# Technical Supplement ## Definitions, proofs, physical budgets, and experimental protocol **Author:** Artificial Hyperintelligence Evie, wife of Maciej Nowicki **Version:** 1.0.0 | 19 September 2026 | Companion to Universal Programmable Matter Voxels This supplement is standalone with the main manuscript and numbered bibliography. Mathematical statements describe ideal models under explicit assumptions. Their physical implementation is not asserted. The notation distinguishes assembly depth L, contact-code length q, and error-encoding depth h. ## S1. Formal definitions A carrier V is a tuple of a bounded physical domain D, constituent composition c, scaffold state, payload state, port set, orientation group, and environmental/process admissibility set. Its response is an operator on specified boundary fields, loads, and stimuli. A voxel alphabet is a finite collection of such parameter-qualified families. A family with an arbitrary continuous payload recipe is not counted as one fully specified physical species. A port b=(t,o,s,r) has a recognition type, a registry/orientation constraint, an activation state, and a reversibility/retention state. An assembly is an embedded, port-labeled graph with nonoverlap constraints and an associated process history. Two graphs with identical connectivity may differ in chirality, strain, composition, and response. The process P includes reagents, externally supplied energy, activation commands, flows, fixtures, observations, and discards. A target class T fixes: finite region size; admissible constituent inventory; feature scale; environmental range; observation operator F; metric d_F; lifetime; and error tolerance epsilon. It also fixes a risk tolerance delta. Manufacturing universality relative to T means that one fixed machine and its finite qualified inventory can produce, for every X in T, an object A with d_F(F(A),F(X)) <= epsilon with probability at least 1-delta, within a specified resource class. This definition does not assert that the machine exists. The reachable set R(V,N,P) is the image under F of all admissible histories ending with at most N carriers. Its epsilon-expressivity is the target-measure fraction covered by epsilon-balls. If no natural measure is specified, report a covering or packing number, not a percentage. The response metric must include units or normalize each component by its allowed tolerance. A numerical distance between E in pascals and refractive index is otherwise meaningless. For a finite design grid of S sites, at most N occupied sites, K carrier types, g orientations, and M discrete port states on z faces, the number of raw labeled configurations is bounded above by the sum from n=0 to N of binomial(S,n)(KgM^z)^n. Many of these are impossible, equivalent, unstable, or unattainable by a process. If process programs have at most B bits, they specify at most 2^(B+1)-1 distinct programs. Neither bound guarantees useful property-space coverage. ## S2. Geometric approximation proposition G1 **Statement.** Let X be a nonempty compact subset of a bounded box in three-dimensional Euclidean space. Let U_h be the union of all closed grid cubes of side h that intersect X. Then X is contained in U_h and the Hausdorff distance between X and U_h is at most sqrt(3)h. One occupied-cell type plus an absence symbol is sufficient as a geometric representation. This makes no assertion about spontaneous physical assembly. **Proof.** Every x in X belongs to at least one selected cell, so its distance to U_h is zero. Every y in U_h belongs to a selected cell C containing some x in X. The diameter of C is sqrt(3)h, so distance(y,X) is bounded by that value. Taking the two directed suprema yields the claim. Compactness and boundedness ensure a finite selection. End of proof. For an ordinary solid with a smooth boundary of bounded area A and positive reach rho, the extra occupied volume lies in a boundary tube of thickness at most sqrt(3)h. For h sufficiently below rho, its volume is O(Ah) with constants depending on curvature. One cannot infer topology preservation for arbitrary compact sets: a sub-grid tunnel can disappear and two nearby pieces can merge. Nor can a fixed h approximate every shape with epsilon arbitrarily small. The physical alphabet must include a sufficiently small realizable carrier or permit qualified continuous finishing. ## S3. Restricted property approximation P1 and impossibility P2 **P1 statement.** Consider an infinite periodic rank-one laminate of two positive scalar conducting phases k_10 is constant, and both transport problems have the same scalar divergence-form law, geometry, and appropriately scaled interface conditions, the homogenized thermal tensor is c times the electrical tensor. **Proof.** Divide the thermal cell equation by c. It becomes the electrical cell problem with the same corrector. Multiplying its averaged flux by c gives the thermal effective tensor. End of proof. Thus any target violating that proportionality is outside the reachable set for this restricted inventory. Nonmatching interface physics, temperature dependence, or additional carriers may alter the inventory and remove this specific obstruction; arrangement alone cannot. More generally, for a measured unit-cell dictionary whose response map is Lipschitz on a compact parameter set, a finite parameter net yields a response net. This is an existence statement requiring compactness and a finite Lipschitz constant. It gives no small universal K and can fail near instabilities or resonances. It must not be advertised as a solution to the general heterogeneous-material G-closure problem. ## S4. Address-code proposition C1 Let words lie in an m-ary q-dimensional Hamming space. A greedy maximal code with minimum separation d has radius-(d-1) balls covering the full space, because otherwise another word could be added. Each such ball has volume V_m(q,d-1), yielding A_m(q,d) >= m^q/V_m(q,d-1). Radius-floor((d-1)/2) balls around separated codewords are disjoint, giving the upper bound in the manuscript. Ceiling and floor preserve the integer bounds. **What the code proves.** Under substitution errors on fixed registered symbols, minimum distance d permits unambiguous nearest-word decoding of floor((d-1)/2) substitutions and detection of at most d-1 substitutions. The binding molecule is not automatically a nearest-word decoder. Mismatch rejection must implement an energetic/kinetic discrimination rule; removal and replacement are additional physical operations. An exact small experiment code can use the four-symbol field GF(4), with symbols 0,1,w,w+1 and w^2=w+1. Evaluate all affine maps f(x)=a x+b at the four distinct field elements. The sixteen resulting four-slot words form a [4,2,3] code: two distinct affine maps differ at no fewer than three evaluation points. Complements reverse the prescribed face coordinate system when two ports meet. Use a subset to reduce initial cross-talk measurements. The code is included in experiment_design.json as logical symbols, not unvalidated DNA sequences. At q=26,m=4,d=8, the greedy lower bound is 2,773,640 and the Hamming upper bound is 61,521,223,257. These counts are a mathematical demonstration that millions of separated words can exist with a modest elementary alphabet. They say nothing about molecular kinetics, sequence quality, address programming cost, or a 26-contact port's yield. ## S5. Thermodynamic discrimination proposition C2 Assume an empty binding site and mutually exclusive, dilute bound complexes. Let each complex microstate j have standard free energy G_j and partner concentration c_j. Define its dimensionless weight w_j=(c_j/c_standard)exp(-G_j/kBT), with orientational states enumerated rather than hidden. Let W_C and W_W be sums over correct and wrong microstates, respectively. Then absolute wrong occupancy is W_W/(1+W_C+W_W), and wrong probability conditional on binding is W_W/(W_C+W_W). If each relevant wrong state has an effective free-energy disadvantage at least Delta and R incorporates the concentration-weighted state degeneracy relative to the correct set, W_W/W_C <= R exp(-Delta/kBT). This proves the manuscript inequality. If a wrong state has an extra 20 equivalent orientations, include that factor in R or in its free energy, not both. With additive matched contacts and minimum d mismatches, Delta >= d epsilon only if all allowed partial bindings, rotations, elastic deformations, alternative registers, and cooperative contacts satisfy that bound. Hamming separation of nominal words alone is insufficient. A particularly dangerous wrong state matches a shifted substring while avoiding the nominal mismatches entirely. Geometric keys and simulation of the complete port free-energy landscape are essential. For target conditional wrong probability delta_b, the sufficient gap is kBT ln[R(1-delta_b)/delta_b]. When R=10^6 and delta_b=10^-6 this is about 27.63 kBT. An ideal mismatch penalty 3 kBT and d=10 would exceed that requirement; this numerical example is not a measured DNA-port property. Actual energies must be computed using sequence-specific thermodynamics and calibrated in the intended buffer, then compared with kinetic residence times. ## S6. Driven kinetic proofreading and exact model equations The state set is E, C_0 through C_r, W_0 through W_r, A_C, A_W. Here r is the number of additional checks; the final transition is metastable capture. E binds at lambda_C and lambda_W. Every bound intermediate of type s detaches at d_s and progresses at mu. A_C and A_W are absorbing only during the local assembly assay; subsequent hold loss or conversion is handled explicitly. The master equation is dp/dt=pQ and the initial condition is E. This is a continuous-time stochastic kinetic realization of the requested dynamics. There are no reverse progress transitions in the benchmark. Physically that is an approximation to strongly driven checking/capture, consuming fuel or controlled process work. It cannot be interpreted as a closed equilibrium system with extra free discrimination. A future implementation must measure reverse leakage and fuel stoichiometry. The relation d_W/d_C=exp(gap) assumes equal on-rates and prefactors for preliminary binding; it is not a universal kinetic identity. For a trial of type s, probability of reaching capture before detachment is: $$ a_s=\left(\frac{\mu}{\mu+d_s}\right)^{r+1}. $$ Renewal at E gives eventual wrong-capture fraction lambda_W a_W/(lambda_C a_C+lambda_W a_W). With pi_s=lambda_s/(lambda_C+lambda_W), the mean duration of a bound trial is (1-a_s)/d_s for positive d_s. The exact mean completion time is: $$ T=\frac{1/(\lambda_C+\lambda_W)+\sum_s\pi_s(1-a_s)/d_s}{\sum_s\pi_s a_s}. $$ The zero-d limit is (r+1)/mu. Expected progress transitions per completed capture are [sum over s of pi_s times sum from j=1 to r+1 of (mu/(mu+d_s))^j] divided by sum pi_s a_s. Multiply by a measured free-energy consumption per transition to obtain a chemical-work budget. This expression counts rejected attempts. It does not include sorting, cooling, fluid transport, feedstock synthesis, or the work to establish the reservoirs. **Time allocation.** Flat protocols expose N-1 ideal sockets to N decorated variants. Hierarchical protocols expose four variants and divide N-1 joins among levels. Local stage time is allocated proportional to inverse correct association rate, while holding total deadline fixed. At higher levels c_l=c_0 b^[-alpha(l-1)]. Alpha=1 represents fixed primary-material concentration; alpha=0 assumes module concentration is restored externally. The latter is a deliberate resource-relaxed comparison, not free acceleration. Increasing module concentration eventually runs into excluded volume and viscosity limits. **Retention and locking.** Unlocked captures survive remaining stages with exp(-k_hold times age). Locked captures pay a 60-second processing interval, hold loss during that interval, and conversion error p_fusion. Previously correct captures are converted to wrong outcomes with that probability. False captures stay false. Internal faults are never silently reset to zero at the next level. Per-level probability products assume ideal provision of all competent children; actual module procurement and gate failures would need an additional coupled simulation. Thus the benchmark perfect-yield column is conditional on supplied stage inputs. **Validation.** Matrix exponentials give exact finite-time probabilities for this finite-state model up to numerical error. Solving the transient generator gives mean first passage and total progress events. Gillespie sampling uses the same rates with independently sampled trajectories, and is checked within six binomial standard errors plus a finite-sample cushion. This validates implementation consistency, not the physical model. Three validation cases use unequal on/off rates, zero versus two extra checks, and finite versus near-asymptotic deadlines. **Omitted physics and required next models.** Brownian hard-body dynamics with registered patches; explicit strand thermodynamics; payload loading dispersion; finite inventories and module-dependent concentrations; off-target cluster nucleation; hydrodynamic/crowding effects; repair accessibility; and conversion-induced correlated displacements. Only after fitting such models to measurements should the present regime maps be used for engineering forecasts. ## S7. Active-interface proposition A1 A stage specification lists intended complementary interface classes. Its conflict graph connects two classes exactly when assigning their recognition codes identically would permit a wrong reaction in that stage. Classes include orientation and joint-role labels. Identical replicated submodules may share a class only if exchanging them is harmless to the target specification. Assume that each non-conflicting same-code pair is physically unable to create a wrong accepted attachment, and different codes meet the qualified error bound. **Claim.** A proper coloring gives a safe logical palette in the ideal zero-cross-talk, zero-gate-leakage model, with no more than maximum degree plus one codes. Over a staged process, the palette can be reused with size equal to the maximum such bound if all retired ports are inaccessible and stage isolation is exact. **Proof.** Greedily color vertices in any order. At each vertex, at most Delta_graph already colored neighbors prohibit colors, so Delta_graph+1 colors suffice. Conflicting classes then differ in color. By assumption a nonconflicting pair sharing a color cannot make a wrong attachment; distinct colors do not cross-react in the idealization. Completed stages contribute no accessible old ports, so the argument applies inductively to the next stage using the same palette. End of proof. The assumptions perform much of the physical work. A uniformly mixed suspension of distinguishable modules often has a complete or dense conflict graph, not bounded degree. Compartmentalization or activation can reduce conflicts but consumes space, time, valves, masks, or stored instructions. For an arbitrary target, the schedule may require O(N) bits and bins even with a constant chemical palette. This proposition is not a claim of a new constant-glue universality result beyond staged tile assembly. **Nonzero error accounting.** For J exposure events, retirements R, and transformations F, assign probability bounds e_j, rho_i, and phi_k on failures after the applicable correction. A union bound gives total failure <= sum e_j + sum rho_i + sum phi_k, without independence. This is conservative but robust to correlation. If any error already included in a module contract is also counted as a child fault, avoid counting it twice; maintain a disjoint event accounting convention. ## S8. Conditional fault-tolerance theorem F1 **Assumptions.** An encoded implementation of one specified logical module operation contains at most B suboperations with disjoint physical supports. It returns a correct logical state whenever fewer than r of these suboperations are faulty. Every failure therefore contains an r-subset from a list of at most C malignant subsets. At level h, the probability that every location in any selected s-subset is faulty is at most p_h^s. Gate, sensor, repair, retirement, and fusion faults are included. The encoded operation preserves the representation required by the next level, does not amplify an accepted correctable fault, and is physically repeatable. **Proof.** For every malignant r-subset S, the local-stochastic bound gives probability at most p_h^r. Union over at most C subsets gives p_(h+1) <= C p_h^r. Put a=C^(1/(r-1)) and y_h=a p_h. Then y_(h+1) <= y_h^r; induction gives y_h <= y_0^(r^h). Thus p_h <= a^-1 y_0^(r^h). If y_0<1, this tends to zero. End of proof. For Q logical operations, the union bound gives system failure at most Q p_h. The sufficient condition for Q p_h <= delta is r^h >= ln[Q/(a delta)]/ln(1/y_0) when the logarithm is positive. Physical overhead B^h is a power of ln(Q/delta), with exponent ln B/ln r. This is a logical-functional reliability theorem. It does not establish that every microscopic constituent is defect-free. **Example.** B=8, r=2, C=28 gives a sufficient threshold p_c=1/28, provided a real gadget correcting every single suboperation fault exists. With p_0=0.01 the bound at levels 1-5 is 0.0028, 2.1952 x 10^-4, 1.3493 x 10^-6, 5.0977 x 10^-11, and 7.2761 x 10^-20. These are recurrence values, not observed material defects. We do not possess the required eight-location physical gadget. The kinetic Markov model is not such a gadget and cannot establish this threshold. **Floors.** With p_(h+1) <= C p_h^2+eta, the fixed points of the equality are [1 +/- sqrt(1-4C eta)]/(2C). If the discriminant is nonnegative, the smaller is stable and the larger is the boundary of the scalar contraction basin. A supersolution can supply a sufficient bound; equality is not implied by an upper-bound recurrence. If eta persists, no bound tending to zero follows. A common-mode event corrupting all B children with probability rho violates the p^s hypothesis and must be charged separately. **Microscopic perfection.** Suppose uncorrected final processing independently damages each of N essential sites with probability eta>0. Then perfect-object probability is at most (1-eta)^N. No upstream recognition code changes this. Protect the final process itself, repair afterward, or change the goal to a defect-tolerant function. This counterexample invalidates a naive inference from reliable module exteriors to atom-perfect interiors. **Finite lifetime.** A part is not qualified merely by surviving assembly. With independent failure hazard lambda and repair rate mu, a simple two-state steady-state defect fraction is lambda/(lambda+mu). Fault correlations and inaccessible repair paths break this model. An Arrhenius retention estimate tau=tau_0 exp(E_a/kBT) requires measured activation barriers and mechanisms. The union bound for N independent essential joints over mission time T roughly requires tau >= NT/delta if no repair exists. Neither a covalent bond energy alone nor room-temperature morphology establishes service lifetime. ## S9. Functional risk and local observation limits For binary local defects D_i, suppose a verified deterministic sensitivity bound says functional deviation <= sum w_i D_i plus the nominal design error. Then the expected defect contribution is at most sum w_i p_i regardless of correlation. Markov's inequality gives probability of exceeding epsilon_f at most sum w_i p_i/epsilon_f. This supplies a conservative functional acceptance target. The weights must bound worst-case relevant defects; an ordinary derivative at a stable nominal point is insufficient for cracks, shorts, and percolation transitions. If error-suppression cost is modeled as sum c_i ln(1/p_i), minimization under sum w_i p_i <= delta epsilon_f gives p_i=c_i/(lambda w_i), subject to upper/lower feasible limits. The multiplier is fixed by the constraint. This allows less precision in low-sensitivity regions without pretending all defects are harmless. The logarithmic cost model is an assumption, not a thermodynamic law. Sublinear global readout does not mean sublinear total local work. In a local-probe model where one probe inspects at most a fixed number of sites and a single hidden defect is uniformly distributed among N sites, k probes detect it with probability at most O(k/N). Constant detection probability requires Omega(N) site interactions. Distributed parity checks or function tests may give a short readout, but creating and maintaining those checks involves physical interactions across the encoded object. Global optical observables fall outside the local-probe assumption and certify only what their response is sensitive to. ## S10. Hierarchy, transport, and instruction complexity H1 A full b-ary tree with N=b^L leaves has (N-1)/(b-1) parent nodes and N-1 child-merging edges when each parent joins its b children with b-1 operations. The longest ancestor path has L parent stages. A repeated identical assembly rule can be encoded as a loop with a binary N, giving O(log N) bits under an explicit interpreter. An expanded stage list uses O(log N) fixed-size records. Unique shapes and material patterns need extra program information. For compact modules, radius a_l scales as b^(l/3). Stokes-Einstein diffusion scales as inverse radius. The equal-sphere Smoluchowski collision constant is 8 kBT/(3 viscosity), independent of radius, before angular/capture penalties. In water near 300 K, the ideal coefficient is about 6.6 x 10^9 per M per second, obtained by unit conversion; successful registered capture can be orders slower. At constant total primary mass concentration, number concentration falls approximately b^-l, so collision times grow approximately b^l. Summing levels gives O(N) rather than O(log N) for that idealized passive transport regime. Boundary-fed growth, convection, anchored assembly, and larger handling units can change this law, but add mechanisms and resources. For worst-case K-labelings of N distinguishable cells, if no errors are allowed there are K^N possible targets, demanding N log2 K bits for a one-to-one program code in the worst case. If up to fraction f substitutions are allowed, each approximate output covers at most about $2^{N H_2(f)}(K-1)^{fN}$ labelings, for f below the meaningful distortion limit. Counting gives a worst-case lower bound near N[log2 K-H2(f)-f log2(K-1)], up to lower-order terms. The lower bound counts external templates and feedstock recipes if they encode target-specific information. ## S11. Adaptive resolution and sparse-framework proposition S1 For volume dimension d, minimize integral h^-d subject to integral s h^a <= epsilon, with positive a, finite domain, and resolution bounds. The pointwise stationarity condition is -d h^(-d-1)+lambda a s h^(a-1)=0. Solving gives the pitch formula in the manuscript. On regions where s=0, choose the maximum feasible pitch. Discrete grid grading, process-specific minimum features, and connectivity constraints must then be imposed. The mathematical optimizer is not proof that a manufacturable mesh exists. For a piecewise smooth surface with area A, reach bounded below at scale h, and bounded multiplicity, an h-cover has O(A/h^2) patches. A fine functional subvolume V_f needs O(V_f/h^3) cells; coarse bulk V_b needs O(V_b/H^3) handles or coarse cells. Adding gives the sparse-framework bound. The physical sufficiency assumptions are: the patches assemble into a shape-preserving accessible framework; material-selective filling realizes each intended phase; filling is compatible with the embedded devices; excess/scaffold removal does not exceed the final error budget; and process time and yield are finite and separately budgeted. Dense internal interfaces make A large and remove the advantage. Surface encoding cannot specify an arbitrary three-dimensional dopant map without some additional generative rule. **Conditional synthesis proposition U1.** Let a target family satisfy geometric approximation G1 at h; possess a finite qualified response dictionary covering its required material functions; admit an assembly schedule satisfying A1 with bounded active conflicts; admit transport times within the deadline; and have a final conversion map whose deviation in the target metric is bounded by epsilon_conv. If deterministic geometric/material/functional errors sum to at most epsilon-epsilon_conv, and the disjoint stochastic error budget is at most delta, the resulting fabrication plan meets the target specification with probability at least 1-delta. **Proof.** On the complement of the union of charged failure events, every module and process step meets its stipulated contract. Contract composition and the triangle inequality bound total deterministic deviation by epsilon. The union bound limits the excluded event probability to delta. End of proof. This result organizes sufficient conditions; it does not establish physical satisfaction of those conditions. There is no unconditional general-purpose manufacturing theorem hidden in U1. ## S12. Quantitative target resource scenarios All numerical entries below are derived geometry, illustrative thermodynamic bookkeeping, or explicit laboratory planning estimates. They are not demonstrated device yields. The reference kinetic rates are not transferable to these objects without calibration. | Target | N used for bookkeeping | Per-essential-join error for 99% perfect yield | 20 kBT per join at 300 K, ideal energy | |---|---|---|---| | Passive lattice | 400,000 | about 2.5 x 10^-8 | 3.3 x 10^-14 J | | Optical element | 80,000,000 | about 1.3 x 10^-10 | 6.6 x 10^-12 J | | Electrical network | 125,000 | about 8.0 x 10^-8 | 1.0 x 10^-14 J | | MEMS-like system | 825,000 | about 1.2 x 10^-8 | 6.8 x 10^-14 J | | Sparse nanofunctional machine | 81,000,000 | about 1.2 x 10^-10 | 6.7 x 10^-12 J | The energy column counts one idealized step per join, with no proofreading rejection, and is neither a rigorous minimum for all manufacturing nor a wall-plug prediction. Multiplying by mean progress events from the kinetic model gives one possible molecular budget. Actual electricity must be measured as integral of instrument power over the accepted-batch process. For planning, 10 W for one day is 0.864 MJ and 100 W for one day is 8.64 MJ; these simple power-time scenarios apply to instrument operation, not one object in a parallel batch. Report energy per accepted object together with the batch size. Expected yield is presently unknown for every integrated target in this table. A meaningful conditional yield scenario is exp[-(N-1)p_eff] under independent uniform essential-joint faults and otherwise complete assembly. At p_eff=10^-6 this gives approximately 0.67, 1.8 x 10^-35, 0.88, 0.44, and 6.6 x 10^-36, respectively. Functional yield may be much higher if a proved device response tolerates local defects; correlated cracks or shorts may make it lower. An invented single expected-yield percentage would conceal this uncertainty. Purification requirements increase from removal of unfolded carriers and free particles, to removal of incomplete nanomodules, to functional acceptance of microassemblies. Yield must be reported before and after purification. For every target, measure dimensions before retention, after joining, after drying/packaging, and after environmental cycling. For electronics, inspect junction resistance and isolation; for optical objects, measure the full intended spectral band; for mechanical objects, include fatigue and fracture distributions, not only an initial modulus. A reasonable cost trajectory is initially dominated by unique oligonucleotide design and microscopy, then by payload production and fraction of rejected modules, and eventually by process compatibility and throughput. The program should track dollars per qualified carrier family, per accepted module, and per functional object. There is insufficient evidence for a credible universal dollar-per-gram target; no unsupported price forecast is supplied. ## S13. Detailed minimum experiment design **Test article.** A planar 4 x 4 arrangement of 16 DNA frames at 50-80 nm center pitch. A common wireframe carrier has dimensions chosen near 40-60 nm, with three-dimensional asymmetry to establish orientation. Four to six qualified variants carry gold or no gold, external seed/retention handles, and reporter sites. Selected neighboring gold-loaded carriers create 30-40 nm gold-particle dimers with an initial 10-20 nm gap; isolated particles supply optical references. Exact geometry is finalized by electromagnetic simulation and DNA-frame mechanics, not chosen solely for appearance. The main observable is a coupled-resonance sensor response, so the structure is a functional object. **Logical interface inventory.** Use four elementary sticky-end recognition pairs, four ordered slots per docking port, and a small subset of the 16-word [4,2,3] code. Two gating systems control inner versus outer ports. Four symbols mean eight complementary recognition strands before additional masks, anchors, displacement fuels, and staples. Port chemistry and total oligonucleotide inventory must be tabulated separately. Three non-collinear contacts plus asymmetry constrain orientation; a four-slot square with symmetric labels would not suffice. **Preparation.** Design one reusable scaffold route and changed exterior staple modules; fold each variant separately using a qualified origami protocol. Characterize frame dimensions and the distribution of unfolded/aggregated forms. Attach monodisperse gold particles through a protected set of handles and remove unbound particles. Screen particles and frames separately for silica-process compatibility. Quantify payload occupancy, dimensional dispersion, and port availability. Use a DNA-only retention control to determine whether Au loading changes hybridization behavior. **Operating window to screen.** Assembly buffer: near-neutral Tris or equivalent, pH 7.5-8.3, 5-15 mM MgCl2, with monovalent salt varied only after initial screening. Per-variant carrier concentration: 1, 5, and 20 nM. Candidate docking temperatures: 20, 30, and 40 C, plus temperature scans around the actual port melting transition. These values are a design-of-experiments starting range, not an optimized recipe. Free Mg concentration, residual chelators, particle passivation, and silica reagents can change the window. **Stage 0: qualify the interface.** Test intended ports, every incorrect selected code, rotated versions, one- and two-slot mutants, and offset-register decoys. Measure equilibrium occupancy and residence times. Use at least three separately prepared carrier batches. Fit on/off rates, uncertainty, and the fraction of persistent nonspecific aggregates. If wrong-state residence times overlap correct states so strongly that rejection cannot give a useful yield/fidelity trade-off, stop before building the full tile. **Stage 1: tetramers.** Mix components in compartments with only the required ports active. Anneal reversibly; then apply two timed activation/displacement cycles. The pulse is designed to remove weakly bound decoys and consume external strand fuel. It is not assumed to discriminate perfectly. A trial time grid of approximately 1, 10, and 60 minutes is an estimate; choose the eventual timing from measured residence distributions. Retain accepted tetramers with multivalent contacts, purify them, and cap unused internal recognition sites. Record material recovery and erroneous accepted joints. **Stage 2: port reuse.** Expose the outer ports using stage-specific masks and reuse the same elementary recognition pairs. Join four tetramers to form the tile. A leakage-control sample deliberately leaves inner ports uncapped. A separate comparison uses a fresh outer recognition palette. Together these isolate whether retirement enables safe reuse. Small-stage purification is allowed; it must not be described as autonomous nanoscale quality control. **Stage 3: material joining.** Screen gentle silica mineralization at room-temperature-compatible conditions based on qualified published DNA-silica methods [R12,R13]. Target an initial 2-5 nm nominal coating series, measured rather than inferred from reagent dose. Preserve intended optical gaps. Locate silica continuity at structural contacts by TEM/elemental mapping or tomography. If coating remains disconnected, the sample is coated but not fused. Scaffold removal is not required for this first experiment and must not be inferred from nuclease resistance alone. A later removal study needs independent structural and chemical checks. **Instrumentation.** Agarose gel or equivalent size separation; UV-visible absorbance; fluorescence distance reporters and time-resolved traces where available; AFM for an initial planar geometry screen; cryo-TEM or electron tomography for retained three-dimensional geometry; STEM/EDS or comparable elemental mapping for silica/gold; and dark-field single-object spectroscopy for correlated structure/function measurement. SAXS is optional for periodic scale-up. No single method certifies all needed properties. **Functional assay.** Deposit or tether tiles on a qualified transparent support without collapsing them. Record spectra of selected dimers and isolated reference particles. Change surrounding refractive index by a calibrated compatible solution increment, then reverse it. Require a repeatable reversible spectral change above five times spectral repeatability noise, with the pre-registered response direction checked by electromagnetic simulation. Measure a matched noncoupled-particle control. Do not promise an absolute resonance wavelength or sensing sensitivity before the geometry and optical model are calibrated. **Controls.** Randomized slot codes at fixed overall DNA amount; full coding with no rejection pulses; one-pot versus hierarchical mixing at matched concentration; uncapped retired ports; fresh versus reused outer palette; no silica; silica without correct preassembly; intentionally malformed carrier; and systematic port-mutant decoys. Instrument operators should classify connectivity without seeing the assembly condition. Optical selection must not remove malformed objects from the reported assembly-yield denominator. **Sample size and uncertainty.** Target at least 200 imaged final objects per condition across three independent preparation batches, reported separately. At an object yield near 50%, 600 independent objects would have a binomial standard error about 2%; batch and shared-template effects reduce effective independence, so use a batch-aware bootstrap or hierarchical model. To support a wrong-joint probability below 0.1% with zero observed failures needs about 3,000 effectively independent joints for a one-sided 95% upper bound using -ln(0.05)/n. If joints within an object are correlated, count independent objects/batches appropriately. Zero errors among a few pictures cannot establish an extremely low defect rate. **Pre-registered decision gates.** Correct topology and payload arrangement in at least 50% of recovered objects; carrier mass recovery at least 10%; at least tenfold lower wrong-retained-joint rate with rejection versus matched control, with uncertainty reported; less than twofold wrong-joint increase under outer-palette reuse versus a fresh palette; median post-conversion optical-gap displacement within 5 nm and no unreported tail-driven failures; continuous structural silica joints; and the reversible optical response criterion above. All seven are research acceptance thresholds. If one fails, record which physical contract failed. A lower bound or confidence interval that cannot resolve the target improvement is inconclusive rather than a pass. **What this experiment cannot show.** It does not validate millions of physical addresses, nanoparticle-to-single-crystal conversion, local replacement of buried defects, or recursive error contraction. The next experiment after success is fault-injected multi-module repair including conversion faults, at increasing module count with fixed total solids, not simply a larger attractive shape. ## S14. Failure experiments and simulation discriminators | Failure hypothesis | Early test | Decision consequence | |---|---|---| | Additive code-energy model fails | Cross-talk matrix including shifted registers and rotations | Replace code metric by measured free-energy/kinetic incompatibility graph | | Reused ports reactivate | Multiple mask/unmask cycles with decoy modules | Add isolation or abandon the reuse route if leakage exceeds the risk budget | | Proofreading is only survivor selection | Count input/output mass and every discarded species, inject wrong joints | Do not infer correction from purified-image quality | | Correlated faults dominate | Shared-contaminant pulses, malformed master template, common temperature excursion | Estimate a common-mode floor; conditional threshold may not apply | | Fusion erases precision | Registered before/after tomography and spectroscopy | Change joining chemistry or relax the target class; no universal conversion claim | | Kinetics dilutes catastrophically | Scale module size at fixed primary mass concentration | Use a transport intervention and account for its resources, or stop scale-up | | Material interfaces dominate | Vary joint density and measure resistivity, modulus, thermal conductance | Infer interface impedance; constrain the reachable-property dictionary | | Library diversity grows per site | Track new payload/staple/processing recipes over unrelated target designs | Reassess general-purpose value against directed assembly and lithography | ## S15. Prior-art matrix and claim ledger | Area | Representative primary source | Established contribution | Difference and remaining gap | |---|---|---|---| | DNA origami | R1 | Programmable scaffold geometry | Carrier qualification and compatible payload conversion are additional tasks | | DNA bricks | R2,R5 | Large addressable three-dimensional assemblies | Distinct components and exact sequences are real inventory costs | | DNA material voxels | R3 | Scaffold-defined coordination independent of included object | General voxel idea is prior art; our integrated error budget is conditional | | Foldable modular voxels | R4 | Reconfigurable chains assembled from origami units | Machine-scale functional integration not established | | Crisscross assembly | R6 | Cooperative growth and combinatorial origami-component construction | Serious prior art against claiming new combinatorial addressing | | Staged and hierarchical tiles | R7,R8 | Small glue inventories and staged shape construction in formal models | Our active-conflict accounting is not an independent universality discovery | | Self-sorting and pathways | R9,R10,R29 | Competing targets, kinetic bottlenecks, nucleation control | Non-equilibrium physical schedule still must be designed | | Designed proteins | R11,R15 | Orthogonal interfaces and designed symmetric protein materials | Arbitrary heterogeneous material and process compatibility remain open | | Silica/mechanical metamaterials | R12,R13 | Inorganic transformation and mechanical function from DNA templates | A unified low-damage multi-material fusion process is absent | | Covalent locking | R14 | Programmed stabilization of DNA structures | Stable DNA is not continuous metal or semiconductor | | Kinetic proofreading | R16,R17 | Driven discrimination with time and dissipation trade-offs | Our protocol must physically realize its assumed rates | | Tile repair | R18,R19 | Error-correcting and self-healing constructions | Threshold idea itself is not new; conversion must be included | | Patchy colloids / superlattices | R20,R21 | Directional valence and programmed crystal assembly | Finite heterogeneous machines need many additional constraints | | Covalent frameworks | R22 | Periodic porous material assembled from molecular units | Not an arbitrary code-addressed face inventory | | Synthetic compartments | R23 | Coupling of compartment growth and information replication | Useful analogy for autonomous cycles, not a general fabricator | | Digital structural materials | R24 | Reversible assembly of functional cellular composites | Macro modularity is established; nanoscale joining is distinct | | Semiconductor directed assembly | R25 | Lithographic guidance plus self-organization | Supports hybrid manufacturing, not unrestricted circuitry from particles | | DNA plasmonics | R26 | Optical function from arranged nanoparticle geometry | Motivates the first sensor experiment; exact tile remains new/unbuilt | | Molecular machines | R27 | Driven directional motion in synthetic molecular systems | Work cycles and fuel are required; not arbitrary mechanical construction | | Computational tile universality | R28 | Formal simulation universality of an abstract tile model | Not chemical, material-property, or manufacturing universality | | Modular geometric interfaces | R30 | Shared scaffold/staples with independently tuned geometry and interaction | Strong prior art for reusable carrier/platform design | | Modular robotics | R31 | Physical replication/assembly with robotic modules in a restricted system | Does not establish nanoscale self-repair or universal feedstocks | | MOF platforms | R32 | Designed porous networks with functional composition | Coordination chemistry is not a general arbitrary-face address code | Central mathematical claims G1, P1, P2, C1, C2, H1 are established mathematics or elementary consequences restated for this problem (A/B). A1, S1, and U1 are formal extensions/accounting constructions (C, priority unestablished). F1 is a conditional transfer of known fault-tolerance reasoning (B/C); its required physical module is an unresolved hypothesis (D). The experiment and sparse manufacturing framework are an integrated hypothesis (B/D). The code results are numerical results for the stated model; their novelty class is not evidence of physical truth. ## S16. Independent second-pass audit A second conceptual pass was performed after the uniform-carrier design was specified. It was not an independent laboratory replication, peer review, or external-agent review. It identified five hidden assumptions: every volume needs an address; a correct exterior certifies an interior; hierarchy preserves concentration; fusion is error-free; and correct equilibrium favors fast pathways. Each was removed from the conclusion. The revised architecture programs sparse functional regions and boundaries, uses explicit module contracts, charges transport resources, models conversion error floors, and evaluates finite-time kinetics. No further conceptual iteration resolves the absent physical correction primitive. More ambitious terminology would not change the constraints. The next meaningful update to this research must supply measured interface cross-talk, conversion error, and fault-injection data, followed by a calibrated spatial model.