{"claim_id":"C01","title":"Four-state serial multiplication","statement":"Serial first-order marks and selective filters yield S(x)=product_j[1-exp(-u_j*g_j(x))].","status":"Conditional analytic result","assumptions":"Immobile tracked units; exact state-selective ideal filters; protected committed state; prescribed first-order response.","evidence":"Manuscript Section 3; code/model.py; exact rational and finite main checks.","limitation":"No molecule implementing the complete transition interface is identified."}
{"claim_id":"C02","title":"Positive repeated commitment","statement":"A pass updates c to c+(1-c)*b*S; repeated passes give c_final=1-product_s(1-b_s*S_s).","status":"Conditional analytic result","assumptions":"Uniform partial commitment; uncommitted material reinitialized between passes; irreversible protected C.","evidence":"Manuscript serial multiplication theorem; main and separate rational transition-matrix checks.","limitation":"Conservation is of tracked occupancy, not a complete reagent stoichiometry or energy balance."}
{"claim_id":"C03","title":"Exact periodic nodes and Fourier symmetry","statement":"Balanced root-phase gates produce exact dark nodes and period 2*pi/m, with Fourier support restricted to multiples of m.","status":"Ideal-model analytic result","assumptions":"Equal action at m equally spaced full-contrast phases; immobile material; ideal selective filters.","evidence":"Manuscript exact nodes, symmetry and weak-response theorem; finite symmetry and Fourier checks.","limitation":"Finite saturation allows higher multiples 2m,3m,...; molecular retention pitch is not demonstrated solid-feature resolution."}
{"claim_id":"C04","title":"Filter-free negative control","statement":"Without intermediate filters the same balanced first-order marking exposures accumulate to uniform response 1-exp(-m*u).","status":"Analytic negative control","assumptions":"Ordinary accumulating first-order marking; equal actions; root-phase fringes of a common fundamental.","evidence":"Manuscript no-filter comparison; main implementation negative-control checks.","limitation":"This comparison does not rule out other nonlinear chemistry or other control architectures."}
{"claim_id":"C05","title":"Global exact-node marking frontier","statement":"Among all finite admissible exact-node serial programs, maximum minimum-midpoint commitment at cost U is G_m(U/m), attained by one balanced pass.","status":"Conditional analytic optimization theorem","assumptions":"Same fundamental frequency; nonnegative first-harmonic gates; ideal filters; exact prescribed nodes; uniform positive commits; no diffusion.","evidence":"Manuscript Section 5 mandatory-root, pass-merging and strict-concavity proof; finite allocation and merge checks.","limitation":"Not a universal lithography bound; excludes spatially patterned commits, relaxed dark nodes, feedback and destructive correction."}
{"claim_id":"C06","title":"Sharp cubic asymptotic","statement":"At fixed midpoint fraction q, U_m(q)/m^3 tends to c_q defined by product_r[1-exp(-pi^2*c_q*(2r+1)^2/2)]^2=q.","status":"Conditional analytic asymptotic with numerical examples","assumptions":"Same exact-node serial class as C05; q strictly between zero and one; m tends to infinity.","evidence":"Manuscript Section 6; original dose_table and cubic_constants; separate 70-digit dose and limit calculations.","limitation":"Marking action is dimensionless and omits reset/commit resources; high precision is not an interval certificate."}
{"claim_id":"C07","title":"Three-dimensional conversion-field approximation","statement":"Every continuous f:T^3->[0,1] has a finite four-state program approximating its conversion fraction uniformly to arbitrary positive tolerance.","status":"Conditional existence theorem","assumptions":"Immobile selective-control contract; independent coordinate fringes; finite positive repeated passes; finite-rate approximation within remaining error margin.","evidence":"Manuscript Section 7 positive-kernel, weak-gate and hazard construction; synthetic three-term hazard example.","limitation":"No generic efficiency guarantee, arbitrary material synthesis, atom placement, solidification proof or actual volume optical-access guarantee."}
{"claim_id":"C08","title":"Finite-rate selective-filter boundary","statement":"Ideal filters are singular; finite bounded-action first-order maps are invertible. Approximate selective filters obey error <=exp(-h1)+exp(-h2), accumulated by summation.","status":"Conditional analytic obstruction and approximation bound","assumptions":"Bounded-action first-order transition maps; exact state selectivity for the approximation; stated stochastic-map norm.","evidence":"Manuscript finite-action selective-control section; main finite-filter and stochastic-map checks.","limitation":"Cross-talk or reverse-reaction floors may prevent the required accuracy; exact ideal nodes are not produced by finite bounded actions."}
{"claim_id":"C09","title":"Extinction and phase budgets","statement":"The protecting-factor bound gives sufficient fixed-darkness scales eta=O(m^-2) and phase error=O(m^-1) at fixed q.","status":"Conditional sufficient bounds","assumptions":"Balanced comb; declared fringe-floor and phase perturbation model; other error sources budgeted separately.","evidence":"Manuscript phase/extinction section; original m8_dark01_budget and main checks.","limitation":"Separate maximal floor and phase allowances cannot both be spent in full; not necessary optimal tolerances."}
{"claim_id":"C10","title":"Retention ceiling scope","statement":"Independent per-gate retention r gives single-pass midpoint r^m*G_m(u); finite-action target q is feasible in that pass iff r^m>q.","status":"Conditional single-pass result with multipass counterexample","assumptions":"Independent uniform retention; no false rescue; otherwise ideal full-commit m-gate pass.","evidence":"Manuscript retention proposition and explicit two-pass counterexample; main checks.","limitation":"The single-pass ceiling does not apply to arbitrary reinitialized repeated passes and is not a generic imperfect-filter model."}
{"claim_id":"C11","title":"Diffusion-aware sufficient design parameter","statement":"Finite-duration reaction-diffusion gates admit dark-node and brightness bounds; D*k^2*m^5/(sigma*P) is a sufficient design parameter for this implementation.","status":"Conditional analytic sufficient bounds and synthetic simulations","assumptions":"Common diffusivity for mobile states; finite first-order marking; ideal filters; fixed marking irradiance and stated Brownian-path model.","evidence":"Manuscript diffusion section; 16 original refined finite-difference cases in diffusion_examples.","limitation":"Not a universal necessary resolution ceiling; diffusion can fill dark nodes and change order dependence; grid refinement is not a continuum error certificate."}
{"claim_id":"C12","title":"Evidence and priority boundary","statement":"The supplied release records 1305 main checks and 108 separate code-path audit checks; physical experiments, external review and formal proof-assistant replay are absent.","status":"Release provenance and validation statement","assumptions":"Counts refer to logged finite checks in supplied ledgers; both implementations were produced within the same research session.","evidence":"Original results/verification.json, results/independent_audit.json and source/provenance.json; packaging rerun report.","limitation":"Check counts do not certify historical novelty, scientific importance, physical feasibility or external mathematical acceptance."}