{
  "compiler": "### 8.4 The v3 gauge-aware compiler pass\n\nBefore material allocation, the compiler checks whether the declared function is invariant under a positive common conductance multiplier. If absolute current, power, delay, finite external loading, or an unscaled junction matters, it retains the absolute R6 contract or expands the specification; it does not discard those requirements.\n\nFor the quotient-compatible class it adds the following instructions to the inherited growth grammar: provision a stable comparison witness; create bounded-degree local pair-comparison links; reserve capacity and guard bands; reconstruct relative log responses with a finite confidence budget; reduce the feasible-scale interval; select its lower endpoint; execute bounded incremental repairs; retain both material and comparison access; issue a common-epoch completion acknowledgement; seal with the declared margin.\n\nThe module comparisons are the line graph of the physical junction graph: modules sharing a junction can be compared through local switches. An additional short link connects one module to the stable witness. The target family and all comparison links are generated by the fixed interpreter. The seed does not contain the dense covariance matrix used for offline numerical evaluation. A uniform radius bound and operating budgets are charged as part of the compiler/environment specification.\n\nThe scalar G2 optimization is exact linear-time arithmetic. G3 is a conservative feasibility pass. It does not make general developmental compilation easy, supply an autonomous general CAD compiler, or solve the chemistry of the primitives. The emitted v3 JSON capsules record the allowed functional quotient explicitly; a quotient must never be inferred merely because it increases simulated yield.\n\nFor linear elastic or hydraulic analogues, the same scale idea can apply to a suitably declared static response with prescribed boundary data. It transfers only if the actual constitutive law, couplings, and actuator reachability preserve the assumed positive quadratic form. Arbitrary materials, nonlinear motion, and dynamic devices do not follow.\n\n",
  "experiments": "## 13A. New v3 experiments and what they establish\n\n### 13A.1 Controlled changes from the v2 model\n\nThe new suite uses the same nonperiodic nearest-neighbor role grammar and grown service-tree geometry: an 8 by 8 planar network with 112 scalar modules and a 5 by 5 by 5 spatial network with 300 modules. Each module is represented by its post-conversion scalar response, initially uniform on [0.65,1.35] in witness-relative units. This bounded disorder model is specified, not fitted to laboratory material data. The v2 microscopic bond-error model and absolute conductance objective are different, so their yields are not directly comparable.\n\nThe v3 nominal target has four external ports. Success requires completed manufacturing and projective response error at most 0.04; the internal voltage map under a unit boundary drive is scored separately. Two-terminal scale equivalence would be vacuous and is not the test. The local G3 log tolerance is 0.04; remaining seal error is bounded by 0.005; diagnostic radius is approximately 0.006; deposition increments lie in [0.007,0.012]; available reserve is 1.05 per module; and the inspection horizon is 128.\n\nThe comparison graphs have 293 and 1,186 edges. A stable, absolutely uncalibrated witness is attached by one extra link. Local paired observations share detector gain. Fresh ratio log-errors are independent Gaussian with single-sample standard deviation 0.012. A simultaneous failure budget of 0.005 requires 326 samples per pair in 2-D and 218 in 3-D. Matching schedules use seven and twelve nonoverlapping probe slots respectively. The one-witness resistance maxima are 2.61535 and 1.64203, which explain why the smaller 2-D object needs more repeats per edge.\n\nThe manufacturing ensemble uses a finite factored estimator; the separately executed local-message estimator provides an implementation route and a measured algorithmic round count. Its latency is deliberately excluded from the ensemble's service-and-acquisition time subtotal. Those subtotals are not total fabrication times. The model preserves material delivery capacity, finite growth layers, diagnostic sampling costs, bounded reserves, and local postorder final closure, but it does not solve diffusion, hydraulics, chemical kinetics, Brownian alignment, or molecular control errors. No new molecular species count or energy measurement is claimed.\n\n### 13A.2 Results from 512 new paired manufacturing runs\n\nEach of eight conditions uses the same 32 underlying initial-state seeds in each dimension. Methods within a seed are correlated comparisons, not additional independent worlds. A completed object is not scored as successful merely because its controller says so: the final four-port response is independently computed from the final physical conductances after sealing.\n\n| Condition | 2-D functional completion | 3-D functional completion | False local certificates, 2-D / 3-D |\n|---|---|---|---|\n| Projective reserve repair | 32/32 | 32/32 | 0 / 0 |\n| Shared detector gain varied per pair/epoch | 32/32 | 32/32 | 0 / 0 |\n| All material conductances multiplied by 0.6 | 32/32 | 32/32 | 0 / 0 |\n| Equally capable conventional ratio controller | 32/32 | 32/32 | 0 / 0 |\n| Fixed unit-scale representative | 0/32 | 0/32 | 0 / 0 |\n| Reserve reduced to 0.4 | 0/32 | 0/32 | 0 / 0 |\n| Spatial differential comparison bias | 0/32 | 0/32 | 29 / 7 |\n| Witness link removed before completion | 0/32 | 0/32 | 0 / 0 |\n\nShared detector gains ranged from exp(-0.7) to exp(0.7) and canceled in the paired ratio. The common-material-scale condition also scales the witness and available conductance increments. It verifies the stated symmetry; it is not a test of arbitrary changes to material chemistry. The equally capable conventional controller has the identical policy by construction and ties. This comparison does not establish superiority over conventional ratio control, lithography, or printing.\n\nThe fixed-representative condition is conservatively rejected by the local feasibility gate because some modules already exceed its allowed conductance. It is not a comparison against every possible conventional fabrication strategy; replacement, subtraction, or a different global certificate could change the result. The insufficient-reserve case is also rejected honestly before repair.\n\nFor the projective policy, mean projective response errors were 0.002221 and 0.001779 in 2-D and 3-D. The worst among all 64 positive baseline runs was 0.005852. Mean maximum absolute voltage-map errors were 0.002015 and 0.002296 for a unit drive; the worst was 0.003493. These are synthetic endpoint scores under the declared static conditions.\n\nMean added-material proxies were 11.4985 and 33.6441, about 29.3% and 29.6% of nominal functional-network material. These proxies use the inherited 0.4 material units per normalized conductance unit; they are not grams or measured deposition energies. Allocated reserve capacity is 105% of nominal functional conductance and is larger than the material actually consumed. Empty reserve geometry, comparator hardware, contacts, and the stable witness also occupy real space and are not assigned measured fabrication costs here.\n\nMean inspection counts were 68.3438 and 69.3438. The conservative strategy acquires roughly 6.53 million and 17.93 million completed ratio samples per object. Its mean service-and-acquisition time subtotals were 454.27 and 968.48 in the model's arbitrary time units, excluding inference and reduction latency. This is a substantial control-resource cost, not evidence that nanoscale manufacture is already fast.\n\nThe final absolute conductance ratios averaged 1.2912 and 1.2947, although the prescribed projective function passed. Under a common material factor of 0.6 they averaged 0.7747 and 0.7768, while the voltage map and projective errors were unchanged to numerical precision. These deliberately different absolute conductances make the allowed specification change transparent. A 32/32 outcome alone has an exact two-sided 95% binomial lower confidence endpoint near 0.891; it does not establish unit population yield.\n\nThe differential-bias test adds a gradient-shaped log error spanning -0.15 to +0.15 across the object. Of its 64 runs, 36 completed with false certificates and failed the independent projective test; 28 exhausted their reserves. Mean cycle-residual energies remained nearly the same as the unbiased cases. The separate noiseless counterexample recovered a wrong field with maximum bias 0.15 and cycle residual approximately 0.00000000000000214. A passed cycle check cannot validate this failure mode.\n\n![New v3 manufacturing outcomes and false certificates](../figures/gauge_results.png)\n\nFigure 7. The new positive cases and mandatory negative controls. Incomplete objects contribute to neither functional completion nor false accepted certificates. The matched conventional condition is an exact policy tie, not an independent invention.\n\n### 13A.3 Numerical checks, local execution, and the reserve transition\n\nThe package includes 28 passing research/software tests: the original 16 and 12 new ones. New checks include 80 independent linear-program comparisons of G2 feasibility and minimum material, 500 adversarial endpoint-noise constructions for G3, covariance identity, common-scale cancellation, an invisible differential-bias example, the local solver's residual bound, and reference-access failure. Floating-point tests are not formal interval proofs.\n\nThe G4 formula was checked against 100 independent numerical integrations; the largest absolute disagreement was about 0.0000000000000197. Two hundred four-port passive-network tests satisfied G1, with the largest projective error about 0.014934, below 0.04. Six thousand independent noise-field draws tested the calibration covariance; the largest relative sampling deviation among the tested diagonal variances was 4.32%, compatible with this finite Monte Carlo check but not a formal distributional calibration test.\n\nThe phase experiment sampled 25,000 arrays: 5,000 each at sizes 4, 16, 64, 256, and 1,024. Each array was evaluated at 31 capacities, producing correlated evaluations within a size. At reserve 0.55 the 64-module result was 295 successes in 5,000 arrays, against an exact probability of 0.060415. At reserve 0.60 all arrays were feasible, as the support-wide theorem predicts. The worst empirical-versus-exact probability difference over the reported grid was 0.01298. The transition was derived before these computations; it was not inferred by fitting a curve.\n\nThe comparison-topology study uses ordinary 1-D, 2-D, and 3-D nearest-neighbor boxes. At 128 nodes in 1-D the worst effective resistance was 127; at a 16 by 16 grid it was 3.60852; at an 8 by 8 by 8 box it was 1.27912. These illustrate established dimension-dependent calibration behavior [P15], not a newly discovered dimension law. All simulations are reproducible from stored seeds, parameters, and code; no laboratory measurements were introduced.\n\n![Calibration noise and sample budgets](../figures/gauge_calibration.png)\n\nFigure 8. Finite-box effective resistance and the sample count needed for the same simultaneous log-error radius. More dimensions help only when the comparison graph supplies the corresponding independent routes. Local solution time remains a separate constraint.\n\n",
  "experiment": "## 14A. The decisive v3 experiment\n\nUse four post-conversion conductor modules arranged as a resistive bridge with separately accessible terminals and independently bounded parallel reserve paths. Choose nonidentical nominal conductances, for example proportional to 1, 2, 1.4, and 0.8, so the intended voltage output is not a trivial zero signal. The bridge output is then about -0.30303 of a unit top-to-bottom drive under the declared branch orientation. Preserve a generic pair-comparison frontend and witness until all modules complete.\n\nThe first gate is a benchtop electronic emulation with switched passive resistors. It can decisively test the numerical representative selection, budgets, paired-gain cancellation, and reference-release logic. It cannot validate the proposed material backend. The second gate substitutes an actual scaffold-compatible post-conversion incremental deposition actuator, characterized before its use in the growth experiment.\n\nCompare projective representative selection against a fixed representative and a matched conventional ratio controller. In separate conditions, vary shared detector gain, deliberately introduce differential arm error, remove the witness link early, and cross the measured reachable-reserve boundary. Verify final function with an independent instrument and the full four-port response under three independent grounded excitations. Measure deposited material, rejected modules, electrode/contact drift, and every intervention. The controller must not receive the final evaluator's conductance ground truth.\n\nG2 is supported if measured reachable intervals predict feasible versus infeasible target bands and the minimum-material endpoint agrees within measurement/actuation error. It is falsified for a proposed backend if coupling, nonmonotone growth, discontinuous percolation jumps, or unstable contacts invalidate those intervals. G3 is supported only after its bounded noise, step, capacity, access, and seal premises are measured. An apparent counterexample with violated premises falsifies that backend or parameterization, not the mathematical implication.\n\nG4 requires a separate ensemble across module counts and controlled bounded disorder. Four modules can test exact reachability and the boundary mechanism; they cannot establish an asymptotic transition. A scale study using 4, 16, and 64 switchable modules is a relatively inexpensive next stage. Replacing those controls by chemistry is a later gate. Sample sizes should follow a prospective power calculation after noise characterization; a few successful bridges cannot establish high-yield manufacture.\n\nThe highest-information negative control is differential bias with a small cycle residual. If the physical system detects it, identify which additional information breaks the G5 ambiguity. If it does not, bound its frequency or keep the corresponding calibration requirement. Neither outcome supports a self-certifying absolute nanofabricator.\n\n",
  "scaling": "### 16.1 Additional v3 scaling costs\n\nThe v3 reduction removes an absolute reference requirement for a scale-free function; it does not remove O(R) generic comparator sites, accessible reserve space, or temporary state. A regular three-dimensional comparison network needs order log(RH/delta) repeated samples per local edge at fixed error radius, hence order R log(RH/delta) total samples per inspection. The graph must remain connected and sufficiently well supplied. The common witness needs stability, even though its absolute conductance need not be known.\n\nSequential neighbor relaxation has a different scaling: on a regular box of side n it takes order n squared times a precision logarithm. If a hypothetical 1-cm object used 100-micrometer modules, it would have n = 100 and approximately one million module positions. With an assumed 1-ms local update time and a logarithmic factor of 14, a simple diffusion-like solve would take order 140 seconds per inspection before constant factors and acquisition time. Seventy inspections would be order 2.7 hours of inference alone. At a 10-micrometer module pitch, the n-squared term is one hundred times larger. These are illustrative scaling calculations, not predicted device timings.\n\nHierarchical calibration, physically fast analog relaxation, parallel module replacement, or fewer diagnostic epochs may improve that cost. They must preserve noise and service certificates and be demonstrated; no assumed exponential communication speedup is charged to the present result. Applying a comparator to every atom is excluded. Functional modules must be sufficiently large and response-simple to amortize the control apparatus.\n\nA common conductance scale changes dissipated power under voltage drive. If thermal or current constraints impose a scale interval, that interval re-enters G2 and needs physical calibration. Functional gauge freedom cannot hide an impossible power budget. A witness that remains stable in a benchtop bridge may drift during centimeter-scale mineralization, drying, or heating, so beta and the diagnostic-error budget must be measured across the actual maturation history.\n\n",
  "audit": "### 17.4 Adversarial audit of the v3 advance\n\n| Objection | Answer and remaining limit |\n|---|---|\n| Is the function made easier after seeing results? | The allowed common-scale symmetry is declared before repair. Absolute objectives stay absolute. |\n| Are two-terminal projective scores vacuous? | Yes; the new demonstrations use four terminals and a separate internal voltage map. |\n| Is scale invariance new? | No. The candidate contribution is the reserve compiler, robust construction, exact model yield boundary, and closure integration. |\n| Does the gauge fix every shared error? | It cancels only permitted common scale. Differential bias can be perfectly cycle-consistent and still wrong. |\n| Is an uncalibrated witness automatically safe? | No. Its stability, access, regime, and relation to actuator units remain explicit assumptions. |\n| Is the reserve threshold a universal phase transition? | No. It is exact for the stated bounded scalar reachability model; finite-noise hardware uses G3's stronger conditions. |\n| Are the reserves and comparators free? | No. The examples use generous capacity and millions of ratio samples. Their apparatus is not chemically implemented. |\n| Is the complete estimator locally executed in every run? | The ensemble uses a finite factored solve. A separate implemented neighbor-message solver is verified, with latency reported separately. |\n| Is functional success equivalent to all local certificates? | No. Local envelopes are sufficient; a specific device may work without them. |\n| Can the comparison network be removed region by region? | Only with a new valid dependency certificate. The implemented policy retains it until all affected modules complete. |\n| Does this solve arbitrary functional matter? | No. It solves a conditional passive scalar-module problem and identifies an experimentally testable backend. |\n\n"
}