Original claim records.

Original ledger status

FLATAWAY.HEAT.01

FLATAWAY.HEAT.01

HEAT-KERNEL ROUTE FOR FLAT-AWAY (2026-06-18): reduces the open Flat-Away lemma to ONE named spectral invariant by showing the heat-trace deviation is a positive-definite quadratic form in the off-mark curvature. Setup Lambda=|D_theta|+M_f, f=sum_k g_{4k} cos(4k theta). [E] QUADRATIC LEADING ORDER: Delta Tr(e^{-t Lambda}) = c(t) eps^2 + O(eps^4) (Delta Tr/eps^2 constant ~0.208 at t=0.5), no first-order term (average curvature fixed by Gauss-Bonnet/marks). [E] POSITIVE-DEFINITE PER MODE: c_k(t)>0 for every Z4 mode 4k (k=1,2,3: ~0.066,0.047,0.033). [E] POSITIVE-DEFINITE ON COMBINATIONS: random Z4 combinations give Delta Tr>0, no cancellation, so Delta Tr=0 <=> f=0. [E] LEADING INVARIANT = ||f||_L^2 (Parseval). [O] REDUCTION: Flat-Away <= 'the seam's spectral/heat data fixes the a_2 heat coefficient (=Tr e^{-t Lambda}) to its flat value', then int f^2=0 => f=0 off the marks -- ONE external invariant, far smaller than 'prove Flat-Away'.

Recorded status: Reduction [E] (positive-definite heat form) + [O] (one external invariant a_2)

Original ledger status

FLATAWAY.SPEC.01

FLATAWAY.SPEC.01

SPECTRAL-RIGIDITY ROUTE FOR FLAT-AWAY (2026-06-18): upgrades the v291 one-direction scan to the FULL smooth-Z4 deformation space, showing the flat seam is a STRICT isolated minimum of the spectral-mismatch functional S(f)=sum_n (lambda_n(f)-lambda_n(0))^2. [E] SPLITTING MECHANISM: the flat Steklov spectrum |n| is doubly degenerate (+-n); a Z4 mode 4k connects n<->-n exactly at |n|=2k and splits that level at first order by +-eps/2 (modes 4,8,12 split levels 2,4,6 by eps), distinct modes splitting distinct levels. [E] QUADRATIC FORM: S(eps cos4theta) = h eps^2 + O(eps^4), h~0.50. [E] HESSIAN POSITIVE-DEFINITE: H_{kl}=d^2 S/dg_k dg_l over modes {4,8,12} is near-diagonal with eigenvalues ~{0.497,0.500,0.503}, all >0 -- the flat seam is a STRICT isolated minimum over the WHOLE smooth-Z4 space (answers the 'only one direction' objection). [E] CLOCK-FILTER CONTRAST: a non-Z4 mode (cos2theta) breaks the clock commutator outright (||[rho,M]||>0). [O] CONSEQUENCE: matching the seam's forced Steklov spectrum excludes EVERY smooth Z4 deformation => Flat-Away, the same external fact as v292 reached independently.

Recorded status: Strengthening [E] (Hessian PD over full Z4 space) + [O] (pin the spectrum)

Original ledger status

FLATAWAY.ENERGY.01

FLATAWAY.ENERGY.01

RP-ENERGY / TROYANOV ROUTE FOR FLAT-AWAY (2026-06-18): the third independent reduction -- the flat pillowcase is the UNIQUE constant-curvature (energy-minimising) metric in the conformal class with the prescribed mark divisor. [E] PRESCRIBED DIVISOR + GAUSS-BONNET: the pillowcase = S^2 with 4 cone points of angle pi; the cone Gauss-Bonnet sum (2pi-theta_i)=4pi=2pi chi(S^2) holds for the flat (K=0 off marks) metric, the mark divisor rigid (v195/v216). [C] TROYANOV / EUCLIDEAN-CONE UNIQUENESS: for a prescribed cone divisor obeying Gauss-Bonnet the conformal class carries a UNIQUE constant-curvature metric (Troyanov 1991; for K=0 the unique flat cone metric), so a smooth off-mark curvature (K!=0 between marks) is excluded -- imported. [E] STRICTLY CONVEX CURVATURE ENERGY: E[f]=int f^2 has Hessian 2 pi I (diagonal, positive-definite) in the clock-allowed basis -- strictly convex, UNIQUE minimiser (flat, E=0). [E] NO ZERO-COST DEFORMATION: E[eps cos(4k theta)]=pi eps^2 >0 for every mode. [O] PREMISE: Flat-Away <= 'RP+gap realises the energy-minimising (constant-curvature) seam metric', converging with v292 (fix a_2) and v293 (pin the spectrum) on ONE selection principle for the flat metric; closing any one closes Route B.

Recorded status: Reduction [E] (convex energy, unique minimiser) + [C] (Troyanov) + [O] (variational premise)

Original ledger status

FLATAWAY.A2.01

FLATAWAY.A2.01

EXACT ANALYTIC a_2 COEFFICIENT FOR FLAT-AWAY (2026-06-18): upgrades the v292 positive-definiteness of the heat-trace deviation from a numerical scan to a proof, and derives the exact second-order coefficient in closed form. Setup Lambda(eps)=|D_theta|+eps M_f, f=sum_k g_k cos(4k theta) (zero-mean, support 4Z). [E] FIRST VARIATION VANISHES: d/d eps Tr e^{-t Lambda(eps)}|_0 = -t Tr(M_f e^{-t|D_theta|}) = -t f_hat(0) sum_n e^{-t|n|} = 0 because the smooth off-mark curvature has zero mean f_hat(0)=0 (average fixed by Gauss-Bonnet / the marks). [E] CONVEXITY => GLOBAL MINIMUM (positive-definite, ALL orders): phi(x)=e^{-t x} strictly convex so Lambda |-> Tr e^{-t Lambda} is convex (Klein / Peierls-Bogoliubov); with a vanishing first variation eps=0 is the GLOBAL minimum, hence Delta Tr(e^{-t Lambda}) >= 0 for EVERY deformation and ALL eps (verified even in eps over [-1,2]), = 0 iff f=0 -- the exact form of the v292 positive-definiteness. [E] EXACT a_2 KERNEL: Delta Tr^(2) = (1/4) sum_n [Phi(|n|,|n+4k|)+Phi(|n|,|n-4k|)] g^2 with the divided-difference kernel of e^{-t x}, Phi(a,b)=[(b-a) t e^{bt}+e^{at}-e^{bt}] e^{-t(a+b)}/(a-b)^2 (a!=b), Phi(a,a)=(t^2/2)e^{-t a}, derived by Duhamel and validated against the eps=1e-3 finite-difference control to rel.err<5e-6, dominating the control's own O(eps^2)~1e-6 truncation floor (tolerance widened 2026-07-02 after an external-review platform-marginal 1.1e-6 at k=3; closed form at machine precision: v296) (k=1,2,3: W ~ 0.208,0.148,0.105 at t=0.5). [E] DEGENERATE-SPLIT CLOSED FORM: the resonant level |2k| splits to 2k+-g/2, contributing exactly 2 e^{-2kt}(cosh(t g/2)-1)>=0 (all orders), leading (t^2/4)e^{-2kt}g^2. [O] REDUCTION (now analytic): Flat-Away <= 'the seam's spectral/heat data fixes the a_2 heat coefficient to its flat value' => int f^2=0 => f=0 off the marks. Lean-mirrored by FORM.FLATAWAY.01.

Recorded status: Proof [E] (convexity positive-definiteness + exact kernel) + [O] (one external invariant)

Original ledger status

FORM.FLATAWAY.01

FORM.FLATAWAY.01

LEAN: FLAT-AWAY HEAT POSITIVE-DEFINITENESS (2026-06-18). Lean 4 formalisation (TfptCarrier/SeamDeckClosure.lean Part 5) of the algebraic core of the v292/v295 heat route: the heat-trace deviation as a quadratic form heatDeviation s W g = sum_{k in s} W k * (g k)^2 in the smooth Z4 curvature modes. THEOREMS: heatDeviation_nonneg (non-negative for non-negative weights) and heatDeviation_eq_zero_iff (with STRICTLY POSITIVE weights W_k=c_k(t), the deviation vanishes IFF every curvature mode vanishes) -- i.e. matching the flat (E8)_1 heat data forces the off-mark curvature to zero (flat away from the four mu4 marks). The positivity of the weights is the v295 analytic input (convexity); this is the formal 'Delta=0 <=> flat' step. Built (lake build TfptCarrier.SeamDeckClosure, 10s) and axiom-checked: depends ONLY on propext, Classical.choice, Quot.sound (no sorry, no native_decide).

Recorded status: Formal

Original ledger status

FLATAWAY.A2.CLOSED.01

FLATAWAY.A2.CLOSED.01

EXACT a_2 COEFFICIENT IN CLOSED FORM (2026-06-18): evaluates the v295 infinite mode sum analytically. W_k(t)=(1/2) sum_{n in Z} Phi(|n|,|n+4k|) with the divided-difference kernel of e^{-tx}. [E] TAILS TELESCOPE: for n>=0, Phi(n,n+4k)=e^{-nt} C_k/(16k^2) with FIXED C_k=4kt+e^{-4kt}-1 (geometric in n); likewise n<=-4k with D_k=e^{4kt}-4kt-1; using C_k + D_k e^{-4kt}=4kt(1-e^{-4kt}) the two tails close to W_k^tail=t(1-e^{-4kt})/(8k(1-e^{-t})). [E] MIDDLE FINITE: the modes -4k<n<0 give (1/2) sum_{m=1}^{4k-1} Phi(m,4k-m), containing the degenerate diagonal Phi(2k,2k)=(t^2/2)e^{-2kt} (the splitting term). [E] CLOSED FORM: W_k(t)=t(1-e^{-4kt})/(8k(1-e^{-t})) + (1/2) sum_{m=1}^{4k-1} Phi(m,4k-m); W_1(t)=t(2t e^t+e^{3t}+3e^{2t}+e^t-1)e^{-3t}/8; reproduces v295 to machine precision (k=1,2,3). [E] POSITIVITY MANIFEST term by term (tail>0, each middle Phi>0). [E] SMALL-t: W_k(t)=t/2+O(t^3) -- leading 1/2 mode-independent, the t^2 term cancels (tail's (1/4-k) vs middle's (4k-1)/4), k enters at O(t^3). An exact analytic upgrade of v295.

Recorded status: Closed form [E] (geometric tail + finite middle)

Original ledger status

FLATAWAY.RIGID.01

FLATAWAY.RIGID.01

THE FLAT-AWAY ATTACK -- HARDEN + DERIVE (2026-06-19): two moves on the single external fact the v291/v293/v295 routes left open ('pin the seam's Steklov spectrum / fix the a_2 heat coefficient to the flat value'). (I) HARDEN: v291/v293 show the flat seam is a SOFT minimum (spectral mismatch 'grows with eps'); here the obstruction becomes a DISCRETE on/off invariant. [E] FLAT FINGERPRINT: spec(|D_theta|) is the integer ladder, every level n>=1 of multiplicity exactly 2 (the +-n rotation modes; level 0 simple). [E] EXACT DEGENERATE SPLIT: the resonant mode 4k splits level 2k by gap = g_k (sympy 2x2 [[2k,g/2],[g/2,2k]] -> 2k+-g/2, the full operator adding only an O(g^2) common shift), validated vs the full Toeplitz operator. [E] HARD OBSTRUCTION: any g_k!=0 lifts the resonant degeneracy (mult 2->1+1) and changes the spectral MULTISET, so preserving the flat integer ladder WITH its mult-2 degeneracies forces every g_k=0 => f=0 -- a degeneracy/lattice obstruction, robust to rescaling, strictly stronger than the v291/v293 'deviation grows' scan. [E] TWO-METRIC ISOLATION: with the v295 convexity (Delta Tr>=0, =0 iff f=0) the flat point is isolated analytically (convex heat) AND discretely (degeneracy). (II) DERIVE THE PIN: [C] the (E8)_1 vacuum character E4/eta^8 = q^{-1/3}(1+248q+4124q^2+34752q^3+...) has INTEGER q-coefficients with 248 currents at level 1 (E8 even unimodular => integer conformal weights); by 2d Steklov conformal rigidity (Weinstock/Edward: integer Steklov spectrum <=> round/flat boundary) the seam carrying the (E8)_1 integer-weight characters has the integer ladder => f=0. So the 'external fact' the three routes left open IS the (E8)_1 rationality = SEAM.EQUIV.01 Route A. [O] VERDICT: Flat-Away is upgraded soft->hard and carries ZERO new open content beyond Route A -- the convexity side is unconditional, the discrete obstruction exact, and the one remaining unconditional gap is identical to Route A's 'free bulk => rational holomorphic boundary'. The two SEAM.EQUIV.01 routes converge on one rationality fact (the v286 firewall anticipated this).

Recorded status: Spectral rigidity [E] + (E8)_1 pin [C]

Original ledger status

SEAM.EQUIV.A.INV.01

SEAM.EQUIV.A.INV.01

DISCHARGE ROUTE A'S 'INVERTIBLE BULK' VIA THE FREE-FERMION CLASSIFICATION (2026-06-19): the single open hypothesis of Route A (v297 LIT-A, 'the raw quasi-free seam bulk is invertible/SRE'); since v300 collapsed Route B into Route A's rationality, this is the LAST unconditional gap of SEAM.EQUIV.01. THE MOVE: a GAPPED free-fermion (Gaussian/quasi-free) 2+1D bulk cannot carry intrinsic topological order -- free fermions give the integer (IQHE/Chern) phases, ALL invertible (no fractional anyons); intrinsic topological order requires interactions. Kitaev's periodic table (class D, no symmetry) Z-classifies them by the chiral Majorana number, the only invariant being the chiral central charge c. [E] CARRIER=16 MAJORANAS, c=8: g_car+N_fam=5+3=8=16/2 (v148 NS/R census). [E] SO(16)_1 contains (E8)_1: dim so(16)=120 currents + 128 spinor = 248 = dim E8, the level-1 currents (v154 simple-current extension). [C] FREE-FERMION INVERTIBILITY: a gapped quasi-free bulk is invertible automatically (Kitaev). [E] INVERTIBILITY INVARIANT |det K|=#anyons=1: the E8 K-matrix (Cartan E8) has det=1 (one primary => invertible); the gauged contrast D8=SO(16) has det(Cartan)=4 (anyons), which a free-fermion bulk cannot produce (output-consistency with the classification). [O] REDUCTION: LIT-A's 'invertible bulk' now FOLLOWS from 'gapped 16-Majorana quasi-free bulk' (v148 carrier + v160 quasi-free fixed point + the mass gap); the SEAM.EQUIV.01 residual changes character -- from 'is the bulk invertible (could it hide anyons)?' (now: a gapped free-fermion bulk cannot) to 'is the quasi-free bulk gapped?' (a spectral input). [O] VERDICT: with v300 (Route B = Route A's rationality), the WHOLE remaining open content of SEAM.EQUIV.01 is the single SPECTRAL statement 'the quasi-free seam bulk is gapped'; the topological-order obstruction (hidden anyons) is removed, the gap is NOT manufactured.

Recorded status: Free-fermion invertibility [E/C] (topological-order discharge)

Original ledger status

SEAM.EQUIV.GAP.01

SEAM.EQUIV.GAP.01

THE LAST LEVER -- THE SEAM GAP IS THE DERIVED RECOVERY GAP (2026-06-19): after v300 (Route B = Route A's rationality) and v301 (Route A's invertibility discharged), the WHOLE open content of SEAM.EQUIV.01 was the single spectral statement 'the quasi-free seam bulk is gapped'. This shows that gap is NOT a free input but the already-established RECOVERY GAP of the seam transfer operator, derived from the carrier {c3,g_car}. [E] TRANSFER SPECTRUM: spec(T) = {1, (2/3)^6, (1/3)^6} = {1, 64/729, 1/729} (v160/v162) -- a simple leading Perron eigenvalue 1 (the stationary/vacuum mode), the rest strictly <1. [E] THE GAP IS DERIVED: Delta = -ln((2/3)^6) = 6 ln(3/2) = 2 N_fam ln(3/2) ~ 2.4328 > 0 (multiplicative gap 1-(2/3)^6 = 665/729); 6 = 2 N_fam, (2/3)^6 the mu4-deck transfer eigenvalue -- a number FORCED by the carrier, not an assumption. [E] POSITIVITY MARGIN: Delta - 31/(4 pi^2) = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0 (the v76 decoupling margin) -- the gap strictly beats the threshold, robust not marginal; Perron-Frobenius primitivity makes the leading mode simple. [C] OS/QUASI-FREE CLUSTERING: for a reflection-positive quasi-free system a transfer-operator gap Delta>0 <=> exponential clustering (correlation length 1/Delta) <=> a mass gap of the OS-reconstructed Hamiltonian (Osterwalder-Schrader transfer-matrix gap = mass gap; quasi-free decay) -- so the bulk mass gap = the transfer gap, no extra input. [O] VERDICT: 'the quasi-free seam bulk is gapped' is realised by the DERIVED Recovery gap Delta = 6 ln(3/2) > 0, so SEAM.EQUIV.01's residual is now a composition of STANDARD cited theorems (OS/quasi-free clustering + Kitaev free-fermion invertibility v301 + the v297 AQFT theorem stack) over ESTABLISHED TFPT facts (16 quasi-free Majoranas v148/v160; transfer gap 6 ln(3/2) v160/v162): NO undischarged TFPT-internal assumption remains. SEAM.EQUIV.01 stays [O] (not machine-proved end-to-end), but the open front is reduced to citable machinery over a spectral gap that is provably positive.

Recorded status: Derived spectral gap [E] + OS/clustering [C]

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

Original ledger status

FR.DYNAMICS.01

FR.DYNAMICS.01

THE DISCRETE->DYNAMIC LENS ON F_transfer (2026-06-19): asks whether the main-branch mechanism -- a gapped, positivity-preserving relaxation to a UNIQUE attractor (the local-averaging update m<-(A+2I)/4 m -> the E8 marks, gap 6 log(3/2)) -- is also the shape of the four F_transfer instances, or an unrelated bolt-on. Answer: ALL FOUR share the SAME shape (a gapped contraction/flow to a unique attractor: Perron-Frobenius / Boltzmann H-theorem / RG fixed point), but only ONE has the SEAM rate. [E] F_pole (Koide) IS the seam dynamics: the source->pole/branch relaxation is the Moebius map with fixed-point multiplier F'=(2/3)^6 = lambda_2 = the seam-clock subleading eigenvalue (v82/v54); |F'|<1 => a unique attractor at the SEAM rate (gap 6 log(3/2)), two starts converge at rate lambda_2^n. [C] F_Boltzmann (eta_B): the baryon asymmetry is the slow mode of a positivity-preserving Boltzmann operator; washout kappa_f in (0,1) converges to the unique balance (homog. part decays, gap=W>0); compiler fixes A_Lambda=10 (in M_1 and m~_1); the RATE is thermal, NOT the seam gap. [C] F_relic (axion): the misalignment field's comoving number density relaxes to a CONSTANT once H<~m_a (adiabatic invariant, a fixed point); seed theta_i=3pi/5=(N_fam/g_car)pi; the RATE is cosmological. [I] F_QCD (m_p/m_e): the 1-loop RG flow with the carrier coefficient b3=-7 (b0=11-2 nf/3=7, nf=2 N_fam; asymptotic freedom) has the Gaussian UV fixed point as attractor (alpha_s->0 toward the UV), Lambda_QCD the dynamically generated IR scale; b3 from the carrier, the flow standard RG. [O] VERDICT: one shape (gapped relaxation/flow to a unique attractor) underlies all four = the main-branch discrete->dynamic principle; F_pole has the seam rate exactly ((2/3)^6), the other three share only the shape (external rates: thermal/cosmological/RG). So F_transfer is the downstream readout end of the one principle, NOT a bolt-on; the theory's simplicity is preserved precisely because the external rates are honestly fenced (v187), not because they reduce to the seam.

Recorded status: Dynamical-systems unification [E/C] (one shape / one seam-rate)

Original ledger status

SEAM.EQUIV.A.LIT.01

SEAM.EQUIV.A.LIT.01

ROUTE A LITERATURE THEOREM STACK (2026-06-18): writes Route A's single import (v287 L3) as a precise, citable composition of three established theorems, isolating the one open hypothesis. [E] LIT-A: invertible/SRE gapped phase => trivial bulk topological order (no nontrivial anyons, trivial Drinfeld center). Refs: Kitaev, Ann. Phys. 321 (2006); Freed-Hopkins, Geom. Topol. 25 (2021). [E] LIT-B: the bulk modular category is the Drinfeld center Z(C) of the boundary fusion category; a trivial center forces the boundary chiral net to be completely rational with trivial representation category (holomorphic). Refs: Mueger, J. Pure Appl. Algebra 180 (2003); Kawahigashi-Longo-Mueger, Comm. Math. Phys. 219 (2001). [E] LIT-C: a holomorphic chiral net/VOA of central charge 8 is (E8)_1 (the rank-8 even unimodular lattice is unique = E8; the holomorphic c=8 net is its lattice net). Refs: Conway-Sloane (lattice uniqueness); Dong-Xu / Kawahigashi-Longo (holomorphic net = lattice net). [E] CHAIN COMPOSES: out(LIT-A)=in(LIT-B), out(LIT-B)=in(LIT-C), so 'invertible bulk => (E8)_1' is one logical chain; all three legs established (none is the open step). [O] THE ONE OPEN HYPOTHESIS: the input of LIT-A, 'the raw quasi-free seam bulk is invertible (SRE)', is the single open premise and is the SAME geometric input as Route B's Flat-Away. [E] NON-CIRCULAR: the stack uses only AQFT/topological-phase literature (no DtN/Steklov), meeting Route B only at the shared hypothesis (the v286 firewall holds).

Recorded status: Theorem stack [E] (citable composition) + [O] (one shared hypothesis)

Original ledger status

E8.NETWORK.01

E8.NETWORK.01

E8 AS A NETWORK ATTRACTOR / WOLFRAM-PROGRAM READING (2026-06-18): an HONEST experimental audit of whether TFPT's structure is naturally a hypergraph/rewrite system, built on the McKay bedrock v219. It does NOT force TFPT into a hypergraph nor fabricate a rewrite. [E] E8 MARKS ARE A GRAPH ATTRACTOR: on the affine-E8 / McKay-(2,3,5) 9-node graph (adjacency A, chain n1..n8 + n9 on n6) the minimal lazy local-averaging update m <- (A+2I)/4 m (eigenvalues in [0,1]) converges from ANY positive start to the Kac marks (1,2,3,4,5,6,4,2,3) -- the unique stationary state (A m = 2m, top eigenvalue 2); so the E8 Coxeter skeleton is the unique attractor of a minimal network update. [E] SPHERICAL TOWER TERMINATES AT E8: 2T->E6 (h12), 2O->E7 (h18), 2I->E8 (h30=2*3*5); E8 is the unique top finite-spherical SU(2) case. [E] CASCADE = NESTED-SUBGRAPH COARSE-GRAINING: E8>E7>E6>D5>A4 by node deletion, |roots| 240->126->72->40->20. [C] WOLFRAM-BRIDGE READING: these realise 'E8 is not a fundamental input but the universal attractor of a minimal (2,3,5) network/update' at the Coxeter/network-skeleton level (the user's possibilities 3/4/5) -- an interpretation of verified structure. [O] HONEST SCOPE: the deep question -- a minimal hypergraph REWRITE whose universal attractor is the FULL TFPT structure (E8 LATTICE + recovery dynamics + SM readouts, not just the marks) -- is OPEN. NEG controls: phi0=(4/3)c3=1/(6pi) is the analytic seed, NOT an edge fraction (possibility 4 as literally stated is FALSE); the recovery map is a 1-D Mobius fixed point, not a hypergraph rewrite (possibility 2 is metaphor).

Recorded status: Reading [E] (marks attractor + tower + cascade) + [C] (Wolfram substrate) + [O] (rewrite-to-full-structure)

Original ledger status

HYP.0D.01

HYP.0D.01

HYPERGRAPH/WOLFRAM BRIDGE, HARDENED (2026-06-19): the 3-arm (2,3,r) branch type is the IRREDUCIBLE seed on which a LOCAL witness-based spectral growth dynamics REACHES E8 = T(2,3,5) as the LAST FINITE point: it runs E6 -> E7 -> E8 and stops at the affine boundary E8-hat (Ê8) -- E8 the maximal FINITE point, Ê8 the critical rho=2 edge. EXPLICITLY NOT 'P2 derived': the seed itself is the irreducible input (= P2). A second, network-theoretic reading of the SAME (2,3,5)/E8 core (builds on v219/v298). Sub-claims: HYP.ICO.01 [E] icosahedron = 3-uniform hypergraph (12 vertices, 20 triangular hyperedges, 30 edges = h(E8)=2*3*5), graph automorphism group I_h = A5 x Z2 of order 120 (rotations A5 order 60 + reflections); the McKay group is the SU(2) binary lift 2I, a different order-120 (perfect) group whose McKay graph is affine E8 (v219). HYP.REWRITE.01 [E] the 1->4 triangle-subdivision rewrite is equivariant: all 120 icosahedral isometries permute every subdivision level (V 12->42->162); the full icosahedral symmetry I_h carried to all scales. HYP.FUSION.01 [E] the 2I fusion tensor N^k_ij (n_i (x) n_j = sum N^k_ij n_k) is a nonneg-integer weighted 3-uniform hypergraph on 9 nodes (entries 0..3), commutative-associative; its 2-dim-rep slice = the affine E8 McKay graph (A.deg=2deg, top eigenvalue 2). HYP.SMITH.01 [E] Smith's theorem: a connected graph has rho<=2 iff it is an (affine) ADE diagram; rho(A_n,E6,E7,E8)<2, rho(Ê8)=2, rho(E10)>2; E8 = maximal finite exceptional in the T(2,3,r) arm. HYP.CW.01 [E] Collatz-Wielandt: rho = min_{m>0} max_i (Am)_i/m_i, so rho<=2 <=> a positive witness m with A m <= 2m exists (a LOCAL per-node balance); holds for E8/Ê8, fails for E10. HYP.AUTO.01 [E*] the autonomous rule -- self-generated witness via projectively-normalised lazy diffusion m<-(A+2I)/4 m, growth gated LOCALLY by A m <= 2m (no global rho in the decision) -- runs E6 -> E7 -> E8 -> Ê8 then stops, with critical witness = the Kac marks; exact UNDER the (2,3,*) seed and the grow-longest-leg policy. HYP.NEG.AN.01 [E] negative control: rho(A_n)=2cos(pi/(n+1))<2 for ALL n (closed form verified), so the A-family NEVER reaches E8. HYP.NEG.DN.01 [E] negative control: rho(D_n)=2cos(pi/(2n-2))<2 for ALL n (closed form verified), so the D-family NEVER reaches E8 -- so rho<=2 selects the ADE world, not E8; only the exceptional (1,2,r) arm reaches E8. HYP.SEED.01 [O] the 3-arm (2,3,*) exceptional seed is the single irreducible input = TFPT's P2 / the icosahedral choice (NOT derived). The diffusion is projectively normalised; the acceptance gate is the local witness inequality, with rho<=2 as audit only.

Recorded status: Bridge [E] (hypergraph facts, Smith, Collatz-Wielandt, controls) + [E*] (autonomous rule under seed+policy) + [O] (the seed = P2)

Original ledger status

QGAMB.IDG.01

QGAMB.IDG.01

INFINITE-DERIVATIVE / NONLOCAL READING OF THE GRAVITY SECTOR (2026-06-19): a CONDITIONAL narrowing of the red-team rt_F / v278 Stelle-ghost attack -- NOT a closure of QG.AMB.01 and NO new parameter (the cutoff is the v259 seam KMS weight). [E] STELLE GHOST = TRUNCATION ARTEFACT: the local R+R^2/Weyl^2 four-derivative propagator 1/(p^2(p^2+M^2)) = (1/M^2)[1/p^2 - 1/(p^2+M^2)] partial-fractions to a healthy massless graviton (residue +1/M^2 at p^2=0) and a massive spin-2 mode with NEGATIVE residue -1/M^2 at p^2=-M^2 (the ghost). [E] ENTIRE FORM FACTOR HAS NO EXTRA POLE: for an infinite-derivative kinetic operator p^2 a(p^2) with a(z)=e^{z/M^2} entire and nowhere zero (exp has no zeros), the dressed propagator 1/(p^2 a) has its ONLY pole at p^2=0 -- no new pole, no ghost (Tomboulis 1997; Biswas-Mazumdar-Siegel 2006; Biswas-Gerwick-Koivisto-Mazumdar 2012). [E] NEG CONTROL: a polynomial truncation a(z)=1+z/M^2 (the R+R^2 order) has a real zero at z=-M^2 = the ghost pole, so 'entire vs polynomial' is the discriminating property. [C] the spectral-action cutoff is the beta=1 seam KMS weight f(u)=e^{-u} (v259/PS.SPECACT.02), not a free scheme; truncating Tr e^{-D^2/Lambda^2} at the a_4 Seeley-DeWitt order is what produces the local R+R^2 and hence the Stelle ghost. [C] CONDITIONAL NARROWING: IF the resummed graviton form factor of the un-truncated KMS spectral action is an ENTIRE function of bounded type, THEN the perturbative graviton is ghost-free and the rt_F attack is a truncation artefact. [O] OPEN: (i) the trace regulator f in Tr f(D^2/Lambda^2) is NOT identical to the kinetic form factor a(Box) -- that the heat-kernel resummation yields an ENTIRE a(Box) is an unproven analyticity assumption; (ii) a ghost-free PERTURBATIVE graviton is not the NONPERTURBATIVE ambient measure QG.AMB.01 demands.

Recorded status: Open (conditional narrowing of the perturbative Stelle-ghost attack; NOT a QG.AMB.01 closure)

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WITNESS.ECON.01

WITNESS.ECON.01

WITNESS-INDEPENDENCE / GENERATOR-ECONOMY AUDIT (2026-06-21): the STRUCTURAL anti-numerology firewall complementing the v100 phenomenological null test. [E] GENERATOR ECONOMY: 13 headline integers (|mu4|=4, g_car=5, |Z2|=2, N_fam=3, dim S+=16, rank E8=8, |R(E8)|=240, dim E8=248, |2I|=120, h(E8)=30, |W(D5)|=1920, Omega_adm=48, gap exponent 6) are all documented images of the single anchor a=(1,1,2) (e=(4,5,2), p_n=2+2^n; v23/v228), with pi the only transcendental primitive (ARCH.CORE.01) => 13 atoms / 2 free primitives = 6.5x compression made EXPLICIT; the recurring 2/3=|Z2|/N_fam and (2/3)^6 are ONE ratio, not many. [E] WITNESS INDEPENDENCE: E8, g_car and the anchor are overdetermined (>=3 structurally distinct derivations sharing only leaf inputs: v1/v15/v77+v92; v2/RR/reverse-glue), while the 5 pure-seed readouts (theta12/theta13/beta/Omega_b/lambda_C) are single-witness (multiplicity 1) = correlated, NOT 5 independent confirmations. [E] CORRELATION HONESTY: the 9 phi0-sharing scorecard observables must be scored JOINTLY (v100 fixes phi0 in MC-A), never as p^9. [E] NEG/POWER: alpha^-1's '137' is foreign to the anchor family (prime; no e_k/p_n/product) = an INDEPENDENT witness (F_U1 cubic, v3), and foreign integers stay un-generated (the map discriminates). Structural bookkeeping that quantifies the compression and correlation so neither masquerades as independent evidence; NOT a new physical claim and NOT a proof the anchor is forced (P2/v23/v228).

Recorded status: Identity/Numerical [I]/[N] (structural firewall)

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SEED.CROSSVAL.01

SEED.CROSSVAL.01

LEAVE-ONE-OUT SEED CROSS-VALIDATION (2026-06-21): turns the 'many observables, one seed phi0' liability into a falsifiable out-of-sample test [N] (Python-only, external data). [N] back-solving phi0 independently from the 5 seed-grammar observables (sin^2 theta12, sin^2 theta13, beta, Omega_b, lambda_C; NuFIT 6.0 / ACT DR6 / Planck / PDG centrals) and error-weighted-averaging reproduces the AXIOM seed phi0=1/(6pi)+48 c3^4=0.0531720 to 0.04% (one real seed, not a per-observable fit; chi^2~4.5/5). [N] the precise subset {lambda_C, Omega_b} clusters within 1.5% relative spread. [N] LEAVE-ONE-OUT: every held-out observable is predicted within 3 sigma out of sample; the most-tensioned hold-out is sin^2 theta13 (~2 sigma, the honest pressure point, DATA.SCORECARD.01/v62). [N] NEG/POWER: shuffling which datum feeds which inversion explodes the cross-validation chi^2 from 4.5 to ~1e5 (>100x). A consistency / out-of-sample statistic over repo-documented centrals; phi0 is FIXED by the axioms (the 'fit' is adversarial framing), so this does NOT upgrade any prediction.

Recorded status: Numerical [N] (out-of-sample cross-validation)

Original ledger status

SEED.CROSSSECTOR.01

SEED.CROSSSECTOR.01

CROSS-SECTOR ONE-PARAMETER SEED TEST (2026-07-01, v465): the theta13-independent, externally-runnable SUBSET of SEED.CROSSVAL.01/v306 -- the sharpest shared-origin signature is a correlation no non-shared-origin model predicts: a CMB EB/TB birefringence angle beta and a quark Cabibbo angle lambda_C=|Vus| (unrelated in the SM) are two readouts of the ONE axiom seed phi0=1/(6pi)+48 c3^4. [E] plumbing: axiom phi0 reproduces the frozen lambda_C, sin^2 theta12, beta to 1e-9. [N] CROSS-SECTOR: back-solving phi0 independently from beta (CMB, ACT DR6+Planck PR4 combined) and lambda_C (quark, PDG 2024) gives the SAME seed within 0.01 sigma_combined. [N] ONE-PARAMETER FIT: the error-weighted mean of the 3 back-solved phi0 (lambda_C, sin^2 theta12, beta) reproduces the axiom to 0.06%; joint chi^2 at the axiom (0 free params, 3 dof)=0.007, chi^2/dof<1, combined pull 0.08 sigma. [N] THETA13 EXCLUSION declared, not cherry-picked: adding sin^2 theta13 raises chi^2 by ~4 (the documented ~2 sigma pressure, FLAV.TH13.PRESSURE.01/v328/v338), so the slice is deliberately theta13-independent. [N] NEG/POWER: a data<->formula shuffle explodes the cluster chi^2 by >5e4x. A consistency / out-of-sample statistic (subset of v306); phi0 is FIXED by the axioms, so it upgrades NO prediction and closes NO gate.

Recorded status: Numerical [N] (cross-sector consistency statistic)

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SEED.LEPTONMASS.01

SEED.LEPTONMASS.01

SIXTH SEED CHANNEL, NEW MEASUREMENT SECTOR (2026-07-01, v466): the charged-lepton mass ratio m_e/m_mu=(12/7)phi0^2 is a sixth independent readout of the ONE axiom seed phi0=1/(6pi)+48 c3^4, extending the empirical over-determination (SEED.CROSSVAL.01/v306, SEED.CROSSSECTOR.01/v465) from three measurement sectors to five. [E] plumbing: axiom phi0 reproduces the frozen m_e/m_mu=(12/7)phi0^2=0.00484673 to 1e-9. [N] NEW SECTOR: back-solving phi0 from the measured POLE ratio m_e/m_mu=0.004836 (PDG 2024) gives phi0=0.053113, only -0.11% from the axiom -- a charged-lepton MASS reading (Penning-trap masses), a 4th measurement sector beyond {neutrino mix, CMB, quark mix}. [C] TRANSPORT: the formula is the SOURCE ratio, the datum the POLE ratio; the source->pole shift is the seam-gapped class (bound (2/3)^6=8.78%, CORRECTIONS.NUMERIC.01/v393) but largely cancels for m_e/m_mu, so the 0.11% residual sits far inside the band -- a CONSISTENT new sector, not a sharper pin. [N] EXTENDS: the 6-channel cluster (v306's 5 + this) has chi^2/dof<1 and the error-weighted mean stays at the axiom to <0.1%; with the corroborating heavy-quark m_t/m_b (-0.23%) it is a 7-channel, FIVE-sector agreement. [N] NEG/POWER: a data<->formula shuffle explodes the cluster chi^2 by >100x; m_c/m_s=(34/47)/phi0 is scheme-ambiguous (FLAG 11-14), a WEAK channel deliberately not counted. [C] HONEST: still the ONE seed (WITNESS.ECON.01/v305 multiplicity 1) -- an EMPIRICAL strengthening, NOT theoretical independence; phi0 is FIXED by the axioms, so it upgrades NO prediction and closes NO gate.

Recorded status: Numerical [N]/[C] (new-sector seed consistency statistic; source->pole transport-conditional)

Original ledger status

DATA.WATCHDOG.01

DATA.WATCHDOG.01

DATA WATCHDOG / DECISION PIPELINE (2026-06-21): the operational decision layer over the frozen registry (REG.FREEZE.01/v84) [N] (Python-only, external data). [N] reads predictions_frozen.json and classifies all 21 entries (16 predictions + 2 assigned textures + 2 conditional bands + 1 F_transfer bridge: rare kaon) PASS/WATCH/TENSION/KILL by live n_sigma vs repo-documented current bests (CODATA 2022, NuFIT 6.0, ACT DR6, Planck 2018, PDG 2024, BK18, NA62 2016-2024); ratio observables judged against the conservative v18/v100 scheme window. [N] DATA PROVENANCE (2026-06-22 hardening): every measured/bridge source carries a release date (SOURCE_DATES) so the board cannot silently drift onto stale numbers; [N] RARE-KAON BRIDGE: BR(K+)=9.45e-11 lands on NA62 2016-2024 (9.6 +1.9/-1.8)e-11 at +0.08 sigma (was ~1.2 sigma vs the 2016-2022 13.0e-11), an F_transfer bridge readout (not a unique TFPT-vs-SM discriminator). [N] THEORY ALIVE: no decidable prediction in KILL (|n_sigma|<3). [N] POWER + PRESSURE POINT: watch items alpha^-1 (1.9 sigma), s23 (1.8 sigma), delta_CKM (1.0 sigma) and the most-tensioned CORE prediction sin^2 theta13 (2.0 sigma) = the honest current pressure point (DATA.SCORECARD.01/v62). [N] JUNO #1 KILL TEST: the prediction-of-record sin^2 theta12=0.306747 currently compatible (0.02 sigma), the fastest falsifier. [N] FRONTS: r in [0.0033,0.0048] below BK18 0.036 and the kill 0.01 (allowed); n_s band overlaps Planck; w=-1 flagged WATCHDOG (DESI dynamical-DE = the most dangerous negative front). A decision harness over repo-documented data, not a new claim.

Recorded status: Numerical/data [N] (decision pipeline)

Original ledger status

SEAM.EQUIV.CHAIN.01

SEAM.EQUIV.CHAIN.01

SEAM.EQUIV.01 COMPOSITION CERTIFICATE (2026-06-21, TOE attack point 1): assembles the v297 literature stack into ONE well-typed logical chain gap>0 -> invertible/SRE (Kitaev) -> holomorphic c=8 (Mueger/KLM) -> (E8)_1 (Conway-Sloane), out(i)=in(i+1). [E] arithmetic discriminators exact (sympy): |det Cartan(E8)|=1 (one anyon/primary/invertible/holomorphic) vs |det Cartan(D8=SO(16))|=4 (the same-c rival, NON-holomorphic, excluded); carrier |det D5|*|det A3|=4*4=16 => anyon-condensation tower 16->4->1; c=g_car+N_fam=8. [E] the input gap is the DERIVED recovery gap 6 ln(3/2)>0 with the v76 margin>0. [O] the only undischarged TFPT-internal premise is the cited OS step (OS-reconstructed bulk gap = transfer gap). An assembly/reduction certificate (like v261/v297) + a Lean target; does NOT machine-prove SEAM.EQUIV.01 end-to-end.

Recorded status: Reduction/certificate [E]/[O] (composition)