique on the computable sector up to exactly ONE named flat direction, and the two routes provably share the executable OS skeleton 5/5 with all divergences control-typed (NEW row SEAM.EQUIV.SKELETON.01 [E]) -- the open structure of the parent is now concentrated on the SAME named extension question from both sides (N1/N4 on the MMST side, the global quantisation + OS bridge on the twistor side), with the executable levels verified to agree. Parent logic unchanged: closed if MMST.01 OR TWISTOR.01 closes; both stay [O]; parent stays [O].

Recorded status: Contract [E] (typing + firewall + taxonomy) + [O] (the named theorem)

Original ledger status

SEAM.EQUIV.A01

SEAM.EQUIV.A01

ROUTE A (AQFT) DEPENDENCY CHECKER (2026-06-18): for the open Gral arrow 'free Gaussian RP bulk => holomorphic single-sector boundary net', types the 5-lemma AQFT chain and isolates the ONE missing standard theorem (does NOT prove the arrow). L1 [E] RawRPSeam --OS--> CAR/quasi-free boundary system (v155 quasi-free boundary, v160 Gaussianity from the Pfaffian, v240 GNS/OS, v175 net existence + full-cone RP). L2 [E] gap + chirality => gapped chiral bulk (gap Delta=6log(3/2)>0 v64; c=8 chiral net v156/v157). L3 [O] invertible/SRE gapped Gaussian bulk => single-sector boundary (mu-index 1) -- THE ONE MISSING STANDARD THEOREM: an invertible (SRE) gapped bulk has no bulk topological order, hence a single-sector boundary net; the AQFT form of det K=1, to be imported/proved externally. L4 [E] mu-index 1 + c=8 => holomorphic (v234 holomorphy<=>1 sector). L5 [E] holomorphic c=8 => (E8)_1 (v83/v277/v281, unique even unimodular rank-8; SO(16) the det-4 non-holomorphic counterexample). VERDICT: 4/5 lemmas discharged, so Route A reduces SEAM.EQUIV.01 to ONE cited standard theorem (invertibility => single-sector), not a diffuse programme.

Recorded status: Dependency checker [E] (4/5 discharged) + [O] (the one external theorem)

Original ledger status

SEAM.EQUIV.B01

SEAM.EQUIV.B01

ROUTE B (DtN) FULL-L^2 SUB-PRINCIPAL Z4 BLOCK-DIAGONALITY (2026-06-18): proves (numerically, on the full Toeplitz operator) the advisor's 'probably provable' step, lifting v201/v284 from the finite H^1 block to the full L^2 operator. Setup: the seam DtN Lambda = |D_theta| + M_f with <n|M_f|n'> = f_{n-n'} (boundary curvature f), rho = diag(i^n). [E] CHARACTER ORTHOGONALITY: a mark-sourced curvature f(theta)=sum_{j<4} g(theta - j pi/2) has Fourier coefficient f_m = g_m sum_{j<4}((-i)^m)^j = 4 g_m [4|m], so support ONLY on modes m==0 (mod 4) (verified nonzero modes {0,+-4,+-8,+-12}). [E] FULL-L^2 BLOCK-DIAGONALITY: on the full mode range -50..50 the Toeplitz M_f connects only equal clock-characters, so Lambda commutes with rho, ||[rho,Lambda]||~2e-15 -- the all-orders statement on L^2, lifting the finite block (v201/v284). [E] NEGATIVE CONTROL: a generic (non-mark-local) curvature gives ||[rho,Lambda]||=4.44>0. [E] LEAN CONSISTENCY: the operator form of SeamDeckClosure.geom_sum_fourth_root + markLocal_blockDiagonal. [O] THE SHARPER RESIDUAL: why is the raw seam sub-principal term mark-local (a sum of mu4-orbit sources)? -- the curvature-locality half of SEAM.EQUIV.01.

Recorded status: Reduction [E] (full-L^2 lift, lifts v201/v284) + [O] (mark-locality residual)

Original ledger status

MARKLOCAL.RAW.01

MARKLOCAL.RAW.01

RAW MARK-LOCALITY DECOMPOSITION (2026-06-18): decompose the ONE remaining Route-B residual after v288 ('why is the raw seam sub-principal term mark-local?') into an explicit 5-lemma chain, discharge the four standard/established lemmas and isolate the single open analytic lemma (Flat-Away). The best closing lever for Route B; does NOT close mark-locality. CHAIN (raw RP seam => mark-local sub-principal term): L1 [C] CONIC PARAMETRIX -- the Steklov/DtN of a conic surface splits as Lambda = |D_theta| (principal) + M_f (sub-principal = multiplication by boundary curvature f) + delta-sources at the cone points (b-calculus/conic parametrix, Melrose; imported). L2 [O] FLAT-AWAY -- THE ONE OPEN ANALYTIC LEMMA: reflection positivity + the mass gap + the four branch marks force the seam curvature to VANISH away from the four marks (the flat-pillowcase realisation, the curvature-locality heart of SEAM.EQUIV.01). L3 [E] ORBIT-SOURCE -- the four marks form a single mu4 orbit {0,pi/2,pi,3pi/2} (v195 Lefschetz/character marks + order-4 clock; v216 Gauss-Bonnet 4 cone points deficit pi). L4 [E] FOURIER-SUPPORT -- a curvature sourced on the mu4 orbit has discrete Fourier support ONLY on modes m==0 mod 4 (v264/v288; verified by FFT, with a non-orbit negative control showing the ORBIT structure, not just '4 marks', gives mark-locality). L5 [E] FULL-OPERATOR -- hence M_f is Z4 block-diagonal and [rho,Lambda]=0 on the full L^2 (v288), so omega o rho = omega (Tomita-Takesaki, v198). VERDICT: 4/5 discharged (L1 conditional-by-literature, L3/L4/L5 exact); Route B reduces to exactly L2 (Flat-Away) -- if L2 falls, Route B closes and SEAM.EQUIV.01 needs only the holomorphy half (Route A).

Recorded status: Decomposition [E] (chain + L3/L4/L5) + [C] (L1 literature) + [O] (L2 Flat-Away)

Original ledger status

SEAM.ADVERSARY.01

SEAM.ADVERSARY.01

ROUTE-B RED-TEAM: Z4 BLOCK-DIAGONALITY != MARK-LOCALITY (2026-06-18). Constructs the strongest adversary against Route B -- a SMOOTH Z4-symmetric off-mark curvature f(theta)=eps*cos(4 theta) -- and shows HONESTLY (no manufactured pass): [E] the adversary is Z4-symmetric (Fourier support only on {+-4} subset 4Z) but NOT mark-local (f(pi/4)=-eps != 0), a smooth off-mark curvature rather than a sum of mu4-orbit sources; [E] it PASSES the v288 commutator test ||[rho,Lambda_adv]||=0 EXACTLY (the same pass as a genuine mark-local curvature, since i^n - i^(n+-4)=0) -- so [rho,Lambda]=0 follows from Z4-invariance, which is WEAKER than mark-locality, and the commutator alone CANNOT exclude this modulus; [E] BUT it shifts the DtN spectrum off the flat integer ladder |n| (max |dlambda|=0.155 at eps=0.3) and the heat trace Tr(e^{-t Lambda}) (delta=0.019 at t=0.5) -- the spectral zeta/heat coefficients that fix the KMS weight beta and the (E8)_1 character. [O] VERDICT: the smooth Z4 off-mark curvature is a GENUINE candidate modulus the commutator misses but the spectral data excludes, so a proof of Flat-Away (v289 L2) must invoke spectral/RP-energy/heat-kernel constraints, not just Z4-invariance. v288 proves the commutator lift; it does NOT pin off-mark flatness -- that is exactly the open Flat-Away lemma, sharpened to 'no smooth Z4 off-mark modulus survives the spectral data'.

Recorded status: Red-team [E] (adversary passes commutator, shifts spectrum/heat) + [O] (consequence)

Original ledger status

FLATAWAY.RP.01

FLATAWAY.RP.01

THE FLAT-AWAY LEMMA AS A NAMED MINI-THEOREM (2026-06-18). States the one open link of v289 precisely and types its three proof routes (opened by the v290 red-team). LEMMA: given RawRPSeam with the mass gap, chirality and the four mu4 branch marks, the SMOOTH curvature density in the DtN sub-principal symbol VANISHES away from the branch divisor (the seam is flat away from the four marks). It is the bottleneck because v288 lifts the Z4 block-diagonality to L^2 GIVEN a mark-sourced (flat-away) sub-principal term, and v290 shows a smooth Z4 off-mark curvature passes the commutator yet shifts the spectrum -- so Flat-Away is exactly the gap and must be closed with spectral/energy/heat input. THREE ROUTES: [E] SPECTRAL-RIGIDITY -- scanning f=eps*cos(4 theta), the DtN-spectrum deviation from the flat integer ladder is 0 ONLY at eps=0 and grows with |eps|, so the forced KMS spectrum has the flat seam as its unique match (lever, conditional on pinning the spectrum); [E] HEAT-KERNEL -- the heat trace Tr(e^{-t Lambda}) deviation is 0 ONLY at eps=0, so matching the flat/(E8)_1 heat data forces the smooth curvature to vanish; [C] RP-ENERGY -- RP+gap is a Dirichlet-energy minimisation in the Troyanov conformal class with prescribed cone angles, whose unique minimiser is the constant-curvature (flat-away) metric (Troyanov; cf. v284). VERDICT: three INDEPENDENT routes to ONE lemma; closing any one closes Route B (and with Route A's holomorphy half, SEAM.EQUIV.01). All three remain open handles, not proofs.

Recorded status: Contract [O] (lemma + verdict) + [E] (spectral/heat uniqueness) + [C] (Troyanov route)