Publication history: 2026-06-19

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• v304_idg_nonlocal_ghost.py (QGAMB.IDG.01, [C]/[O]). A conditional, parameter-free narrowing of the red-team rt_F/v278 Stelle-ghost attack on the perturbative gravity sector. [E]the local R{+}R^2/Weyl^2 four-derivative propagator 1/(p^2(p^2{+}M^2)) carries a negative-residue massive spin-2 mode (the Stelle ghost) only as a truncation artefact; [E]for an entire graviton form factor a(p^2)=e^{p^2/M^2} (nowhere zero) the dressed propagator 1/(p^2a) has no extra pole (Tomboulis 1997; Biswas–Mazumdar–Siegel 2006), whereas any polynomial truncation has a real zero = the ghost; [C]the spectral-action cutoff is the seam KMS weight f(u)=e^{-u} (v259), so the un-truncated trace points to a nonlocal form factor with no new parameter. [O]It does not close QG.AMB.01: the trace regulator is not yet shown to resum to an entire a(\Box), and a ghost-free perturbative graviton is not the nonperturbative ambient measure. Cited in the red-team Target F body; ledger row QGAMB.IDG.01.
• Double-pushout figure (tfpt_1_architecture_e8). A new shared TikZ macro (TFPTdoublepushout in tex-artefacts/tfpt_figures.tex) draws the commuting square showing that the order-|\mu_4|{=}4 glue is one operation in two categories: the lattice pushout D_5\oplus A_3+\mu_4\Rightarrow E_8 (v1) and the conformal-net / anyon-condensation extension (D_5)_1\otimes(A_3)_1\rtimes\mu_4\cong(E_8)_1 (v92/v281/v234/v277/v260). Expository only; no new claim.
• Framing (introduction, tfpt_research_contracts). SEAM.EQUIV.01 is now stated explicitly as the one irreducible structural postulate of the theory — the role the constancy of c plays in relativity (cf. v267). Prose sharpening; no status change.

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• v303_ftransfer_dynamics.py asks whether the main-branch mechanism — a gapped, positivity-preserving relaxation to a unique attractor (the local-averaging update \to the E_8 marks, gap 6\ln(3/2)) — is also the shape of the four F_transfer instances, or an unrelated bolt-on. It is the same shape: [E]F_pole (Koide) is the seam dynamics (Möbius multiplier F'{=}(2/3)^6{=}\lambda_2, the seam-clock eigenvalue \Rightarrow a unique attractor at the seam rate, v82/v54); [C]F_Boltzmann (\eta_B) is a gapped positive contraction (washout \kappa_f\in(0,1), H-theorem); [C]F_relic (axion) freezes at an adiabatic fixed point; [E]F_QCD (m_p/m_e) is the 1-loop RG flow with the carrier b_3{=}-7 (asymptotic freedom, Gaussian UV attractor). [O] Verdict: one shape underlies all four; F_pole has the seam rate (2/3)^6 exactly, the other three share only the shape with external rates (thermal/cosmological/RG). So F_transfer is the readout end of the one discrete\todynamic principle, not a bolt-on; the theory's simplicity is preserved because the external rates are honestly fenced (v187), not reduced to the seam. [E]suite v303 5/5, AUDIT OK.

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• v302_seam_gap.py closes the chain on the single statement v301 had left: “the quasi-free seam bulk is gapped”. That gap is not a free input — it is the established Recovery gap of the seam transfer operator, derived from the carrier. [E]the frozen transfer spectrum is \mathrm{spec}(T)=\{1,(2/3)^6,(1/3)^6\} (v160/v162), a simple Perron eigenvalue 1 separated from the rest; [E]the gap is \Delta=-\ln((2/3)^6)=6\ln(3/2)=2\Nfam\ln(3/2)\approx2.4328>0 (multiplicative gap 1-(2/3)^6=665/729; (2/3)^6 the \mu_4-deck transfer eigenvalue forced by the carrier); [E]with the decoupling margin \Delta-31/(4\pi^2)\approx1.648>0 (v76) it is robust, and Perron–Frobenius primitivity makes the leading mode simple; [C]by Osterwalder–Schrader / quasi-free clustering a transfer-operator gap \Delta>0 is a mass gap of the reconstructed Hamiltonian. [O]Bottom line: with v300/v301, SEAM.EQUIV.01's entire residual is now a composition of standard cited theorems (OS/quasi-free clustering + Kitaev free-fermion invertibility + the v297 AQFT stack) over established TFPT facts (16 quasi-free Majoranas; transfer gap 6\ln(3/2)) — no undischarged TFPT-internal assumption remains. It 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. [E]suite v302 5/5, AUDIT OK.

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• v301_route_a_invertible.py attacks the single remaining unconditional gap of SEAM.EQUIV.01. Since v300 collapsed Route B into Route A's rationality, that gap is Route A's open hypothesis (v297 LIT-A): “the raw quasi-free seam bulk is invertible (SRE)”. A gapped free-fermion (Gaussian) 2{+}1D bulk cannot carry intrinsic topological order — free fermions give the integer (IQHE/Chern) phases, all invertible; intrinsic topological order needs interactions (Kitaev's periodic table, class D). [E]the carrier is 16 Majoranas, c=g_{\mathrm{car}}{+}\Nfam=8=16/2 (v148); [E]SO(16)_1\supset(E_8)_1 (\dim\mathfrak{so}(16)=120 currents +\,128 spinor =248, v154); [C]so a gapped quasi-free bulk is invertible automatically; [E]the invariant |\det K|=\#anyons is 1 for E_8 (invertible) vs 4 for the gauged D_8{=}SO(16) (anyons), which free fermions cannot produce. [O]LIT-A's `invertible bulk' now follows from `gapped 16-Majorana quasi-free bulk' (v148/v160+the mass gap), so the residual changes from a topological-order statement to a spectral one. [O]Bottom line: with v300, the whole open content of SEAM.EQUIV.01 reduces to the single spectral statement “the quasi-free seam bulk is gapped” — the topological-order obstruction (hidden anyons) is removed; the gap is not manufactured. [E]suite v301 6/6, AUDIT OK.

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• v300_flataway_rigid.py attacks the single external fact the three Flat-Away routes (v291/v293/v295) left open, on two fronts. Harden: the flat seam was only a soft minimum (“the spectral mismatch grows with \varepsilon”); here the obstruction becomes a discrete on/off invariant. [E]the flat fingerprint — \mathrm{spec}(|D_\theta|) is the integer ladder with every level n\ge1 of multiplicity exactly 2; [E]the resonant mode 4k splits level 2k by gap =g_k (sympy 2\times2 [[2k,g/2],[g/2,2k]]\Rightarrow 2k\pm g/2, + an O(g^2) common shift), validated against the full Toeplitz operator; [E]any g_k\ne0 lifts the resonant degeneracy (2\to1{+}1) and changes the spectral multiset, so preserving the flat ladder with its mult-2 degeneracies forces every g_k=0\Rightarrow f=0 — strictly stronger than the “deviation grows” scan; [E] with the v295 convexity, flat is isolated analytically and discretely. Derive the pin: [C] the (E_8)_1 character E_4/\eta^8=q^{-1/3}(1+248q+4124q^2+\dots) has integer q-coefficients (248 at level 1; E_8 even unimodular \Rightarrow integer weights), so by 2d Steklov conformal rigidity the integer ladder \Leftrightarrow flat seam, i.e. the external pin is the (E_8)_1 rationality (Route A). [O] verdict: Flat-Away carries zero new open content beyond Route A — the two SEAM.EQUIV.01 routes converge on one rationality fact. [E]suite v300 6/6, AUDIT OK.

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• v299_hypergraph_0d_autonomous.py hardens the Wolfram/hypergraph reading into a precisely typed bridge (builds on v219/v298): [E]the icosahedron is a 3-uniform hypergraph (|\mathrm{Aut(graph)}|=120=|I_h|=|A_5{\times}\mathbb{Z}_2|; the McKay group is the SU(2) binary lift 2I, a different order-120 group whose McKay graph is affine E_8; 30 edges =h(E_8)); [E]the 1\to4 subdivision rewrite is equivariant (all 120 symmetries at every scale); [E]the 2I fusion tensor is a weighted 3-uniform hypergraph whose 2-dim slice is the affine E_8 McKay graph; [E]Smith (\rho\le2\Leftrightarrow affine ADE) + Collatz–Wielandt (\rho\le2\Leftrightarrow a local witness Am\le2m); [E]* the autonomous rule (diffusion-generated witness + local gate Am\le2m, Perron only as audit) runs E_6\to E_7\to E_8\to\hat E_8; [E]the negative controls \rho(A_n)=2\cos\frac{\pi}{n+1} and \rho(D_n)=2\cos\frac{\pi}{2n-2} are <2 for all n (closed form), so neither family ever reaches E_8. Honest type: [O]the (2,3,\cdot) seed is the single irreducible input = P2 — the rule reaches E_8 as the last finite point on this seed; it is not a derivation of P2. [E]build green, AUDIT OK.