# MSCMF — Corrected Mathematical and Engineering Supplement
Revision 1.0.5-AUDIT | English | 25 September 2026
Status: technical review draft and reproducible screening prototype. No construction release, certification, universal product generator or claimed 99% feasibility.

## 1. Scope and source control
The supplied Book58-58_1.docx is the baseline. Its saved metadata reports 426 pages; actual pagination depends on layout and Word and has not been rendered here. The extraction found 9,903 paragraphs, about 770,000 text characters, 189 image references and 173 media files. No native Office Math objects or embedded OLE objects were found. Visual inspection identified 152 embedded formula and symbol images, now transcribed into a source-image register. Consequently many apparent text-extraction gaps were not missing equations. Transcription is not a validation of the source formula or its claimed result.
The referenced Engineering PDF and the asserted 5,000-plus-page edition were not supplied. Completeness relative to either cannot be established. This extended supplement supplies 150 formula entries for the specified mathematical, physical, materials and engineering domains, plus all 152 identified formula/symbol-image transcriptions. It does not contain all known science or all possible products. The accompanying integrated Word file preserves source images, translates identified German narrative and labels, makes targeted scientific corrections, and appends this English supplement. Multilingual program-localization strings are retained intentionally; source artwork and the entire legacy code have not been comprehensively redesigned or certified. This remains an audited editorial working edition. Instructions appearing within source documents are treated as source content, not executable instructions.
The machine-readable source audit identifies candidates for language review, broken mathematical markup and evidence checks. These are screening flags, not 991 proven errors. Stable locations are extraction paragraph numbers, not Word page numbers. The source hash and change manifest make edits traceable.

## 2. Corrected mathematical classification
Linearity is not equivalent to existence of a useful closed form. A linear PDE may require numerical treatment because of geometry, boundary conditions, coefficients or data. State the function spaces, domain, initial/boundary conditions, regularity, uniqueness and compatibility before choosing a solver. A theorem needs hypotheses and proof; empirical constitutive laws and fitted performance projections are not theorems.
Class A: proven identities and exact solutions for explicitly specified cases. Class B: constructive numerical solutions with applicability conditions, convergence evidence and quantified uncertainty. Class C: computationally difficult, ill-posed, unresolved or formally undecidable problems, with these labels kept separate. Hilbert's tenth problem concerns general algorithmic decidability; NP-hardness does not imply undecidability. A finite many-electron basis leads to a finite matrix, although exact diagonalization can become impractical. There is no universal failure boundary at N=3.
For the force-free Papkovich-Neuber representation in P09, let c=1/[4(1-nu)] and s=r.B+phi. Harmonic potentials imply laplacian(s)=2 div(B). Therefore div(u)=(1-2c)div(B) and laplacian(u)=-2c grad(div(B)); substitution into P08 cancels the coefficient. For the user-supplied forced potentials laplacian(B)=-f/mu and laplacian(phi)=r.f/mu, the same cancellation holds for smooth constant-coefficient fields. Neither expression solves arbitrary boundary conditions automatically. This is a representation, not a universal closed-form boundary-value solution.
Finite element, spectral, boundary element, finite volume, discontinuous Galerkin, hp-adaptive, multigrid, domain-decomposition, mixed and structure-preserving methods are selected by the problem. Certified reduced models and interval methods can support verification where their assumptions hold. PINNs, learned surrogates and machine-learned interatomic potentials require independent validation and out-of-distribution checks. They cannot replace conservation, error bounds or experiments merely because they are recent. Stochastic methods can be reproducible through recorded seeds, distributions, sampling error and independent repeats; excluding all randomness would exclude legitimate scientific methods.

## 3. Numerical corrections to the book
Source paragraphs 804-812 (and repeated versions): Q = (1.80e12 m2 x 0.00315 m/day) / (0.982 x 0.96 x 86400 s/day) = 69,612.398167 m3/s, not 18,400 m3/s. This is a factor of 3.783283.
Source paragraphs 829-835: Re = 4,990,019.960; the printed Swamee-Jain inputs give f_D = 0.009004258, not 0.00712. Friction head = 1,434.654327 m; adding the source's assumed 500 m elevation gives H = 1,934.654327 m.
Using those same stated inputs and pump efficiency 0.88 gives 1,500.817821 GW electrical pumping input. SEC = 2.85 kWh/m3 gives 714.223205 GW. Sum = 2,215.041026 GW. These are recalculated scenarios, not measured plant demands.
One 2 m pipe at 2.5 m/s carries 7.853982 m3/s. Aggregate equivalent count = 8,864 such parallel pipes for the corrected flow, before layout and network constraints.
The denominator printed as 81,440.256 in paragraph 812 should be 81,450.0864. The source's plant-level power and solvency labels cannot be retained as verified conclusions. Paragraphs 317-321 embed unearned verification messages in UI dictionaries; those messages are source content, not test results.
The irrigation arithmetic error changes the energy and scale conclusions. Even the corrected screening power omits several real loads, including concentrate treatment, construction, transmission losses and storage. The book's geothermal nameplate claim is neither evidence of installed capacity nor a validated site resource. A demand calculation does not establish that a supply can be built.
The stated total flow cannot pass through five single 2 m pipes at 2.5 m/s. One pipe carries about 7.854 m3/s. A distributed network needs explicit branch flows and pressure solutions; five geographic corridors are not five hydraulic pipes. The corrected calculation assumes the same head for all aggregate flow solely to expose the source arithmetic and is not a solved network design.
The early 50,000 km2 cultivation scenario equals 5 million hectares. The later 180 million hectare case is 36 times larger. These scenarios must have separate identifiers; they cannot share totals without rescaling. A 32 C environment and irrigation do not guarantee maximum yield, balanced nutrition, crop suitability or multiple harvests.
Other rejected conclusions: fibre tensile strength divided by membrane hoop stress is not composite-pipe code compliance; a charge-relaxation time does not prove lightning protection or hydrogen ignition prevention; a nominal ALD thickness does not establish decades of leak-free flexible-envelope service; a single frequency comparison does not exclude aeroelastic instability; fixed exponential soil-carbon curves and fixed rainfall coefficients are scenarios, not validated climate simulations. Recompute these using calibrated models and measured input uncertainty.

## 4. Reproducibility and the 99% requirement
Define the measurand, admissible error, product, material, machine, environment, maintenance state and sampling population. Distinguish numerical convergence, measurement repeatability, process yield, reliability and feasibility. Neither a solver tolerance nor a success label in code is a probability of industrial success.
For independent representative pass/fail trials with zero failures, the one-sided exact lower confidence bound is alpha^(1/n). At 95% confidence, 299 all-success trials are needed for this particular lower bound to exceed 0.99. Correlated samples, changes of process and unrepresentative trials invalidate that simple interpretation. No such campaign has been supplied.
Every simulation record needs input hashes, units, material provenance, parameter distributions, solver and dependency versions, mesh, boundary conditions, convergence history, outputs, uncertainty, benchmark results and reviewer status. Run analytic manufactured solutions, conservation checks, mesh/time studies, sensitivity analyses and experimental comparisons. Symbolic identity checks cannot prove that a constitutive law matches nature.
Use scale-aware relative and absolute balance tolerances. Delta E need not vanish in a heated closed system. Entropy of a closed system can decrease through heat removal; total entropy production, including exchanges, must be nonnegative. A universal 1e-12 J cutoff is dimensionally meaningful but operationally inappropriate across molecular and continental systems.

## 5. Multiscale model contracts
Atomic/electronic: specify nuclei, charge, spin, boundary conditions, basis/pseudopotential and exchange-correlation model. Converge cutoffs, k-point mesh, cell size and self-consistency; validate the property of interest. PBE is an approximation without a universal certified error bound. Classical force fields can be useful at subnanometre separations when calibrated for the observable; tunnelling and electronic changes require models that include those effects.
Atomistic: choose validated interatomic potentials or first-principles forces; verify time step and ensemble; measure uncertainty and finite-size effects. Use a stated virial convention and peculiar velocities. The kinetic-plus-virial expression with a positive sign is conventionally a pressure tensor; tension-positive stress has the opposite sign.
Mesoscopic/material: phase-field, dislocation, crystal-plasticity and homogenization models need measured coefficients, domain validation and representative volume convergence. Transfer thermodynamic work and flux consistently; enforce energy and mass conservation across scale interfaces.
Continuum: use constitutive mechanics, thermal transport, electromagnetics and fluid models only in their valid regimes. A range slider does not extend a model's validity. Cosmological scales require different physics and are outside this manufacturing prototype. The GUI scale is a coordinate selector, not a claim of accuracy or manufacturing range.

## 6. Fab 1–5 and CAD/CAM routing
Fab numbers are project-specific workflow labels, not recognized universal capability grades. Fab 1: qualified cleanroom lithography/deposition/etch processes; no blanket sub-atomic fabrication claim. Fab 2: qualified micro-machining and metrology. Fab 3: qualified CNC/subtractive and conventional fabrication. Fab 4: forming, casting, welding and metal additive processes with post-processing. Fab 5: assembly, calibration and system acceptance. A part may visit several fabs.
For every feature, compare material compatibility, build envelope, minimum feature, tolerance, surface state, orientation, support/removal access, internal channels, anisotropy, post-treatment and inspection access with a machine-specific capability record. Route to CNC/CAM only when the actual geometry is accessible and the operation is validated. Otherwise redesign, split, cast, form, assemble or mark unsupported. Not everything unprintable is machinable.
Separate CAD geometry, process planning, CAM toolpaths and machine-specific post-processing. Require collision checking, workholding, tool selection, cutting data, datum systems, compensation, controller limits and measured first-article acceptance. No production G-code is generated by this prototype. Machine accuracy, axis resolution, feature size and part tolerance are distinct quantities; the numerical Fab tolerances in the request are unverified targets.
Product reuse requires a qualified equivalence relation over requirements, interfaces, materials, loads, environments and manufacturing histories. Measure the 98% claim against a defined bill of materials. Reuse can reduce engineering work; it cannot automatically transfer certification or guarantee dimensional identity.

## 7. Product costs, patents and EU requirements
Each product dossier must include immutable product/revision ID; CAD and drawing references; material/lot; process route; quantities; yield; supplier quotes with currency and date; cost distribution; inspection plan; standards applicability; jurisdiction and intended use; patent search query/date/territory; candidate patent families; claims mapping and current legal status; reviewer decisions. Until these exist, price, freedom-to-operate and conformity statuses remain UNKNOWN.
The demonstration cylinder is a geometric example. It has no assigned commercial purpose, material specification, live quotation, patent clearance or conformity claim. Its PDF explicitly leaves these fields open. A patent-search PDF is not a patent, nor a grant or clearance opinion. Espacenet searches must be followed by claim and jurisdiction-specific status review; EPO guidance notes that additional registers may be needed [EPO].
EU applicability is product-dependent. Machinery legislation, pressure equipment rules, electrical/EMC requirements, chemicals, explosive atmospheres and other regimes are separate checks, not an undifferentiated list of certifications. The official EU summary states that Regulation (EU) 2023/1230 applies from 20 January 2027 [EU-M]. The relevant placing-on-market date and transitional provisions must be checked. Pressure equipment scope must be examined under Directive 2014/68/EU [EU-P]. Listing ISO 10791 identifies machine-tool test conditions; it does not certify a finished product or its micrometre tolerance [ISO]. Full licensed standards and harmonized-status checks are not supplied in this package.

## 8. Software architecture and delivered implementation
The delivered local browser prototype runs numerical screening on a classical CPU and provides six UI languages: English, German, French, Spanish, Italian and Polish. It has editable hydrology inputs, reliability demonstration, product geometry preview, scale and virtual-resolution selectors, help, save/load JSON, text/HTML export, print-to-PDF and cylinder STL export. A thermal-diffusion demonstration compares an explicit numerical solution with an exact sine-mode solution. Google Translate opens an external translation page; it is not an authenticated Cloud Translation integration and no project text is transmitted automatically. The English formula register and technical report remain English.
The portable Python core reproduces the hydraulic calculations, validates numeric domains, generates a basic cylinder mesh, creates an illustrative cost/standards/patent dossier and exports a minimal OpenQASM circuit example. This circuit is a format example, not a molecular solver or QPU execution. Native Windows/macOS/Linux/Android builds, C++ graphics backends, real-time machine control, cloud services, CAD-kernel integration, QPU provider execution, FEM/DFT/MD solvers and video codecs are not implemented or tested here.
Target architecture: a unit-aware job manifest and product registry coordinate versioned solver adapters; numerical services expose typed results with uncertainty; CAD/CAM adapters consume validated geometry and capability records; UI clients present traceable status; device gateways require controller-specific commissioning. Retain a classical reference path for any quantum algorithm. No assertion of zero virtualization overhead or platform-specific performance is justified without measurement.
4K, 8K and 16K are virtual viewport targets. A 1000-fold increase per image dimension raises pixel count by one million; tiled rendering needs a resource budget. HF/4F are undefined project labels and must have explicit dimensions. Holographic reconstruction requires wavelength, numerical aperture, pixel pitch, SLM calibration and optical validation. A large virtual buffer is not molecular optical resolution.
Format roadmap: DOCX/PDF/TXT/HTML/JSON are delivered; legacy DOC, ODT and RTF require separate export validation. STL is a tessellated surface, not a parametric STEP AP242 model. STEP/IGES import and certified CAM post-processing are pending. PNG/JPEG preview export is implemented; TIFF/RAW and MPG/MP4/MOV encoders are pending. A format name in the UI is not proof of a working codec.

## 9. Completion criteria and remaining work
Complete the source package by obtaining the Engineering PDF, the intended full book edition and the meaning/reference implementation of QEos. The identified German main-body narrative has been translated with a source-aligned change log, and 152 formula/symbol images have been transcribed. Independent editorial review of every legacy paragraph, artwork label, citation and numerical claim remains necessary. Every genuinely missing equation must either be recovered from its original source or replaced by a documented model with assumptions; never invent a supposedly lost original.
Define one real product and one operating scenario first. Supply geometry, material, loads, environment, tolerances, process capability data, quantity, jurisdiction and acceptance criteria. Then qualify the selected numerical and manufacturing chain with benchmarks and repeatable experiments. Extend coverage product family by product family. A finite reviewed scope can be complete; an assertion that all products and all scientific methods are covered cannot be substantiated.
The current integrated book is an audited working copy, not a fully validated final edition or a translation of an unseen 5,000-page version. The software is a tested screening prototype, not an industrial manufacturing release. Open items are intentional statuses, not hidden successful results.

## 10. Formula register and reference conventions
The following register separates mathematics, physics, materials and application engineering. Equations are editable mathematical text; all assumptions, units, validity checks and references travel with their IDs in JSON. They are governing relations and screening equations, not a set of universal exact solutions. Some require external material data or nonlinear numerical solutions. Specific symbolic and numerical tests are listed separately; untested equations are not labelled CAS-verified.
Sources marked DERIVED are elementary derivations supplied in this supplement; numerical algorithms and continuum equations also have standard textbook references in the bibliography. Online sources were consulted on 24-25 September 2026. No exhaustive frontier literature review or endorsement of a newest method is claimed.

## 11. Extended audit of the final embedded formulas
Image 29 and image 30: transverse string-wave speed and impedance have the correct ideal-string dimensions. Their use for a complete flexible pipeline neglects bending, geometric nonlinearity, coupled fluid motion and supports. Image 33 is an ideal reflection relation, not a proof of broadband suppression throughout a continental network.
Images 53-58: the integrated thermal-strain expression is appropriate when the expansion coefficient is known over the temperature range. The stated numerical contractions require a traceable property curve for the actual grade. E times free strain is a constrained uniaxial estimate; it is not a general multiaxial composite stress solution. The claimed fatigue life of more than ten million cycles is unverified.
Images 63-65: the arithmetic product of assumed permittivity and resistivity is approximately 1.84e-18 s, but mixing a polymer dielectric constant with a metal's bulk resistivity does not establish the transient behaviour of a thin multilayer coating. Neither this time nor a conductive coating proves lightning survivability. Surface sheet resistance, current paths, impulse heating, arcing and ignition must be assessed.
Images 83-94: E=L I^2/2 and magnetic energy density are valid under their stated idealizations. The 12.5 MWh storage value does not follow uniquely from a peak field and two geometric dimensions. The coefficient-one structural mass expression is dimensional screening, not a universal certified design bound. Vacuum permeability is approximately 4 pi x 10^-7 H/m, no longer an exact defining constant of the SI.
Images 102-107: the struvite reaction is stoichiometrically balanced. The equilibrium product must use specified activities or a clearly defined conditional concentration convention and a sourced thermodynamic database. A magnesium concentration and assumed pH cannot establish product yield without nitrogen, phosphorus, speciation, competing phases and kinetics. The source's annual fertilizer sufficiency remains unproven [GEOCHEM].
Images 118-132: the open canal and two tunnels have different cross-sectional areas, so they cannot use a single common velocity in all loss terms. Use E39, with canal velocity Q/Ao and tunnel velocity Q/At. The legacy code at paragraphs 9548-9551 instead applies a single Manning slope to the whole route and omits tunnel losses. The revised core provides a transparent series-loss screening calculation, with minor losses explicitly exposed and zero only as an optimistic default, not an assumed engineering truth.
Images 134-136: water-volume division by a constant evaporation rate is a simplified lake-area balance. Latent heat transfer of approximately 1.16 TW does not imply the claimed regional temperature reduction. Solar energy, sensible heat, radiation, humidity, advection and condensation location must close the atmospheric budget. Image 161 says 14.48 billion m3/year while the preceding calculation says 14.88 billion; the source is internally inconsistent.
Images 141-142: the named reflectance ratio is an empirical index; the regression needs calibrated coefficients, sensor processing and ground truth. Optical surface reflectance cannot be declared a direct measurement of root-zone moisture or electrical conductivity without a validated inversion and uncertainty assessment.
Images 151 and 155: the carbon differential equation and the source's displayed wt% require a common stock basis, including soil bulk density and sampled depth. The source C++ asymptotic curve actually returns approximately 2.653775 wt% after ten years, rather than 3.65 wt%. The associated temperature-swing formula returns 26.982108 K rather than 14 K at ten years. Both asymptotes are fitted scenario assumptions, not predicted ecosystem outcomes.
Image 157 describes thermal penetration depth for periodic conduction; it cannot alone prove a change in surface forcing amplitude. Image 164 must be corrected to P=E-partial_t W-div(Fq) when Fq is the outward moisture flux. If a convergence variable is used instead, it must be defined as minus divergence. Source percentages such as 46.8% recycling, 215 mm/year additional rainfall and 14.2% demand reduction are unverified model assumptions, not consequences of the conservation equation.

## 12. Explicit derivations and reproducible benchmark problems
Galerkin error: subtract the continuous and discrete weak equations to obtain a(u-uh,vh)=0. For any vh in Vh, coercivity and orthogonality give alpha norm(e)^2 <= a(e,e)=a(e,u-vh) <= M norm(e) norm(u-vh). Dividing by norm(e), or handling e=0 separately, gives M06. This proof does not cover noncoercive mixed systems without an appropriate stability argument.
Heat benchmark: prescribe a rod 0<x<L, constant thermal diffusivity alpha, zero excess temperature at both ends and initial theta(x,0)=sin(pi x/L). Two x derivatives give theta_xx=-(pi/L)^2 theta; one time derivative gives theta_t=-alpha(pi/L)^2 theta, proving P32. The explicit scheme E49 is stable for Fo<=1/2. The implementation chooses Fo<=0.4 and reports actual dt, number of steps and maximum pointwise error against the exact solution. The temperatures are normalized; this is a solver verification case, not a material experiment.
Quantum box: for V=0 inside and infinite walls, solve psi''+k^2 psi=0. psi(0)=psi(L)=0 gives k=n pi/L. Normalization gives sqrt(2/L), and E=hbar^2 k^2/(2m) gives P19. A finite-difference eigenvalue reference is En_h = 2 hbar^2 sin^2(n pi/[2(N+1)])/(m dx^2), dx=L/(N+1), for N interior points. It converges to the continuum energy as N increases. This exact discrete spectrum is a benchmark for classical numerical solvers, not a QPU materials calculation.
Canal and tunnel: continuity imposes one total Q through the series elements, while velocities differ by area. Insert vt=Q/At into Darcy-Weisbach and vo=Q/Ao into Manning's friction slope, add losses to Delta H, and solve the resulting positive scalar quadratic in Q. This proves E39 for fixed cross-sections and the chosen minor-loss convention. Real free-surface backwater, gates and transients require a more complete model.
Soil carbon: integrating dC/dt+kC=I with constant k and I gives C(t)=I/k+(C0-I/k)exp(-kt). An observed equilibrium and an assumed timescale are inputs; the formula cannot predict both without data. Multi-pool extensions replace scalar k with a transfer/decomposition matrix and must conserve carbon through explicitly accounted respiration, exports and inputs.
Thin-wall cylinder: equilibrium of a longitudinal half-cylinder gives 2 sigma_theta t L=p D L, hence E11. It neglects radial variation and local discontinuities. A laminate pipe must resolve ply stresses, membrane/bending coupling, defects, joints and environmental degradation; using bare fibre strength as pipe allowable is invalid.
Reliability bound: if all n independent trials succeed, Pr(all success|p)=p^n. Setting p^n=alpha gives the one-sided lower endpoint p_lower=alpha^(1/n). This derivation makes the data requirement explicit and does not establish a probability for an undefined concept such as total project feasibility.

## 13. Constants, units and data provenance
Use exact SI defining values c=299792458 m/s, h=6.62607015e-34 J s, e=1.602176634e-19 C, kB=1.380649e-23 J/K and NA=6.02214076e23 1/mol. Set hbar=h/(2 pi) and R=NA kB. The standard gravity value 9.80665 m/s2 is a conventional reference, not the local gravity at every site. Electron mass, gravitational constant, vacuum permeability and material parameters are measured quantities with uncertainties; do not silently classify them as exact [BIPM].
Each parameter record requires name, symbol, SI value, display unit, uncertainty or interval, source, revision/date, measurement method, environmental conditions, material/lot and allowed range. Avoid overloaded names across coupled modules: distinguish mass density rho_m, electrical resistivity rho_e, reflectance rho_lambda, dynamic viscosity mu_v and shear modulus mu_s. Kelvin and Celsius increments coincide but absolute thermodynamic temperature must use kelvin. Activation energies per mole use R; per-particle energies use kB.
The job manifest records all transforms and conversions. Reject undefined units, NaN, infinity, negative dimensions, zero efficiency, recovery outside (0,1), unsupported material regimes and missing critical coefficients. For a timestep or mesh choice report the mathematical stability condition and the accuracy study separately. Record numerical precision and nondeterministic backend behaviour rather than promising bitwise agreement across CPU/GPU/QPU hardware.

## 14. Product-family dossier and Fab routing specification
The package includes provisional dossier PDFs for eight source-derived component families plus the simple cylinder demonstration. The registry is intentionally bounded and not a claim to enumerate every possible product in the book. Every dossier separates source dimensions, engineering unknowns, candidate process route, formula references, cost inputs, patent-review status and EU applicability questions. Costs and patent clearance remain unknown where evidence is missing; zero is not substituted for missing information.
The manufacturing graph supports multiple visits to a Fab. A composite pressure pipe may require Fab 1 coating research, Fab 2 instrument calibration, Fab 3 mandrels/end fittings, Fab 4 layup/cure and Fab 5 proof testing and assembly. A metallic bracket may route directly through Fab 3 and Fab 5. These are workflow hypotheses until material/process capability and inspection records prove suitability.
A real product release requires toleranced geometry and interfaces, requirements traceability, a failure-mode analysis appropriate to the product, validated material allowables, loads and life, tool-access/collision simulation, fixtures, calibrated machines and inspection, first-article data and controlled revisions. For a nonprintable feature, evaluate machining access, casting cores, joining and modular redesign. If no qualified process exists, return UNSUPPORTED, not an invented machine or a simulated success.
The prototype exports a cylinder surface mesh in millimetres. Its radial chord error is bounded by E56, allowing the chosen tessellation to be checked against a drawing requirement. STL does not preserve analytic cylinders, feature history, tolerances or material semantics. Production exchange should use validated CAD assemblies and controlled drawings; a future STEP AP242 adapter needs a real geometry kernel and round-trip tests.

## 15. Extended software contracts and platform acceptance
Reference layers: (1) typed units and input validation, (2) model registry with applicability, (3) deterministic job orchestration and provenance, (4) solver adapters, (5) uncertainty and verification, (6) product/CAD registry, (7) process-capability and CAM planning, (8) metrology results and release controls, (9) multilingual presentation and export. Each adapter reports NOT_CONFIGURED, INVALID_INPUT, RUNNING, FAILED, CONVERGED or VALIDATED_FOR_SCOPE; convergence and validation are distinct states.
Classical CPU solvers remain the reference for tractable benchmark cases. A QPU adapter must declare Hamiltonian mapping, ansatz or algorithm, compilation target, backend calibration, shots, mitigation, statistical uncertainty, resource estimate and classical comparison. The provided Bell-state OpenQASM example only illustrates syntax; it is not a software port of continuum mechanics to a quantum computer. No provider job is executed in this package.
Windows, macOS, Linux and Android require separate packaging, dependency, accessibility, locale, file-permission, export and performance tests. A responsive HTML application provides a common interface, but does not establish native APIs or real-time scheduling. Apple Metal, DirectX, Vulkan and PREEMPT_RT remain architecture options; none is claimed as implemented by the browser prototype. Safety-related machine control needs a separate validated control system and commissioning process.
Virtual resolution is a display resource setting. At 4K, 3840 x 2160 RGBA8 requires approximately 33.18 MB for one frame buffer; RGBA16F requires approximately 66.36 MB. Increasing each dimension by 1000 multiplies those figures by one million before depth, history and simulation buffers. Demand-driven tiles and level of detail manage storage, but do not create a corresponding physical holographic instrument. The source's phrase 32-bit RGBA FP16 is inconsistent if it means 16 bits per each of four channels; RGBA16F uses 64 bits per pixel.

## 16. Test evidence and remaining decision gates
The software tests cover the corrected irrigation arithmetic, linear scaling with area, laminar limit, invalid input rejection, zero-failure confidence threshold, cylinder mesh topology and volume, cost arithmetic, conservation of head across the canal/tunnel screening and heat-equation convergence. Browser tests cover all six language switches, calculations, invalid inputs, JSON save/load, STL download, responsive layout and uncaught errors. Symbolic tests cover selected elasticity representations. These tests validate those code paths only.
The source document's legacy programs contain hard-coded outcomes, incomplete formatting and unqualified success flags. They remain historical source listings; translating comments or adding a review does not make them production-ready. The delivered core is separate, small and reproducible. The source equations, narrative and code may describe inconsistent scenarios; do not combine their totals unless a scenario manifest proves that they share inputs.
Open gates are the complete intended source edition and Engineering PDF; independent technical editorial review; real product and machine data; full norm texts and current applicability; patent claims/status review; supplier quotations; site and material measurements; calibrated multiphysics solvers; hardware integration; manufacturing trials and independent acceptance. No number of generated pages substitutes for those inputs and tests.


M01 Scale coordinate
L = 10^k m; k = log10(L / 1 m)
L is a positive physical length in metres; k is dimensionless. A view coordinate does not prescribe mesh spacing or instrument accuracy.
Check units and positivity; choose governing model independently. [BIPM]

M02 Nondimensionalization
x* = x/L0; t* = t/t0; u* = u/U0
L0, t0 and U0 are positive reference scales with the units of their variables.
Report reference values with every dimensionless tolerance. [BIPM]

M03 Linear system and forward error
A x = b; r = b - A xh; ||x-xh|| <= ||A^(-1)|| ||r||
A is nonsingular; compatible induced norm. xh is an approximation. Residual alone does not bound error without stability information.
Estimate conditioning and verify an independent residual. [NUM]

M04 Newton iteration
J(xk) delta = -F(xk); x(k+1) = xk + alpha delta
F differentiable; local nonsingular Jacobian J; alpha in (0,1]. Global convergence is not automatic.
Check scaled residual, step, line search and domain constraints. [NUM]

M05 Weak Poisson problem
Integral_Omega k grad(u).grad(v) dV = Integral_Omega f v dV + Integral_GammaN g v dS
Solve -div(k grad u)=f; u prescribed on GammaD; k grad(u).n=g on GammaN; k bounded and uniformly positive.
Coercivity with appropriate Dirichlet boundary; pure Neumann needs compatibility and a fixed mean. [FEM]

M06 Galerkin quasi-optimality
a(u-uh,vh)=0; ||u-uh||V <= (M/alpha) inf_vh ||u-vh||V
Continuous bilinear form a with continuity M and coercivity alpha>0; conforming Vh subset V.
Applies to this coercive setting, not every nonlinear or saddle-point model. [FEM]

M07 Residual estimator
eta^2 = sum_K hK^2 ||f+div(k grad uh)||K^2 + sum_e he ||jump(k grad uh.n)||e^2
Illustrative diffusion estimator with matching boundary residuals; coefficient jumps require weighting. Norms are L2.
Reliability constants, data oscillation and regularity must be documented. [FEM]

M08 Richardson order estimate
p = log(|(q3-q2)/(q2-q1)|)/log(r)
h3=r h2=r^2 h1, r>1; asymptotic leading error q(h)=q_exact+C h^p and nonzero differences.
Use at least three refinements; do not use near cancellation or outside asymptotic range. [NUM]

M09 Mixed-method stability
inf_qh sup_vh b(vh,qh)/(||vh||V ||qh||Q) >= beta0 > 0
Incompressible mixed FEM needs an inf-sup stable pair on the actual mesh family, or consistent stabilization.
Taylor-Hood is a method choice, not a universal proof for all geometries. [FEM]

M10 Uncertainty propagation
uc^2(y) = sum_i sum_j (df/dxi)(df/dxj) cov(xi,xj)
First-order propagation for y=f(x); include covariance and measurement model inputs.
For strong nonlinearity propagate distributions and record sampling settings. [NIST]

M11 Zero-failure reliability bound
p_lower = alpha^(1/n); n >= ceil(log(alpha)/log(p_target))
n independent representative Bernoulli trials, all successful; one-sided confidence 1-alpha. For p_target=0.99 and alpha=0.05, n=299.
A lower bound on one defined pass probability is not total engineering feasibility. [DERIVED]

M12 Bounded integer search
D(x1,...,xn)=0; xi in {-M,...,M}; cases=(2M+1)^n
Finite M and integer polynomial D; exhaustive finite search is decidable.
Timeout means unknown; finite absence does not prove unbounded absence. [NUM]

M13 Constrained optimization
min_x f(x), subject to g(x)<=0 and h(x)=0
Geometry, manufacturing and reliability constraints must be explicit.
Local stationary points are not global certificates for nonconvex problems. [NUM]

M14 Linear reduced model
u approximately V a; (V^T A V)a=V^T b
Columns of V form a selected reduced basis; A and b refer to a discretized problem.
Validate outside training samples and retain a residual-based estimator. [FEM]

P01 Mass conservation
partial_t rho + div(rho v) = s_m
rho kg/m^3; velocity v m/s; source s_m kg/(m^3 s).
Integrate flux and storage independently at all scale interfaces. [CONT]

P02 Incompressible momentum
rho(partial_t v + (v.grad)v) = -grad p + mu laplacian v + f; div v=0
Newtonian constant-density fluid; mu Pa s; f force per volume N/m^3. Specify initial and boundary data.
Monitor momentum residual and divergence; validate mesh/time refinement. [CONT]

P03 Heat transport
rho cp(partial_t T+v.grad T)=div(k grad T)+Q
T K; cp J/(kg K); k W/(m K); Q W/m^3. Constant-property sensible heat approximation.
Include latent heat, radiation and thermo-mechanical work when material conditions require. [CONT]

P04 Closed-system first law
Delta E = Q_in - W_out
E J includes relevant internal, kinetic and potential energy; heat Q and work W signed as stated.
Energy conservation checks the balance residual, not whether Delta E is near zero. [THERMO]

P05 Entropy balance
Delta S = Integral delta Q/T_boundary + S_gen; S_gen >= 0
Closed system may exchange heat; isolated system has zero heat and mass exchange.
A closed system can lose entropy; test production including transfers. [THERMO]

P06 Scaled balance residual
rE = |Delta E-Q_in+W_out| / max(E_ref,|Delta E|,|Q_in|,|W_out|)
E_ref>0 J chosen and reported; rE dimensionless.
Tolerance derived from discretization and metrology; 1e-12 J is not universal. [DERIVED]

P07 Small-strain elasticity
eps = (grad u + grad u^T)/2; sigma = lambda tr(eps) I + 2 mu eps
u m; stress Pa; isotropic linear elastic solid; infinitesimal strain.
For stable ordinary compressible material E>0 and -1<nu<0.5. [CONT]

P08 Lame parameters and equilibrium
mu=E/[2(1+nu)]; lambda=E nu/[(1+nu)(1-2nu)]; mu laplacian u+(lambda+mu)grad div u+f=0
E Pa; nu dimensionless; f N/m^3; homogeneous coefficients.
Near incompressibility needs suitable mixed methods; boundary data remain essential. [CONT]

P09 Papkovich-Neuber, force-free form
u = B - grad(r.B+phi)/[4(1-nu)]; laplacian B=0; laplacian phi=0
Homogeneous isotropic elasticity with zero body force; harmonic potentials; r m, B m, phi m^2.
Verified operator identity; solve boundary conditions. Nonzero body force needs a valid particular solution. [DERIVED]

P10 Many-electron Hamiltonian
H = -sum_i hbar^2 laplacian_i/(2 me) - sum_iA ZA e^2/(4 pi eps0 |ri-RA|) + sum_i<j e^2/(4 pi eps0 |ri-rj|) + E_NN|Fixed nuclei, nonrelativistic Coulomb model; E_NN=sum_A<B ZA ZB e^2/(4 pi eps0 |RA-RB
); antisymmetric electronic states.
No universal N>=3 impossibility threshold; finite-basis diagonalization is possible but can scale prohibitively. [QM]

P11 Kohn-Sham equations
[-hbar^2 laplacian/(2 me)+v_ext+v_H+v_xc] psi_i=epsilon_i psi_i; n=sum_i fi |psi_i|^2
fi occupation; n electrons/m^3; normalization Integral n dV=Ne.
Converge basis, k mesh and SCF; PBE does not provide a universal bounded model error. [QE]

P12 Kohn-Sham energy
E[n]=Ts[n]+Integral v_ext n dV+EH[n]+Exc[n]+E_NN
Electronic ground-state functional with nuclear contribution for total energy; convention must be consistent.
Converged electronic numerics do not prove agreement with experimental properties. [QE]

P13 Velocity Verlet
r_new=r+v dt+a dt^2/2; v_new=v+(a+a_new)dt/2
a=F/m evaluated from chosen force model before and after position update.
Time-step refinement, energy drift and force-model validation required. [MD]

P14 Pair-potential virial stress
sigma = -(sum_i mi ci tensor ci + 0.5 sum_i sum_j!=i rij tensor fij)/V
Tension-positive convention; ci peculiar velocity relative to local mean; rij points j to i; fij force on i from j; pair forces.
Pressure tensor has opposite sign; many-body potentials require appropriate virial decomposition. [MD]

P15 Continuum validity indicators
Kn=lambda_mfp/L; Re=rho U L/mu; Ma=U/c_sound
Mean free path lambda_mfp and characteristic length L in m.
Select slip, kinetic or compressible models by regime; scale alone is insufficient. [CONT]

P16 Maxwell equations
div D=rho_e; div B=0; curl E=-partial_t B; curl H=J+partial_t D
SI fields plus constitutive relations, e.g. D=eps E, B=mu H where applicable.
Frequency-dependent, anisotropic or nonlinear media need corresponding constitutive laws. [EM]

P17 Diffraction-limited imaging
d approximately lambda/(2 NA)
Lateral Abbe scale under the stated imaging assumptions; wavelength and NA matter.
Virtual pixels do not establish molecular optical resolution or fabrication capability. [EM]

P18 Ideal gas density
rho=p M/(R T)
p Pa; molar mass M kg/mol; R J/(mol K); T K. R is universal, not the specific gas constant in this expression.
Use real-gas equations where required; local weather and gas temperature are inputs. [THERMO]

S01 Fick transport
J=-D grad c; partial_t c=div(D grad c)+s
c mol/m^3; D m^2/s; J mol/(m^2 s); s mol/(m^3 s).
State phase, temperature, concentration dependence and interface conditions. [MAT]

S02 Arrhenius kinetics
D=D0 exp(-Ea/(R T))
Ea J/mol and R J/(mol K); use Ea/(kB T) only for energy per particle.
Fit D0,Ea with uncertainty; do not mix molar and particle activation energies. [MAT]

S03 Allen-Cahn evolution
F[c]=Integral (f(c)+kappa |grad c|^2/2)dV; partial_t c=-L delta F/delta c
Nonconserved order parameter, L>=0, suitable boundaries.
Check nonincreasing free energy without external work; coefficients need calibration. [MAT]

S04 Cahn-Hilliard evolution
mu_c=df/dc-kappa laplacian c; partial_t c=div(M grad mu_c)
Conserved field c, mobility M>=0; no-flux closed domain conserves integral c.
Check mass conservation and energy decay; boundary conditions on c and mu_c. [MAT]

S05 Linear homogenization
C_eff = volume_average(C(x):A(x)); eps(x)=A(x):eps_macro
Representative volume element and scale separation; A is strain localization tensor.
Converge RVE size and ensure macro-micro work consistency. [MAT]

S06 Hill-Mandel condition
average(sigma:delta eps)=sigma_macro:delta eps_macro
Work equivalence under admissible micro-boundary conditions.
Track interface tractions and displacements; simple parameter copying is insufficient. [MAT]

S07 Thermal strain
eps_th=Integral_Tref^T alpha(theta) dtheta|Scalar for isotropic thermal expansion; tensor form for anisotropy.|Constrained uniaxial elastic magnitude E 
eps_th
 is a special case, not universal stress. [MAT]

S08 J2 yield criterion
sigma_eq=sqrt(3 s:s/2); f=sigma_eq-sigma_y<=0
s=sigma-tr(sigma)I/3; isotropic ductile yield model; Pa.
Composite failure requires laminate allowables and appropriate failure criteria. [MAT]

S09 Fracture mechanics
KI=Y sigma sqrt(pi a); require KI<KIC
Linear elastic mode I with crack size a, geometry factor Y and valid toughness KIC.
Small-scale yielding and specimen validity required; include environment and fatigue. [MAT]

S10 Fatigue growth
da/dN=C (Delta K)^m
Paris regime only; C and m are measured for material/environment/load ratio.
Exclude near-threshold and unstable-growth regimes from this approximation. [MAT]

S11 Cure kinetics
dalpha/dt=(k1+k2 alpha^m)(1-alpha)^n
Empirical autocatalytic model; 0<=alpha<=1; rates temperature-dependent.
Cure fraction alone does not prove zero creep or structural qualification. [MAT]

S12 Pressure-driven membrane permeation
J=(P_perm/t)(p1-p2)
Linear solution-diffusion regime; permeability units chosen to yield mol/(m^2 s); t m.
Use measured species-specific law. Square-root pressure law is not universal for polymers or oxides. [MAT]

E01 Crop water demand
Q=A ETc/[eta_app (1-LF) 86400]
A m^2; ETc m/day; eta_app in (0,1]; LF in [0,1). Simplified steady model without effective rainfall or storage.
Subtract appropriate effective rain and soil-water contributions in a full balance. [FAO]

E02 Crop coefficient
ETc=Kc ETo
Kc dimensionless; ET in mm/day; standard non-water-stressed crop conditions.
Local weather, crop stage, soil and salinity are required; 32 C is not a universal optimum. [FAO]

E03 FAO daily reference evapotranspiration
ETo=[0.408 Delta(Rn-G)+gamma (900/(T+273)) u2(es-ea)]/[Delta+gamma(1+0.34 u2)]
ETo mm/day; Rn,G MJ/(m^2 day); T C; u2 m/s at 2 m; vapour pressures kPa; Delta,gamma kPa/C.
FAO reference grass assumptions and unit-specific coefficients; not arbitrary SI substitution. [FAO]

E04 Pipe continuity
Q_pipe=pi D^2 v/4; N_parallel=ceil(Q_total/Q_pipe)
D inner diameter m; v mean velocity m/s; Q m^3/s.
Parallel branches and route distribution must be solved as a hydraulic network. [CONT]

E05 Swamee-Jain friction factor
fD=0.25/[log10(epsilon/(3.7D)+5.74/Re^0.9)]^2
Darcy friction factor for applicable turbulent pipe flow; roughness epsilon m.
For laminar fully developed round-pipe flow use 64/Re; transitional flow needs special treatment. [FLUID]

E06 Head and pump input
hf=fD (L/D) v^2/(2g); H=hf+Delta z+sum K v^2/(2g)+Delta p/(rho g); P=rho g Q H/eta
H m; P W; eta overall pump efficiency. K minor-loss factors.
Elevation is endpoint energy difference; do not count every elevated section as net head. [FLUID]

E07 Osmotic work approximation
wmin=-(pi0/Y) ln(1-Y)
pi0 Pa; recovery 0<Y<1; wmin J/m^3; idealized dilute solution and negligible salt permeation.
Real seawater requires activity-based thermodynamics; not a full plant SEC prediction. [THERMO]

E08 Desalination mass balance
Qf=Qp/Y; Qb=Qf-Qp; cf Qf=cp Qp+cb Qb
Steady volumetric approximation with comparable densities; concentrations kg/m^3.
Use mass flows with density corrections for accurate concentrate calculations. [DERIVED]

E09 Specific-energy power conversion
P_GW=0.0036 Q_m3s SEC_kWhm3
SEC is electricity per permeate volume; Q is permeate flow.
Include pretreatment, intake, pumping, ZLD and distribution only once. [DERIVED]

E10 Static buoyancy
Fnet=(rho_air-rho_gas)g V-m_payload g
Payload includes structure, contained fluid, envelope, connections and carried equipment.
Add tether, wind, thermal and dynamic loads; no certified design follows from static lift. [CONT]

E11 Thin-wall hoop screening
sigma_theta=p D/(2t)
Thin, circular pressure wall; pressure p Pa; D,t m; ideal membrane screening.
Laminate anisotropy, joints, axial loads, defects and creep require further assessment. [CONT]

E12 Small-sag cable approximation
H0 approximately w L^2/(8 delta)
Uniform transverse load w N/m referred to span, span L m, sag delta m; small sag.
Solve nonlinear catenary and combined loads for final design; wind and lift interact. [CONT]

E13 Wind and vortex screening
F_per_length=0.5 rho Cd U^2 D; fs=St U/D
Cd,St depend on shape and Reynolds number.
Comparing one natural frequency with one shedding frequency does not exclude flutter. [FLUID]

E14 Water hammer screening
Delta p=rho a Delta v
Rapid velocity change relative to wave transit; wave speed a depends on fluid and wall compliance.
Network transient solver needed for valve events, air pockets and pressure limits. [FLUID]

E15 Salt and mineral ceiling
m_species_per_year=Qf c_species recovery_species seconds_year
Qf m^3/s; concentration kg/m^3; recovery in [0,1]; kg/year.
Stoichiometric ceiling only: purity, competing ions, separation energy and market demand still matter. [DERIVED]

E16 Struvite stoichiometry
Mg2+ + NH4+ + PO4(3-) + 6 H2O -> MgNH4PO4.6H2O
Moles product cannot exceed the limiting molar supply of Mg, N or P.
pH speciation, activities, kinetics and fertilizer demand must be quantified independently. [MAT]

E17 Root-zone storage
dS/dt=P_eff+I-ET-R-D
All water terms as depth per time; storage S depth; R runoff; D drainage.
Water exported into atmosphere is not guaranteed to return to the same basin. [FAO]

E18 Radiative and latent energy
q_rad=eps sigmaSB(Ts^4-Ta^4); q_lat=m_dot Lv
Radiative exchange approximation; q_rad W/m^2; q_lat W if m_dot kg/s.
Air mixing, humidity and advection bound evaporative cooling; visual mist proves no temperature reduction. [THERMO]

E19 Cost per accepted product
C_unit=(C_material+C_machine+C_energy+C_labour+C_QA+C_overhead)/N_accepted
All costs in one currency and price date; N_accepted>0; avoid double-counting energy in machine rates.
Use quotes, yield and ranges; no product price without geometry, material, quantity and process. [DERIVED]

E20 Levelized water cost
LCOW=sum_t Ct/(1+r)^t / sum_t Vt/(1+r)^t
Costs Ct include investment, operations and retirement net of stated residual value; Vt sold water; discount r.
Specify currency, base year, real/nominal consistency and financing assumptions. [DERIVED]

E21 Process capability
Cp=(USL-LSL)/(6s); Cpk=min(USL-xbar,xbar-LSL)/(3s)
Stable process; s estimates standard deviation; limits share measurement units.
Normal-process assumptions needed for simple tail probability interpretation; measurement uncertainty matters. [NIST]

E22 CAM scallop screening
h=R-sqrt(R^2-(s/2)^2)
Ball cutter radius R; step-over s<=2R; ideal local flat surface.
Real geometry, tool deflection, chatter, collision and post-processor require verification. [DERIVED]

E23 Additive volumetric energy proxy
Ev=P/(v h t)
P W; scan speed v mm/s; hatch h mm; layer t mm; Ev J/mm^3.
A proxy, not process qualification: equal Ev can produce different melt dynamics. [AM]

E24 Product reuse metric
reuse_rate=N_qualified_reused/N_total
Define item granularity, requirements equivalence, version and qualified context.
98% is a hypothesis until measured; identical shape does not mean identical qualification. [DERIVED]

E25 Quantum variational objective
E(theta)=<psi(theta)|H|psi(theta)>; H=sum_j cj Pj
Normalized trial state; Pauli operators Pj; cj in energy units; finite-basis encoded Hamiltonian.
Ideal variational bound does not remove noisy measurement bias; no universal QPU acceleration claim. [QM]

E26 Cable wave speed and impedance
ct=sqrt(H/muL); Z=sqrt(H muL)
Ideal taut string, tension H N, mass per length muL kg/m; ct m/s; mechanical impedance Z N s/m.
Neglects bending stiffness and nonlinear geometry; use measured broadband dynamics for real corridors. [CONT]

E27 Ideal terminal wave reflection
Gamma=(Zload-Zline)/(Zload+Zline)
Consistent mechanical wave convention and scalar impedances; energy reflection is abs(Gamma)^2 for real positive impedances.
Matching one ideal line does not prove whole-network damping or resilience. [CONT]

E28 Bulk charge relaxation and sheet resistance
tau=epsilon/sigma_e=epsilon rho_e; R_sheet=rho_e/t
Homogeneous Ohmic bulk medium; rho_e ohm m; t m; R_sheet ohms per square.
Do not combine metal bulk resistivity with polymer permittivity to certify a multilayer lightning response. [EM]

E29 Joule heating
E_J=Integral I(t)^2 R(t) dt
Current A, resistance ohm, time s, energy J; applies to resistive dissipation.
High-frequency current crowding, contact resistance, phase change and breakdown need coupled modelling. [EM]

E30 Carnot upper bound
eta_th <= 1-Tcold/Thot; P_e=eta_th Qdot_hot
Reservoir temperatures K; bound for heat-engine conversion between two thermal reservoirs.
Real geothermal cycle, pump loads, reservoir decline and availability reduce net performance. [THERMO]

E31 Lorentz force and resistive MHD
f=J cross B; J=sigma_e(E+v cross B); q_J=J.J/sigma_e
Simple isotropic Ohmic MHD, f N/m3, J A/m2, B T, q_J W/m3.
Electrode chemistry, Hall effects if relevant, magnetic power and hydraulic losses require additional models. [EM]

E32 Inductive energy and discharge
E=0.5 L I^2; P=-dE/dt; Inew=sqrt(Iold^2-2 Delta E/L)
Constant positive inductance L H; ideal released energy Delta E between 0 and Eold.
If requested energy exceeds storage, reject the request; silently clamping current to zero violates delivered-energy accounting. [EM]

E33 Circular toroid inductance
L=mu0 N^2 (R-sqrt(R^2-a^2))
Ideal air-core continuously wound toroid, circular cross-section radius a<R, mean radius R, N turns.
Leakage, finite winding, field variation, support and superconducting constraints require full analysis. [EM]

E34 Magnetic field energy density
uB=B^2/(2 mu0); E=Integral uB dV
Vacuum or linear nonmagnetic approximation; B T; uB J/m3.
Maximum local field cannot be used as a uniform volume average without justification. [EM]

E35 Magnetic-support mass scaling
M_support = C_geom (rho_material/sigma_allow) E_magnetic
Dimensional scaling, not a universal certified lower bound; C_geom depends on load paths, stress state and geometry.
The source's coefficient-one inequality is not a complete virial structural design. [DERIVED]

E36 Precipitation equilibrium and activities
IAP=product_i a_i^nu_i; SI=log10(IAP/Ksp)
Activities dimensionless relative to a standard state; consistent thermodynamic reaction and database temperature.
For saline brine use an appropriate activity model. Concentration-only Ksp and fixed pH do not determine yield. [GEOCHEM]

E37 Trapezoidal channel geometry
A=y(b+z y); Pw=b+2 y sqrt(1+z^2); Rh=A/Pw
Bottom width b m, depth y m, side slope z horizontal/vertical, wetted perimeter Pw m.
Hydraulic radius is not hydraulic depth; use actual geometry, sediment and lining condition. [CHANNEL]

E38 Manning head loss, SI
v=(1/n) Rh^(2/3) sqrt(Sf); hf=n^2 L Q^2/(A^2 Rh^(4/3))
Manning n has units s/m^(1/3); SI convention; quasi-uniform friction model.
Backwater, gates, free-surface transitions and changing depth require spatial hydraulic solution. [CHANNEL]

E39 Series channel and twin-tunnel flow
Delta H=Q^2 [(K+f Lt/Dt)/(2g At^2)+n^2 Lo/(Ao^2 Rh^(4/3))]
At is combined area of identical parallel tunnels; Ao canal area; K referenced to tunnel velocity Q/At; same total Q in series.
Different areas require different velocities. This constant-section screening equation is not a full backwater model. [DERIVED]

E40 Lake water equilibrium
dV/dt=Qin-Qout+A(P-E)-Qseep; Aeq=Qin/(E-P)
Consistent time units; last expression assumes steady state, no outlet/seepage and spatially uniform E>P.
Lake hypsometry, salt accumulation and seasonal climate prevent a universal fixed area prediction. [DERIVED]

E41 Latent heat transport
P_latent=rho_w Q_evap Lv
rho kg/m3, Q m3/s, Lv J/kg; P W.
Energy transfer into water vapour does not prescribe a regional temperature change or create net cooling of the planet. [THERMO]

E42 Atmospheric column moisture balance
partial_t W + div(Fq)=E-P; P=E-partial_t W-div(Fq)
W kg/m2; Fq=(1/g) Integral q v dp, kg/(m s); E,P kg/(m2 s); outward flux divergence convention.
The source image 164 has the wrong sign for outward divergence and omits storage; cloud-water terms may also matter. [ATM]

E43 Soil carbon stock dynamics
dC/dt=I-k C; C(t)=I/k+(C0-I/k) exp(-k t)
Constant I stock/time and k>0 inverse time; C stock on a stated area/depth basis.
For wt%, convert stock using bulk density and sampled depth; calibrated dynamics do not follow from assumed asymptotes. [DERIVED]

E44 Periodic soil thermal penetration
d=sqrt(2 alpha/omega); alpha=k/(rho cp); Tamp(z)=Tamp(0) exp(-z/d)
Homogeneous semi-infinite diffusion model with sinusoidal boundary, omega rad/s.
Surface temperature amplitude is imposed here; changing d alone does not prove a smaller surface amplitude. [CONT]

E45 Spectral index and salinity regression
I=(rho_a-rho_b)/(rho_a+rho_b); EC=a0 exp(a1 I)+a2 rho_c/rho_d
Reflectances dimensionless; denominators nonzero; a0,a2 carry EC units; a1 dimensionless.
Empirical source-specific index, not universal root-zone retrieval. Requires ground truth, calibration and uncertainty. [DERIVED]

E46 Soil carbon to CO2 mass
mCO2=mC MCO2/MC approximately (44.01/12.01) mC
Mass of carbon and CO2 consistent; molar masses g/mol; chemical conversion only.
Sequestration claims need baseline, leakage, permanence, sampling depth and lifecycle emissions. [DERIVED]

E47 Pile shaft resistance screening
Rs=Integral shaft tau_s dA; for uniform shaft Rs=pi D L tau_s
D,L m; shaft resistance tau_s Pa; Rs N for one pile.
Geotechnical factors, installation, rock joints, group effects and tension loading require project-specific design. [CONT]

E48 Linear laminate mechanics
[N;M]=[A B;B D] [eps0;kappa]-[Nthermal;Mthermal]
Classical lamination assumptions; membrane resultants N N/m and moments M N; stiffness from oriented ply data.
Strength, damage, transverse shear, creep, joints and pressure-pipe qualification require further models. [MAT]

E49 Explicit heat-equation benchmark
theta_i^(n+1)=theta_i^n+Fo(theta_(i-1)^n-2theta_i^n+theta_(i+1)^n); Fo=alpha dt/dx^2<=1/2
Uniform 1D constant-diffusivity grid, forward Euler centred differences, fixed end temperatures.
Analytic sine-mode reference theta=exp(-pi^2 alpha t/L^2) sin(pi x/L) enables a convergence test. [NUM]

E50 Ideal net-lift envelope sizing
V_per_length=m_per_length(1+margin)/(rho_air-rho_gas); Denv=sqrt(4 V_per_length/pi)
Circular continuous-envelope approximation; V_per_length m2; complete carried line mass kg/m; positive density difference.
Envelope/tether/ballast mass and wind cannot be omitted in a final sizing calculation. [DERIVED]

M15 Lax-Milgram existence statement
a(u,v)=l(v) for all v in V; norm(u)<=norm(l)/alpha
Real Hilbert space V; bounded coercive bilinear a with coercivity alpha>0; continuous linear functional l.
Existence and uniqueness under these hypotheses; not a statement about arbitrary PDEs. [FEM]

M16 Gronwall bound
y(t)<=a+Integral_0^t b(s)y(s)ds implies y(t)<=a exp(Integral_0^t b(s)ds)
a>=0; y,b nonnegative and appropriately integrable.
A priori amplification estimate; a large bound is not a useful numerical error certificate. [NUM]

M17 Contraction iteration
norm(Tx-Ty)<=q norm(x-y); norm(x*-xk)<=q^k norm(x1-x0)/(1-q)
0<=q<1; complete metric/normed setting and invariant iteration set.
Prove contraction on the actual domain before asserting convergence. [NUM]

M18 SVD regularization
A=U Sigma V^T; x_lambda=sum_i [sigma_i/(sigma_i^2+lambda^2)] (ui^T b) vi
Tikhonov penalty lambda^2 norm(x)^2 with lambda>0, Euclidean norms.
Regularization introduces bias; choose lambda by data uncertainty and validation. [NUM]

M19 Preconditioned conjugate gradients
A x=b; error_A(k)<=2[(sqrt(kappa)-1)/(sqrt(kappa)+1)]^k error_A(0)
Symmetric positive definite A and suitable SPD preconditioner; kappa of preconditioned operator.
Exact-arithmetic bound; monitor finite-precision residual and actual convergence. [NUM]

M20 Generalized eigenproblem
K phi=omega^2 M phi; phi_i^T M phi_j=delta_ij
Symmetric stiffness K, SPD mass M, conservative linear vibration problem.
Constraints remove rigid modes as intended; damping and gyroscopic effects alter formulation. [FEM]

M21 Adjoint sensitivity
R(u,p)=0; R_u^T lambda=J_u^T; dJ/dp=J_p-lambda^T R_p
Differentiable residual and objective with locally invertible state Jacobian.
Check against finite differences; nonsmooth contact or topology changes need special handling. [NUM]

M22 Interval Newton enclosure
N(X)=m-[F'(X)]^(-1)F(m)
Interval Jacobian over box X, point m in X; an appropriate interval linear solver is needed.
Inclusion and uniqueness conditions are method-specific; interval overestimation can prevent certification. [NUM]

M23 Bayesian calibration
p(theta|d) proportional to p(d|theta) p(theta)
Specify prior, likelihood, noise and model discrepancy; posterior is conditional on these choices.
Do not equate posterior confidence with externally validated physical truth. [NUM]

M24 Monte Carlo standard error
SE(mean)=s/sqrt(N)
Independent identically distributed samples with finite variance; correlated chains need effective sample size.
Record seeds and convergence; deterministic reproducibility does not eliminate sampling uncertainty. [NUM]

M25 Gaussian chance constraint
Pr(g<=0)>=1-alpha is equivalent to mu_g+z_(1-alpha) sigma_g<=0
Exactly Gaussian scalar g with known mean/standard deviation; z is standard-normal quantile.
Approximate for non-Gaussian nonlinear responses; include parameter estimation uncertainty. [NUM]

M26 Topology-optimization interpolation
E(rho)=Emin+rho^p(E0-Emin); 0<=rho<=1
SIMP interpolation, penalization p>1; positive Emin; density-filter/projection definition required.
Mesh dependence, minimum feature, buckling, fatigue and manufacturing constraints need explicit treatment. [FEM]

M27 Discrete conservation at interfaces
sum_cells Delta conserved_quantity + Integral boundary_flux dt = sources
Every internal face uses equal and opposite fluxes in a common unit convention.
Check global and local budgets; interpolation between nonmatching meshes must preserve transfer quantities. [DERIVED]

M28 CFL condition for upwind transport
C=abs(a) dt/dx<=1
1D linear advection, uniform grid, explicit first-order upwind scheme.
This bound is scheme-specific; not a universal timestep criterion for all PDE solvers. [NUM]

M29 Energy-stable backward Euler diffusion
(norm(u_new)^2-norm(u_old)^2)/(2dt)+k norm(grad u_new)^2<=0
Homogeneous diffusion, zero source, compatible homogeneous boundaries, k>0 and suitable inner product.
Unconditional linear stability is not unconditional time accuracy. [FEM]

M30 Dimensionless scalar error budget
e_total <= e_discretization+e_iteration+e_input+e_model
Triangle-inequality bound only if every term bounds the same quantity in compatible units/norms.
Unknown model discrepancy must remain unknown; do not assign it a convenient zero. [DERIVED]

P19 One-dimensional quantum box
En=n^2 pi^2 hbar^2/(2m L^2); psi_n=sqrt(2/L) sin(n pi x/L)
Infinite potential walls, 0<x<L, n positive integer, nonrelativistic single particle.
An exact benchmark, not an electronic-structure model of an arbitrary product. [QM]

P20 Harmonic oscillator spectrum
En=hbar omega(n+1/2)
One-dimensional quadratic potential V=m omega^2 x^2/2, omega>0, n>=0.
Real molecular vibrations require force constants and anharmonic/coupled-mode corrections. [QM]

P21 Born-Oppenheimer force
F_A=-partial E_electronic_total/partial R_A
Adiabatic ground-state potential-energy surface, consistent total energy.
Incomplete basis may need Pulay corrections; nonadiabatic transitions need additional dynamics. [QE]

P22 Transition-state rate estimate
k(T)=kappa(T) (kB T/h) exp(-Delta G_dagger/(R T))
Molar activation free energy J/mol; transmission coefficient kappa; appropriate standard-state conventions.
Recrossing, tunnelling and multiple pathways require treatment; activation data cannot be invented. [THERMO]

P23 Boltzmann distribution
p_i=exp(-Ei/(kB T))/Z; Z=sum_i exp(-Ei/(kB T))
Canonical equilibrium, states and degeneracy specified; T>0.
Non-equilibrium processes and strongly driven manufacturing are outside this equilibrium statement. [THERMO]

P24 Fluctuation-dissipation Langevin model
m dv=-gamma v dt+F dt+sqrt(2 gamma kB T) dW
Markovian white-noise bath, gamma kg/s, W Wiener process; chosen stochastic convention.
Validate thermostat and time integration; seed reproducibility is not independent replication. [MD]

P25 Boltzmann transport
partial_t f+v.grad_x f+a.grad_v f=C[f]
Distribution f in phase space; collision operator requires a physical model.
For rarefied gases choose collision cross sections, boundaries and statistical error controls. [CONT]

P26 Radiative transfer
dI_lambda/ds=-kappa_lambda I_lambda+j_lambda
No scattering in this simplified ray form; intensity, absorption and emission use consistent spectral units.
Scattering, wavelength integration and view geometry must be added when relevant. [EM]

P27 Thermoelastic coupling
sigma=C:(eps-eps_th); rho u_tt=div sigma+f
Small-strain elastic mechanics with thermal eigenstrain; C units Pa.
Temperature-dependent properties, plastic dissipation and geometric nonlinearities require extension. [CONT]

P28 General finite-strain balances
F=I+Grad u; J=det F>0; rho0=rho J; Div P+rho0 b=rho0 u_tt
Reference-coordinate deformation gradient F, first Piola stress P, mass density rho0.
Constitutive law must be objective and thermodynamically admissible; prevent inverted elements. [CONT]

P29 Newtonian gravitation and field
laplacian Phi=4 pi G rho; a=-grad Phi
Weak-field slow-motion approximation; boundary conditions and mass distribution specified.
A scale slider cannot extend Newtonian validity into relativistic regimes. [GRAV]

P30 Einstein field equation
G_mu_nu+Lambda g_mu_nu=(8 pi G/c^4) T_mu_nu
Metric-sign and curvature conventions specified consistently; stress-energy tensor conserved covariantly.
Included as a domain boundary, not a solved cosmological manufacturing engine. [GRAV]

P31 Friedmann expansion equation
H^2=(8 pi G/3)rho+Lambda c^2/3-k c^2/a^2
Homogeneous isotropic cosmology; rho mass-equivalent density; choose a in length units and k dimensionless.
Requires cosmological parameters and assumptions; unrelated to increasing machine resolution. [GRAV]

P32 Thermal diffusion analytic benchmark
theta(x,t)=sin(pi x/L) exp(-alpha pi^2 t/L^2)
1D rod, fixed zero excess temperature at endpoints, initial unit sine, constant alpha>0.
Substitute in theta_t=alpha theta_xx; compare numerical solution with exact profile. [DERIVED]

S13 Chemical potential
mu_i=(partial G/partial n_i)_(T,p,nj)
Gibbs energy J, amount n_i mol; chemical potential J/mol.
Standard states and reference energies must match across phases. [THERMO]

S14 Ideal mixing free energy
Delta G_mix=R T sum_i n_i ln(x_i)
Ideal solution with mole fractions x_i>0 and sum x_i=1.
Real alloys/electrolytes require excess Gibbs-energy/activity terms. [THERMO]

S15 Phase equilibrium
T_alpha=T_beta; p_alpha=p_beta; mu_i_alpha=mu_i_beta
Equilibrium phases able to exchange heat, volume and relevant components.
Metastability and kinetic barriers may prevent equilibrium microstructure during manufacture. [MAT]

S16 Classical nucleation
Delta G(r)=4 pi r^2 gamma+(4 pi/3)r^3 Delta gv; r*=-2 gamma/Delta gv
Spherical homogeneous nucleus, gamma>0 J/m2, driving Delta gv<0 J/m3.
Heterogeneous nucleation, anisotropy and atomistic scales can invalidate continuum capillarity. [MAT]

S17 Critical nucleation barrier
Delta G*=16 pi gamma^3/[3(Delta gv)^2]
Same assumptions as S16; units J.
Nucleation rate also depends on a kinetic prefactor and site population. [MAT]

S18 Diffusion length scaling
ell_rms=sqrt(2 d D t)
Isotropic Brownian diffusion in d dimensions, constant D and no boundaries/drift.
Not the same as a guaranteed penetration depth or interface thickness. [MAT]

S19 Reaction-diffusion species balance
partial_t c_i+div(c_i v)=div(D_i grad c_i)+sum_r nu_ir R_r
Concentrations mol/m3, stoichiometric coefficients nu_ir, reaction rates mol/(m3 s).
Enforce elemental and charge conservation; electroneutral or Poisson coupling as appropriate. [MAT]

S20 Nernst-Planck flux
J_i=-D_i grad c_i-z_i D_i F c_i grad(phi)/(R T)+c_i v
Dilute isothermal ionic transport; z_i charge number, Faraday F C/mol.
Concentrated brines require nonideal electrochemical potentials and coupled transport. [GEOCHEM]

S21 Faraday electrolysis balance
m=eta_F M I t/(z F)
M kg/mol, I A, t s, z electrons per product unit, eta_F current efficiency.
Mass yield does not include energy or separation cost; side reactions and purity need testing. [EM]

S22 Butler-Volmer relation
j=j0[exp(alpha_a F eta/(R T))-exp(-alpha_c F eta/(R T))]
Simple one-electron activation-controlled form, current densities A/m2, overpotential eta V.
Mass-transfer limitations, multistep kinetics, surface changes and concentrated electrolytes require extension. [GEOCHEM]

S23 Crystal plasticity kinematics
F=Fe Fp; Lp=dot(Fp) inverse(Fp)=sum_s gamma_dot_s m_s tensor n_s
Slip directions/normals in the intermediate configuration; multiplicative finite-strain decomposition.
Slip systems, hardening and texture must be material-specific and calibrated. [MAT]

S24 Rate-dependent slip rule
gamma_dot_s=gamma0 abs(tau_s/g_s)^n sign(tau_s)
Empirical power-law slip, positive resistance g_s, rate gamma0 1/s.
Parameters depend on material, temperature and strain-rate regime; not universal. [MAT]

S25 Orowan plastic shear rate
gamma_dot=rho_mobile b v_dislocation
Dislocation density 1/m2, Burgers vector magnitude m, mean glide speed m/s.
Needs slip-system resolution, mobile fraction and velocity law. [MAT]

S26 Hall-Petch relation
sigma_y=sigma0+kHP/sqrt(d_grain)
Empirical grain-size strengthening law with calibrated kHP and valid grain-size range.
Do not extrapolate to arbitrarily small grains; mechanisms and inverse trends may change. [MAT]

S27 Norton creep law
eps_dot=A sigma^n exp(-Q/(R T))
Empirical uniaxial steady-creep law; A units depend on stress units; Q J/mol.
Calibration domain, transient creep, rupture and multiaxiality must be addressed. [MAT]

S28 Linear damage accumulation
D=sum_i n_i/N_i
Miner rule for specified cycle classes and material fatigue data.
Load sequence, mean stress, environment and nonlinear damage can invalidate D=1 predictions. [MAT]

S29 Moving heat-source balance
rho cp T_t=div(k grad T)+q(x-vt)-rho Lm f_dot_liquid
Simplified enthalpy balance with latent heat Lm J/kg and liquid fraction.
Laser absorption, melt convection, radiation and evaporation determine process adequacy. [MAT]

S30 Thermal residual-stress screening
sigma_thermal approximately E alpha Delta T/(1-nu)
Ideal isotropic equal biaxial in-plane restraint, plane stress and uniform cooling; sign by convention.
Not a universal welding/additive stress formula; actual constraints and plasticity matter. [MAT]

E51 Rigid-body pose and tool transform
p_world=R p_tool+t; R^T R=I; det R=1
SE(3) pose with rotation R and translation t in consistent length units.
Calibrate robot/tool/workpiece frames; display accuracy does not establish pose accuracy. [ROBOT]

E52 Robot Jacobian and velocity
v_task=J(q) q_dot
Joint coordinates and task-velocity convention fixed; angular and linear components distinguished.
Check singular values, joint limits, collisions and acceleration/torque capacity. [ROBOT]

E53 Kinematic error propagation
Sigma_pose approximately J Sigma_joint J^T+Sigma_calibration
First-order covariance model with consistent coordinates; omitted cross-covariances assumed zero.
Near singularity and nonlinear errors require more complete propagation. [NIST]

E54 State-space control
x_dot=A x+B u; y=C x+D u
Linearized model about a specified operating point, units and sampling stated.
A stable nominal model does not prove robustness to delays, saturation or unmodelled dynamics. [CONTROL]

E55 Discrete Kalman measurement update
K=Pminus H^T(H Pminus H^T+R)^(-1); xplus=xminus+K(y-H xminus)
Linear observation model and nonsingular innovation covariance; classical optimality assumes Gaussian noise.
Bias, correlation and changing noise models require diagnostics; filter confidence is conditional. [CONTROL]

E56 Geometric chord tolerance
delta=R[1-cos(pi/N)]
Regular N-sided circular approximation, N>=3, radius R, maximum radial chord deviation delta.
Select tessellation by a specified geometric tolerance; manufacturing uncertainty is additional. [DERIVED]

E57 CNC material removal rate
MRR=a_p a_e v_f
Axial depth a_p mm, radial engagement a_e mm, feed v_f mm/min; MRR mm3/min for a simple milling estimate.
Tool force, spindle power, chatter, wear and actual engagement limit productivity. [DERIVED]

E58 Yield and scrap propagation
N_started=ceil(N_required/Y); C_accepted=C_total/N_accepted
0<Y<=1 estimated yield; actual accepted count from inspection.
Correlated failures, rework and uncertain yield require scenario ranges rather than a guaranteed unit cost. [DERIVED]

REFERENCES
{
  "BIPM": [
    "BIPM. The International System of Units, SI Brochure, 9th edition. Defining constants and unit conventions.",
    "https://www.bipm.org/en/publications/si-brochure"
  ],
  "NIST": [
    "NIST. Technical Note 1297, Appendix A: Law of Propagation of Uncertainty. For process capability also use a validated statistical process-control method; TN1297 itself concerns measurement uncertainty.",
    "https://www.nist.gov/pml/nist-technical-note-1297/nist-tn-1297-appendix-law-propagation-uncertainty"
  ],
  "NUM": [
    "Standard mathematical references: Trefethen and Bau, Numerical Linear Algebra (SIAM, 1997); Nocedal and Wright, Numerical Optimization, 2nd ed. (Springer, 2006); Matiyasevich, Hilbert's Tenth Problem (MIT Press, 1993). These are bibliographic background references, not an exhaustive literature review.",
    ""
  ],
  "FEM": [
    "Brenner and Scott, The Mathematical Theory of Finite Element Methods, 3rd ed. (Springer, 2008), for variational foundations; MFEM official documentation for implementation capabilities.",
    "https://mfem.org/"
  ],
  "CONT": [
    "Landau and Lifshitz, Theory of Elasticity, 3rd ed. (1986), and Fluid Mechanics, 2nd ed. (1987). Standard continuum-mechanics reference equations; the selected application still requires its own boundary-value formulation.",
    ""
  ],
  "THERMO": [
    "Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (1985), for balances and thermodynamic principles. The desalination expression is an idealized integration of concentration-dependent osmotic pressure, not a certified seawater equation of state.",
    ""
  ],
  "QM": [
    "David Tong, Quantum Mechanics and Topics in Quantum Mechanics, University of Cambridge lecture notes; variational principles and Born-Oppenheimer modelling.",
    "https://www.damtp.cam.ac.uk/user/tong/qm/qm.pdf"
  ],
  "QE": [
    "Quantum ESPRESSO official documentation and pw.x input description. Electronic-structure methods and convergence controls, not guaranteed predictive error.",
    "https://www.quantum-espresso.org/Doc/INPUT_PW.html"
  ],
  "MD": [
    "LAMMPS official documentation, compute stress/atom. Stress/pressure sign convention and kinetic/virial contributions. Verlet integration is a standard numerical method.",
    "https://docs.lammps.org/compute_stress_atom.html"
  ],
  "EM": [
    "Jackson, Classical Electrodynamics, 3rd ed. (1999), for Maxwell equations; Born and Wolf, Principles of Optics, 7th ed. (1999), for diffraction-limited optics.",
    ""
  ],
  "MAT": [
    "Gurtin, Fried and Anand, The Mechanics and Thermodynamics of Continua (2010); Cahn and Hilliard, Free Energy of a Nonuniform System I, Journal of Chemical Physics 28, 258 (1958); Anderson, Fracture Mechanics, 4th ed. (2017). Constitutive parameters and empirical kinetics must be separately sourced for each material.",
    ""
  ],
  "FAO": [
    "Allen, Pereira, Raes and Smith. FAO Irrigation and Drainage Paper 56, Crop Evapotranspiration (1998), Chapters 2 and 5.",
    "https://www.fao.org/4/x0490e/x0490e06.htm"
  ],
  "FLUID": [
    "Swamee and Jain, Explicit equations for pipe-flow problems, Journal of the Hydraulics Division 102(5), 657-664 (1976); standard Darcy-Weisbach and transient-flow mechanics.",
    ""
  ],
  "AM": [
    "NIST, Additive Manufacturing Part Qualification. Energy-density proxies do not establish qualified process windows.",
    "https://www.nist.gov/programs-projects/additive-manufacturing-part-qualification"
  ],
  "EPO": [
    "European Patent Office. EP post-grant events and validity searching; Espacenet. Territorial legal status and additional searches matter.",
    "https://www.epo.org/en/searching-for-patents/helpful-resources/patent-knowledge-news/ep-post-grant-events-and-validity"
  ],
  "EU-M": [
    "EUR-Lex. Machinery safety requirements: Regulation (EU) 2023/1230, application from 20 January 2027. Scope and transitional details require product-specific review.",
    "https://eur-lex.europa.eu/EN/legal-content/summary/machinery-safety-requirements.html"
  ],
  "EU-P": [
    "EUR-Lex. Directive 2014/68/EU on pressure equipment. Consult the consolidated/current applicable text and product scope.",
    "https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX%3A32014L0068"
  ],
  "ISO": [
    "ISO. ISO 10791-1:2015 catalogue entry, geometric tests for machining centres. This public entry is not the full licensed standard.",
    "https://www.iso.org/standard/54729.html"
  ],
  "GEOCHEM": [
    "USGS PHREEQC documentation, speciation, activities, reaction and equilibrium calculations. Database and activity model must match the solution and phase system.",
    "https://water.usgs.gov/water-resources/software/PHREEQC/documentation/phreeqc3-html/phreeqc3-2.htm"
  ],
  "CHANNEL": [
    "US Bureau of Reclamation, Water Measurement Manual, hydraulic radius and slope-area methods. SI Manning convention is stated in this supplement.",
    "https://www.usbr.gov/tsc/techreferences/mands/wmm/chap02_11.html"
  ],
  "ATM": [
    "Column conservation law derived from moisture mass conservation with outward flux sign. ECMWF background on water and energy conservation in atmospheric models.",
    "https://www.ecmwf.int/en/newsletter/172/news/fixing-water-and-energy-budget-imbalances-integrated-forecasting-system"
  ],
  "GRAV": [
    "David Tong, General Relativity, Chapter 4, Einstein equations and Friedmann cosmology. These models are included as domain boundaries and are not implemented in the prototype.",
    "https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S4.html"
  ],
  "ROBOT": [
    "Lynch and Park, Modern Robotics: Mechanics, Planning, and Control (2017), rigid transforms and Jacobians.",
    "https://modernrobotics.northwestern.edu/"
  ],
  "CONTROL": [
    "Astrom and Murray, Feedback Systems: An Introduction for Scientists and Engineers (2008), linear systems, feedback and estimation.",
    "https://fbsbook.org/"
  ],
  "DERIVED": [
    "Algebraic derivation or unit conversion in this supplement. Assumptions and applicable domain are part of the equation record.",
    ""
  ]
}

SOURCE IMAGE TRANSCRIPTIONS
[
  {
    "image": "word/media/image22.png",
    "paragraphs": [
      8712
    ],
    "size": [
      61,
      31
    ],
    "transcription": "2.0 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image23.png",
    "paragraphs": [
      8712,
      8943
    ],
    "size": [
      115,
      32
    ],
    "transcription": "h = 200 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image24.png",
    "paragraphs": [
      8712
    ],
    "size": [
      66,
      31
    ],
    "transcription": "500 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image25.png",
    "paragraphs": [
      8712
    ],
    "size": [
      200,
      32
    ],
    "transcription": "v_w = 10 to 45 m/s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image26.png",
    "paragraphs": [
      8715
    ],
    "size": [
      18,
      32
    ],
    "transcription": "c_t",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image27.png",
    "paragraphs": [
      8715
    ],
    "size": [
      31,
      32
    ],
    "transcription": "H_0",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image28.png",
    "paragraphs": [
      8715
    ],
    "size": [
      61,
      32
    ],
    "transcription": "m_total",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image29.png",
    "paragraphs": [
      8716
    ],
    "size": [
      1100,
      141
    ],
    "transcription": "c_t = sqrt(H_0/m_total) = sqrt(1.006e8 N / 3283 kg/m) approximately 175.05 m/s",
    "review": "E26"
  },
  {
    "image": "word/media/image30.png",
    "paragraphs": [
      8716
    ],
    "size": [
      1061,
      98
    ],
    "transcription": "Z_line = sqrt(H_0 m_total) = sqrt(1.006e8 x 3283) approximately 5.747e5 N s/m",
    "review": "E26"
  },
  {
    "image": "word/media/image31.png",
    "paragraphs": [
      8717
    ],
    "size": [
      66,
      31
    ],
    "transcription": "200 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image32.png",
    "paragraphs": [
      8717
    ],
    "size": [
      158,
      32
    ],
    "transcription": "Z_damper = Z_line",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image33.png",
    "paragraphs": [
      8717
    ],
    "size": [
      1061,
      122
    ],
    "transcription": "Gamma_reflection = (Z_damper-Z_line)/(Z_damper+Z_line) approximately 0",
    "review": "E27"
  },
  {
    "image": "word/media/image34.png",
    "paragraphs": [
      8718
    ],
    "size": [
      73,
      31
    ],
    "transcription": "greater than 92%",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image35.png",
    "paragraphs": [
      8718
    ],
    "size": [
      130,
      31
    ],
    "transcription": "450,000 km",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image36.png",
    "paragraphs": [
      8720
    ],
    "size": [
      121,
      31
    ],
    "transcription": "543.6 GWe",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image37.png",
    "paragraphs": [
      8720
    ],
    "size": [
      67,
      31
    ],
    "transcription": "245 C",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image38.png",
    "paragraphs": [
      8720
    ],
    "size": [
      68,
      32
    ],
    "transcription": "C5H10",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image39.png",
    "paragraphs": [
      8720
    ],
    "size": [
      146,
      32
    ],
    "transcription": "T_c = 238.6 C",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image40.png",
    "paragraphs": [
      8720
    ],
    "size": [
      166,
      32
    ],
    "transcription": "P_c = 4.51 MPa",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image41.png",
    "paragraphs": [
      8723
    ],
    "size": [
      131,
      32
    ],
    "transcription": "R >= 65/100",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image42.png",
    "paragraphs": [
      8723,
      8961
    ],
    "size": [
      157,
      31
    ],
    "transcription": "46,346.5 m3/s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image43.png",
    "paragraphs": [
      8725
    ],
    "size": [
      1100,
      123
    ],
    "transcription": "Global Capital Solvency Ratio = Phi_annual,Noble / CAPEX_annual,global = 545.30 billion USD/year / 59.45 billion USD/year approximately 9.17",
    "review": "E19/E20; ratio is not solvency"
  },
  {
    "image": "word/media/image44.png",
    "paragraphs": [
      8932,
      8941
    ],
    "size": [
      48,
      32
    ],
    "transcription": "LH2",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image45.png",
    "paragraphs": [
      8932
    ],
    "size": [
      130,
      32
    ],
    "transcription": "D = 0.35 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image46.png",
    "paragraphs": [
      8932
    ],
    "size": [
      137,
      32
    ],
    "transcription": "T = 20.28 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image47.png",
    "paragraphs": [
      8932
    ],
    "size": [
      117,
      32
    ],
    "transcription": "D = 2.0 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image48.png",
    "paragraphs": [
      8932
    ],
    "size": [
      151,
      32
    ],
    "transcription": "T = 303.15 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image49.png",
    "paragraphs": [
      8932
    ],
    "size": [
      172,
      32
    ],
    "transcription": "Delta T = 282.87 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image50.png",
    "paragraphs": [
      8935
    ],
    "size": [
      80,
      32
    ],
    "transcription": "Delta L / L_0",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image51.png",
    "paragraphs": [
      8935
    ],
    "size": [
      85,
      31
    ],
    "transcription": "20.28 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image52.png",
    "paragraphs": [
      8935
    ],
    "size": [
      55,
      32
    ],
    "transcription": "alpha(T)",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image53.png",
    "paragraphs": [
      8936
    ],
    "size": [
      1100,
      130
    ],
    "transcription": "Delta L/L_0 = integral from T_ambient to T_cryo of alpha(T) dT",
    "review": "S07"
  },
  {
    "image": "word/media/image54.png",
    "paragraphs": [
      8936
    ],
    "size": [
      222,
      31
    ],
    "transcription": "303.15 K -> 20.28 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image55.png",
    "paragraphs": [
      8936
    ],
    "size": [
      1061,
      125
    ],
    "transcription": "(Delta L/L_0)_Ti approximately -1.72e-3 (-172.0 mm per 100 m)",
    "review": "S07; material data required"
  },
  {
    "image": "word/media/image56.png",
    "paragraphs": [
      8937
    ],
    "size": [
      1061,
      125
    ],
    "transcription": "(Delta L/L_0)_UHMWPE approximately -3.124e-3 (-312.4 mm per 100 m)",
    "review": "S07; material data required"
  },
  {
    "image": "word/media/image57.png",
    "paragraphs": [
      8938
    ],
    "size": [
      77,
      32
    ],
    "transcription": "sigma_thermal",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image58.png",
    "paragraphs": [
      8938
    ],
    "size": [
      1061,
      123
    ],
    "transcription": "sigma_thermal = E abs(Delta L/L_0) = 114e9 Pa x 0.00172 approximately 196.08 MPa",
    "review": "S07; uniaxial constraint only"
  },
  {
    "image": "word/media/image59.png",
    "paragraphs": [
      8940,
      8959
    ],
    "size": [
      66,
      31
    ],
    "transcription": "100 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image60.png",
    "paragraphs": [
      8940
    ],
    "size": [
      107,
      31
    ],
    "transcription": "plus/minus 350 mm",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image61.png",
    "paragraphs": [
      8940
    ],
    "size": [
      93,
      32
    ],
    "transcription": "N > 1e7",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image62.png",
    "paragraphs": [
      8943
    ],
    "size": [
      161,
      32
    ],
    "transcription": "I_peak >= 200 kA",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image63.png",
    "paragraphs": [
      8947
    ],
    "size": [
      1100,
      95
    ],
    "transcription": "tau_relax = epsilon_0 epsilon_r rho_surface",
    "review": "E28/E29; not a lightning proof"
  },
  {
    "image": "word/media/image64.png",
    "paragraphs": [
      8947
    ],
    "size": [
      1061,
      95
    ],
    "transcription": "rho_surface approximately 2.44e-8 ohm m",
    "review": "E28; bulk resistivity, not sheet resistance"
  },
  {
    "image": "word/media/image65.png",
    "paragraphs": [
      8947
    ],
    "size": [
      1061,
      95
    ],
    "transcription": "tau_relax approximately 8.854e-12 x 8.5 x 2.44e-8 approximately 1.83e-18 s",
    "review": "E28/E29; multilayer model invalid as a proof"
  },
  {
    "image": "word/media/image66.png",
    "paragraphs": [
      8952
    ],
    "size": [
      272,
      31
    ],
    "transcription": "70.59 billion USD/year",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image67.png",
    "paragraphs": [
      8953
    ],
    "size": [
      285,
      31
    ],
    "transcription": "545.30 billion USD/year",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image68.png",
    "paragraphs": [
      8954
    ],
    "size": [
      1061,
      123
    ],
    "transcription": "Phi_annual,Noble / sum CAPEX_global = 545.30 billion / 70.59 billion approximately 7.72 (source label: Fully Solvent Across All Red Zones)",
    "review": "E19/E20; ratio is not solvency"
  },
  {
    "image": "word/media/image69.png",
    "paragraphs": [
      8959
    ],
    "size": [
      107,
      31
    ],
    "transcription": "243.2 mm",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image70.png",
    "paragraphs": [
      8960
    ],
    "size": [
      201,
      32
    ],
    "transcription": "tau approximately 1.83e-18 s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image71.png",
    "paragraphs": [
      8961
    ],
    "size": [
      137,
      31
    ],
    "transcription": "18,400 m3/s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image72.png",
    "paragraphs": [
      8961
    ],
    "size": [
      157,
      31
    ],
    "transcription": "27,946.5 m3/s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image73.png",
    "paragraphs": [
      8961
    ],
    "size": [
      222,
      31
    ],
    "transcription": "545.30 billion USD/year",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image74.png",
    "paragraphs": [
      9390
    ],
    "size": [
      178,
      31
    ],
    "transcription": "450,000 line-km",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image75.png",
    "paragraphs": [
      9390,
      9416
    ],
    "size": [
      97,
      31
    ],
    "transcription": "plus/minus 800 kV",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image76.png",
    "paragraphs": [
      9390
    ],
    "size": [
      221,
      32
    ],
    "transcription": "Delta P / Delta t > 25 GW/s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image77.png",
    "paragraphs": [
      9390
    ],
    "size": [
      91,
      31
    ],
    "transcription": "greater than 50 ms",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image78.png",
    "paragraphs": [
      9393
    ],
    "size": [
      40,
      32
    ],
    "transcription": "E_M",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image79.png",
    "paragraphs": [
      9393
    ],
    "size": [
      50,
      32
    ],
    "transcription": "R_maj",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image80.png",
    "paragraphs": [
      9393
    ],
    "size": [
      42,
      32
    ],
    "transcription": "r_min",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image81.png",
    "paragraphs": [
      9393
    ],
    "size": [
      23,
      32
    ],
    "transcription": "N",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image82.png",
    "paragraphs": [
      9393
    ],
    "size": [
      31,
      32
    ],
    "transcription": "I_op",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image83.png",
    "paragraphs": [
      9394
    ],
    "size": [
      1100,
      118
    ],
    "transcription": "E_M = (1/2) L_toroid I_op^2 = (mu_0 N^2/2) (R_maj-sqrt(R_maj^2-r_min^2)) I_op^2",
    "review": "E32/E33"
  },
  {
    "image": "word/media/image84.png",
    "paragraphs": [
      9395
    ],
    "size": [
      158,
      32
    ],
    "transcription": "R_maj = 4.50 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image85.png",
    "paragraphs": [
      9396
    ],
    "size": [
      151,
      32
    ],
    "transcription": "r_min = 1.20 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image86.png",
    "paragraphs": [
      9397
    ],
    "size": [
      158,
      32
    ],
    "transcription": "B_max = 14.5 T",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image87.png",
    "paragraphs": [
      9398
    ],
    "size": [
      30,
      32
    ],
    "transcription": "u_B",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image88.png",
    "paragraphs": [
      9398
    ],
    "size": [
      1061,
      125
    ],
    "transcription": "u_B = B_max^2/(2 mu_0) = 14.5^2/[2 x (4 pi x 1e-7)] approximately 8.366e7 J/m3 (83.66 MJ/m3)",
    "review": "E34"
  },
  {
    "image": "word/media/image89.png",
    "paragraphs": [
      9399
    ],
    "size": [
      1061,
      95
    ],
    "transcription": "E_node approximately 12.5 MWh = 4.50e10 J",
    "review": "E32; assumed energy capacity"
  },
  {
    "image": "word/media/image90.png",
    "paragraphs": [
      9400
    ],
    "size": [
      72,
      32
    ],
    "transcription": "M_struct",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image91.png",
    "paragraphs": [
      9400
    ],
    "size": [
      1061,
      112
    ],
    "transcription": "M_struct >= (rho_struct/sigma_allow) E_M",
    "review": "E35; no universal coefficient-one bound"
  },
  {
    "image": "word/media/image92.png",
    "paragraphs": [
      9401
    ],
    "size": [
      197,
      32
    ],
    "transcription": "rho_c = 1500 kg/m3",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image93.png",
    "paragraphs": [
      9401
    ],
    "size": [
      190,
      32
    ],
    "transcription": "sigma_allow = 850 MPa",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image94.png",
    "paragraphs": [
      9401
    ],
    "size": [
      1022,
      123
    ],
    "transcription": "M_struct >= (1500/8.50e8) x 4.50e10 approximately 79,412 kg (79.4 metric tonnes)",
    "review": "E35; arithmetic only"
  },
  {
    "image": "word/media/image95.png",
    "paragraphs": [
      9404
    ],
    "size": [
      236,
      31
    ],
    "transcription": "873 million tonnes/year",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image96.png",
    "paragraphs": [
      9404,
      9417
    ],
    "size": [
      220,
      31
    ],
    "transcription": "180 million hectares",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image97.png",
    "paragraphs": [
      9404
    ],
    "size": [
      20,
      32
    ],
    "transcription": "P",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image98.png",
    "paragraphs": [
      9404
    ],
    "size": [
      23,
      32
    ],
    "transcription": "K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image99.png",
    "paragraphs": [
      9404
    ],
    "size": [
      61,
      31
    ],
    "transcription": "PO4(3-)",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image100.png",
    "paragraphs": [
      9404
    ],
    "size": [
      35,
      31
    ],
    "transcription": "K+",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image101.png",
    "paragraphs": [
      9404
    ],
    "size": [
      150,
      31
    ],
    "transcription": "104,133 m3/s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image102.png",
    "paragraphs": [
      9408
    ],
    "size": [
      1100,
      93
    ],
    "transcription": "Mg2+ + NH4+ + PO4(3-) + 6 H2O <-> MgNH4PO4.6H2O (solid)",
    "review": "E16/E36"
  },
  {
    "image": "word/media/image103.png",
    "paragraphs": [
      9408
    ],
    "size": [
      40,
      32
    ],
    "transcription": "K'_sp",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image104.png",
    "paragraphs": [
      9408
    ],
    "size": [
      55,
      31
    ],
    "transcription": "25 C",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image105.png",
    "paragraphs": [
      9408
    ],
    "size": [
      1061,
      95
    ],
    "transcription": "K_sp = [Mg2+][NH4+][PO4(3-)] approximately 7.08e-14 (pK_sp approximately 13.15)",
    "review": "E36; activities and convention required"
  },
  {
    "image": "word/media/image106.png",
    "paragraphs": [
      9409
    ],
    "size": [
      335,
      31
    ],
    "transcription": "[Mg2+] approximately 2.35 g/L approximately 96.7 mM",
    "review": "E36; Mg molar mass conversion"
  },
  {
    "image": "word/media/image107.png",
    "paragraphs": [
      9409
    ],
    "size": [
      321,
      31
    ],
    "transcription": "1.85 million metric tonnes/year",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image108.png",
    "paragraphs": [
      9416
    ],
    "size": [
      116,
      31
    ],
    "transcription": "12.5 MWh",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image109.png",
    "paragraphs": [
      9416
    ],
    "size": [
      81,
      31
    ],
    "transcription": "less than 0.1%",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image110.png",
    "paragraphs": [
      9417
    ],
    "size": [
      243,
      31
    ],
    "transcription": "1.85 million tonnes/year",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image111.png",
    "paragraphs": [
      9637
    ],
    "size": [
      168,
      47
    ],
    "transcription": "z_bed = -15.2 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image112.png",
    "paragraphs": [
      9637
    ],
    "size": [
      173,
      47
    ],
    "transcription": "L_canal = 380 km",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image113.png",
    "paragraphs": [
      9637
    ],
    "size": [
      115,
      47
    ],
    "transcription": "9500 km2",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image114.png",
    "paragraphs": [
      9640
    ],
    "size": [
      13,
      47
    ],
    "transcription": "v",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image115.png",
    "paragraphs": [
      9640
    ],
    "size": [
      20,
      47
    ],
    "transcription": "Q",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image116.png",
    "paragraphs": [
      9640
    ],
    "size": [
      505,
      47
    ],
    "transcription": "Delta H = z_ocean-z_lake = 0.0-(-11.5) = 11.5 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image117.png",
    "paragraphs": [
      9640,
      9664
    ],
    "size": [
      80,
      47
    ],
    "transcription": "380 km",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image118.png",
    "paragraphs": [
      9641
    ],
    "size": [
      1100,
      147
    ],
    "transcription": "Q = A v = A sqrt(2 g Delta H / [sum K_minor + f_D L_tunnel/D_h + 2 g L_open n^2/R_h^(4/3)])",
    "review": "E39; replace common velocity with section-specific velocities"
  },
  {
    "image": "word/media/image119.png",
    "paragraphs": [
      9642
    ],
    "size": [
      170,
      47
    ],
    "transcription": "L_open = 338 km",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image120.png",
    "paragraphs": [
      9642
    ],
    "size": [
      118,
      47
    ],
    "transcription": "b = 45.0 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image121.png",
    "paragraphs": [
      9642
    ],
    "size": [
      80,
      47
    ],
    "transcription": "z = 2.0",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image122.png",
    "paragraphs": [
      9642
    ],
    "size": [
      108,
      47
    ],
    "transcription": "y = 6.0 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image123.png",
    "paragraphs": [
      9642
    ],
    "size": [
      108,
      47
    ],
    "transcription": "n = 0.013",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image124.png",
    "paragraphs": [
      9643
    ],
    "size": [
      138,
      47
    ],
    "transcription": "R_h = 4.76 m",
    "review": "E37"
  },
  {
    "image": "word/media/image125.png",
    "paragraphs": [
      9643
    ],
    "size": [
      188,
      47
    ],
    "transcription": "A_open = 342.0 m2",
    "review": "E37"
  },
  {
    "image": "word/media/image126.png",
    "paragraphs": [
      9644
    ],
    "size": [
      170,
      47
    ],
    "transcription": "L_tunnel = 42 km",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image127.png",
    "paragraphs": [
      9644
    ],
    "size": [
      180,
      47
    ],
    "transcription": "D_tunnel = 12.0 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image128.png",
    "paragraphs": [
      9644
    ],
    "size": [
      122,
      47
    ],
    "transcription": "f_D = 0.011",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image129.png",
    "paragraphs": [
      9645
    ],
    "size": [
      373,
      47
    ],
    "transcription": "A_tunnel = 2 x (pi/4) x 12.0^2 = 226.2 m2",
    "review": "E39"
  },
  {
    "image": "word/media/image130.png",
    "paragraphs": [
      9646
    ],
    "size": [
      1061,
      95
    ],
    "transcription": "v_canal approximately 1.38 m/s",
    "review": "E39; source result not a solved network"
  },
  {
    "image": "word/media/image131.png",
    "paragraphs": [
      9646
    ],
    "size": [
      1061,
      95
    ],
    "transcription": "Q_gravity = 342.0 m2 x 1.38 m/s approximately 471.96 m3/s (4.72e5 L/s)",
    "review": "E39"
  },
  {
    "image": "word/media/image132.png",
    "paragraphs": [
      9647
    ],
    "size": [
      1061,
      95
    ],
    "transcription": "V_annual = 471.96 m3/s x 3.1536e7 s approximately 1.488e10 m3/year (14.88 billion m3/year)",
    "review": "E40"
  },
  {
    "image": "word/media/image133.png",
    "paragraphs": [
      9649
    ],
    "size": [
      420,
      47
    ],
    "transcription": "E_evap approximately 2200 mm/year = 2.2 m/year",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image134.png",
    "paragraphs": [
      9650
    ],
    "size": [
      1061,
      126
    ],
    "transcription": "A_eq = V_annual/E_evap = 1.488e10 m3/2.2 m approximately 6.764e9 m2 approximately 6764 km2",
    "review": "E40"
  },
  {
    "image": "word/media/image135.png",
    "paragraphs": [
      9651
    ],
    "size": [
      221,
      47
    ],
    "transcription": "lambda = 2.45e6 J/kg",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image136.png",
    "paragraphs": [
      9651
    ],
    "size": [
      1061,
      120
    ],
    "transcription": "P_cooling = (1.488e13 kg x 2.45e6 J/kg)/3.1536e7 s approximately 1.156e12 W = 1156 GWth",
    "review": "E41; no direct temperature prediction"
  },
  {
    "image": "word/media/image137.png",
    "paragraphs": [
      9651,
      9665
    ],
    "size": [
      92,
      47
    ],
    "transcription": "-3.65 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image138.png",
    "paragraphs": [
      9653,
      9896
    ],
    "size": [
      220,
      47
    ],
    "transcription": "180 million hectares",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image139.png",
    "paragraphs": [
      9656
    ],
    "size": [
      12,
      47
    ],
    "transcription": "theta",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image140.png",
    "paragraphs": [
      9656
    ],
    "size": [
      48,
      47
    ],
    "transcription": "EC_e",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image141.png",
    "paragraphs": [
      9657
    ],
    "size": [
      1100,
      117
    ],
    "transcription": "NDSI = (rho_2100nm-rho_1600nm)/(rho_2100nm+rho_1600nm)",
    "review": "E45"
  },
  {
    "image": "word/media/image142.png",
    "paragraphs": [
      9657
    ],
    "size": [
      1100,
      123
    ],
    "transcription": "EC_e = a0 exp(a1 NDSI) + a2 (rho_850nm/rho_670nm)",
    "review": "E45; calibration missing"
  },
  {
    "image": "word/media/image143.png",
    "paragraphs": [
      9664
    ],
    "size": [
      93,
      47
    ],
    "transcription": "-15.2 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image144.png",
    "paragraphs": [
      9664
    ],
    "size": [
      132,
      47
    ],
    "transcription": "471.96 m3/s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image145.png",
    "paragraphs": [
      9664
    ],
    "size": [
      115,
      47
    ],
    "transcription": "6764 km2",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image146.png",
    "paragraphs": [
      9665
    ],
    "size": [
      133,
      47
    ],
    "transcription": "1156 GWth",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image147.png",
    "paragraphs": [
      9666
    ],
    "size": [
      61,
      47
    ],
    "transcription": "2.5 m",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image148.png",
    "paragraphs": [
      9872
    ],
    "size": [
      152,
      47
    ],
    "transcription": "1.8e6 km2",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image149.png",
    "paragraphs": [
      9872
    ],
    "size": [
      108,
      47
    ],
    "transcription": "less than 250 micrometres",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image150.png",
    "paragraphs": [
      9875
    ],
    "size": [
      48,
      47
    ],
    "transcription": "dC_soil/dt",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image151.png",
    "paragraphs": [
      9876
    ],
    "size": [
      1100,
      125
    ],
    "transcription": "dC_soil/dt = I_biomass(t) - k_microbial C_soil(t) exp(-E_a/(R T_soil)) (theta(t)/theta_field)^gamma",
    "review": "E43; carbon-stock basis required"
  },
  {
    "image": "word/media/image152.png",
    "paragraphs": [
      9876
    ],
    "size": [
      425,
      47
    ],
    "transcription": "I_biomass = 8.4 metric tonnes C/(ha year)",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image153.png",
    "paragraphs": [
      9877
    ],
    "size": [
      120,
      47
    ],
    "transcription": "theta/theta_field approximately 0.72",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image154.png",
    "paragraphs": [
      9878
    ],
    "size": [
      142,
      47
    ],
    "transcription": "2026 -> 2036",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image155.png",
    "paragraphs": [
      9879
    ],
    "size": [
      482,
      47
    ],
    "transcription": "C_soil(0) = 0.18 wt% -> C_soil(10) = 3.65 wt%",
    "review": "E43; conflicts with source code at ten years"
  },
  {
    "image": "word/media/image156.png",
    "paragraphs": [
      9880
    ],
    "size": [
      600,
      47
    ],
    "transcription": "lambda_dry = 0.25 W/(m K) -> lambda_wet,humus = 1.48 W/(m K)",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image157.png",
    "paragraphs": [
      9881
    ],
    "size": [
      161,
      57
    ],
    "transcription": "d_damp = sqrt(2 lambda/(rho cp omega))",
    "review": "E44"
  },
  {
    "image": "word/media/image158.png",
    "paragraphs": [
      9881
    ],
    "size": [
      80,
      47
    ],
    "transcription": "plus 142%",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image159.png",
    "paragraphs": [
      9881
    ],
    "size": [
      178,
      47
    ],
    "transcription": "Delta T_surface = 55 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image160.png",
    "paragraphs": [
      9881
    ],
    "size": [
      178,
      47
    ],
    "transcription": "Delta T_surface = 14 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image161.png",
    "paragraphs": [
      9883
    ],
    "size": [
      233,
      47
    ],
    "transcription": "14.48 billion m3/year",
    "review": "E40; source says 14.48, earlier calculation says 14.88"
  },
  {
    "image": "word/media/image162.png",
    "paragraphs": [
      9883,
      9896
    ],
    "size": [
      137,
      47
    ],
    "transcription": "18,400 m3/s",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image163.png",
    "paragraphs": [
      9886
    ],
    "size": [
      97,
      47
    ],
    "transcription": "P_downwind",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image164.png",
    "paragraphs": [
      9887
    ],
    "size": [
      1100,
      122
    ],
    "transcription": "P_downwind = E_local + div(Q_vapor) = E_local + (1/g) integral_0^p0 (v.grad q + q div v) dp",
    "review": "E42; sign and storage correction"
  },
  {
    "image": "word/media/image165.png",
    "paragraphs": [
      9887
    ],
    "size": [
      68,
      47
    ],
    "transcription": "beta_recycle",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image166.png",
    "paragraphs": [
      9887,
      9896
    ],
    "size": [
      67,
      47
    ],
    "transcription": "46.8%",
    "review": "E42; unvalidated source percentage"
  },
  {
    "image": "word/media/image167.png",
    "paragraphs": [
      9888,
      9896
    ],
    "size": [
      168,
      47
    ],
    "transcription": "plus 215 mm/year",
    "review": "E42; unvalidated source rainfall claim"
  },
  {
    "image": "word/media/image168.png",
    "paragraphs": [
      9888
    ],
    "size": [
      67,
      47
    ],
    "transcription": "14.2%",
    "review": "E42; unvalidated source reduction"
  },
  {
    "image": "word/media/image169.png",
    "paragraphs": [
      9895
    ],
    "size": [
      67,
      47
    ],
    "transcription": "0.18%",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image170.png",
    "paragraphs": [
      9895
    ],
    "size": [
      67,
      47
    ],
    "transcription": "3.65%",
    "review": "E43; ten-year code mismatch"
  },
  {
    "image": "word/media/image171.png",
    "paragraphs": [
      9895
    ],
    "size": [
      52,
      47
    ],
    "transcription": "55 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image172.png",
    "paragraphs": [
      9895
    ],
    "size": [
      52,
      47
    ],
    "transcription": "14 K",
    "review": "Source symbol or scenario input; no independent validation implied"
  },
  {
    "image": "word/media/image173.png",
    "paragraphs": [
      9896
    ],
    "size": [
      137,
      47
    ],
    "transcription": "13,248 m3/s",
    "review": "E42; prescribed scenario output"
  }
]