---
title: "algorithm-composition-wave19 — Mova code model"
description: "Mova code model of algorithms/tests/composition_wave19.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/algorithms/tests/composition_wave19.rs.groupant/"
---
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# algorithm-composition-wave19

The groupant beside algorithms/tests/composition_wave19.rs in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

## Facts

- **kind:** groupant
- **status:** partial source bound model
- **source:** [`algorithms/tests/composition_wave19.rs`](https://github.com/ElamurAI/mova-free/blob/main/algorithms/tests/composition_wave19.rs)
- **source SHA-256:** `894e55be473b27d0…`
- **repository folder:** [`algorithms/tests/composition_wave19.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/algorithms/tests/composition_wave19.rs.groupant)
- **repository commit:** `855da7bd4a7b`

> **Implementation material.** A Mova code model describes this code at one source revision: partial models, finite checks and their stated scope. It is not a reviewed general exposition, and its labels are shown as written, never raised.

## Components

| component | label | where |
|---|---|---|
| **approximant** | partial source model | [forward](#forward) |
| **ast** | partial | `syntax.json` |
| **composition** | partial source model | [composition](#composition) |
| **cost** | partial source model | [cost](#cost) |
| **descriptant** | partial source model | [epoch-memory](#epoch-memory) |
| **invariant** | partial source model | [invariant](#invariant) |
| **tests** | bounded executed | [checks](#checks) |

## Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

| id | type | label | where |
|---|---|---|---|
| `algorithm-composition-wave19-behavior-complement` | behavior_complement | partial source model | [behavior-complement](#behavior-complement) |
| `algorithm-composition-wave19-reverse-reconstruction` | reverse_reconstruction | partial source model | [reverse-reconstruction](#reverse-reconstruction) |
| `algorithm-composition-wave19-quadratic-solvers` | diagonal_quadratic_derivative_relation | partial source model | [quadratic-solvers](#quadratic-solvers) |
| `algorithm-composition-wave19-polynomial-basis` | basis_transport_quadrature_relation | partial source model | [polynomial-basis](#polynomial-basis) |
| `algorithm-composition-wave19-projection-threshold` | normal_cone_and_sign_reconstruction | partial source model | [projection-threshold](#projection-threshold) |
| `algorithm-composition-wave19-merged-moments` | partitioned_centered_moment_relation | partial source model | [merged-moments](#merged-moments) |
| `algorithm-composition-wave19-frequency-certificates` | additive_frequency_interval_relation | partial source model | [frequency-certificates](#frequency-certificates) |
| `algorithm-composition-wave19-histogram-rank` | bin_value_rank_coordinate_relation | partial source model | [histogram-rank](#histogram-rank) |
| `algorithm-composition-wave19-fptas-subsets` | integer_objective_feasibility_fiber_relation | partial source model | [fptas-subsets](#fptas-subsets) |
| `algorithm-composition-wave19-rational-numeric` | integer_certificate_numeric_transport | partial source model | [rational-numeric](#rational-numeric) |

## Model text


### model

## Wave19 native joins

### forward

Eight distinct bounded client relations cover all twenty modules and cross five new families through actual numeric transports. Independent diagonal minimizers, polynomial identities, KKT constants, direct centered sums, literal counts/ranks, exhaustive subset masks and exact integer arithmetic supplement cross-implementation equality. Inputs and error metrics remain tagged; this is not one serial end-to-end computation.

### invariant

Identical indices, polynomial conventions, half-open rank intervals and exact integer capacities are used across each join. Absolute floating tolerances apply only to specified finite fixtures. Supported percent tiers are 1,2,5,10; objective, additive frequency and L1-normalized dot metrics are not interchangeable. Resource failures cannot be hidden as approximate results.

### cost

Eight bounded families; at most n4 solvers with fixed iteration budgets, n24 moments, 800 stream updates, 257 histogram items, n8 exhaustive subsets (256 masks) and 2million DP work caps, constant-size polynomial/projection/dot fixtures. Oracle work, allocations and retained fixture vectors count. No timing or universal accuracy claim.

### composition

Each named test is a separate tagged relation. Composition requires matching coefficient conventions, finite domains, normalization metrics, probability mass or radius, observation coordinates and explicit tolerance assumptions. Error caps apply to their declared metric only; no uniform whole-pipeline percentage follows. Distinct tests do not constitute one end-to-end pipeline. Independent finite or literal oracles remain distinct from cross-implementation equality.

### epoch-memory

2026-10-30 prospective design retained below; no speed or accuracy gain is claimed.

## Prospective independent integration review, 2026-10-30

Saved before harness construction or timings. Eight bounded test families are proposed: diagonal quadratic minimizers and scalar derivative roots; monomial/Chebyshev polynomial transport and integration with affine interpolation; simplex/soft-threshold/l1 reconstruction plus isotonic sum preservation; merged moments and compensated sums; integer knapsack/subset-sum certificates against exhaustive masks; quantized dot products and compensated exact-coordinate products; two frequency summaries against literal counts; histogram intervals against sorted exact ranks. These are tagged client observations, not a serial pipeline.

Every comparison will use an independently grounded exact bounded oracle where feasible: known diagonal minimizer, rational polynomial identity, KKT threshold formula, direct sums/squared deviations, exhaustive subset optimum, integer product sum, literal frequency and sorted rank. Numerical tolerances will be absolute on fixed small finite domains; no finite test becomes a universal floating-point bound. Integer objective loss certificates use integer multiplication and zero optimum handled exactly. No new >10% tier. Allocation/error semantics remain categorical. Specialized complement relations exclude outputs violating named laws; reverse fibers retain all compatible original inputs, with uniqueness asserted only for deliberately injective coordinate reconstructions.

Expected harness cost is modest, dominated by bounded exhaustive masks at <=8 items and solver iteration budgets; no wall-clock claim. All fixture generation is deterministic. The root will integrate the external source and run serial twice debug and twice release plus a compiled adapter negative control.

Benchmark proposal, before measurements: two bounded CPU pairs, Horner versus direct monomial evaluation and knapsack FPTAS versus exact capacity DP. Use seven samples per pair, alternating which baseline executes first by sample parity, compilation excluded; retain both durations, medians and extrema. Pin fixtures/repetitions before run and check outputs before and after. Horner fixture 16 small coefficients and 256 x in [-0.75,0.75], 200 batches per sample; compare <=1e-11 absolute and consume checksums through black_box. FPTAS fixture <=32 items and capacity <=100000; choose fixed deterministic integer weights/values and supported 10% cap, exact capacity DP baseline, <=3 repetitions per sample. Assert chosen subset feasibility and exact integer 90% inequality; record actual loss, exact fallback status if applicable, resource caps and rejection. If proposed API cost makes fixture unsuitable, record a revised prospective before any measurement, not after seeing timing. Either pair may regress; timings do not establish training/GPU/end-to-end gain. Parent runs benchmark only once source/API is frozen.

### checks

Eight named tests pass twice in debug and twice in release. Two compiled adapter controls reject and restore. Two syntax projections match; macro expansion and semantic AST remain absent. See evidence.md.

### catalog

## Wave19 supplied-input relations

Source-derived specifications; bounded native evidence and repeated partial syntax are recorded in evidence.md. No general proof or complete inverse engine is claimed.

### behavior-complement

Type: behavior_complement. Named check: all eight named joins.

On the eight tagged bounded domains define R by the named independent identities and inequalities. Its complement includes mismatched polynomial bases, derivative signs, diagonal coordinates, threshold signs or mass, lost merged counts, wrong variance denominator, frequency bounds excluding literal counts, wrong inclusive-value/exclusive-rank endpoints, unattainable subset objectives, loss exceeding integer budget, and dot certificates normalized by |sum| instead of sum|products|. Invalid inputs, allocation failures and budget statuses are categorical, not numerical tolerance. These tests do not exhaust R's complement or establish universal floating-point accuracy.

### reverse-reconstruction

Type: reverse_reconstruction. Named check: all eight named joins.

Retain every tagged input compatible with shared coordinates and observed identities. Full polynomial coefficients under a fixed-degree basis transform have a unique converse, and known nonzero signs reconstruct l1 coordinates from magnitudes, but arbitrary finite samples do not identify functions. Threshold zeros retain intervals; moments/frequency/histogram observations lose order; objective values retain many subset masks; dot values retain many vector pairs. The bounded exhaustive subset oracle checks nonempty feasible fibers. No general inverse execution engine or composition-wide percentage bound is asserted.

### quadratic-solvers

Type: diagonal_quadratic_derivative_relation. Named check: quadratic_solvers_join_scalar_roots_and_polynomial_values.

Four diagonal systems at n=1..4, diagonal d_i=i+1, known minimizer i-1.5 and b_i=d_i*x_i. CG and diagonal GD coordinates, Horner derivative roots by bisection, and golden minimizers are within absolute 1e-6 of this independent minimizer. Derivative at CG result is within 1e-9. Scalar brackets [-4,4]; root budget100, golden200, CG100, GD10000. This bounded well-conditioned SPD/separable family does not imply nonlinear or ill-conditioned convergence. Cost includes O(n²) CG matvec per iteration, O(n) GD and bounded scalar callbacks.

### polynomial-basis

Type: basis_transport_quadrature_relation. Named check: polynomial_basis_join_derivative_and_quadrature.

Full c0 Chebyshev [2,3,-2,1] is identically monomial [4,0,-4,4]. For 33 dyadic points -1..1 compare both evaluations and independent literal polynomial/derivative to 1e-13 absolute. Integrate Clenshaw on [-1,1], compare to analytic 16/3 within 1e-12; Simpson budget1000 depth20 tol1e-10. Coefficient conventions and interval must match. Equal finitely sampled values do not identify arbitrary functions; basis transform is invertible only at fixed polynomial degree. O(33*4) evaluations plus bounded quadrature.

### projection-threshold

Type: normal_cone_and_sign_reconstruction. Named check: projections_join_threshold_reconstruction_and_mean.

For x=[-3,1,-2,0], radius3 and threshold1, simplex of abs(x) must be [2,0,1,0], l1 projection and soft-threshold both [-2,0,-1,0]. These independent KKT values exclude a shared sign/threshold bug. Three small isotonic fixtures have preserved compensated sum, monotonicity, idempotence and normal-cone inequalities against ten constant feasible vectors -3..6. These inequalities are necessary observations, not a complete global optimality proof. Thresholded zeros admit intervals of antecedents; preserve original sign for nonzero reconstruction. Costs include sort at n4, PAVA and compensated sums.

### merged-moments

Type: partitioned_centered_moment_relation. Named check: merged_moments_join_compensated_centered_sums.

Lengths1..24, values ((7*i)%11)-5, split at floor(n/2). Merge streaming halves; count equals n, mean and population/sample variances match compensated direct sums and squared deviations to absolute1e-12. n1 sample variance is not asserted. Centering and ddof must agree. Mean/variance leave order and generally many samples ambiguous. O(n) updates/direct oracle and O(1) merge with fixture storage O(n).

### frequency-certificates

Type: additive_frequency_interval_relation. Named check: frequency_summaries_join_literal_histogram_certificates.

Four supported tiers1,2,5,10 and deterministic 800-item stream key7 at every third position, else i%137. Both summaries total800. For keys0..140 compare inclusive bounds to independent BTreeMap frequency and verify interval width*100<=800*tier in exact integer arithmetic. This is additive frequency error normalized by total, not rare-key relative error. Counter tables and frequencies lose order. O(800*capacity) worst update costs plus exact map storage; normal BTreeMap allocation behavior remains categorical.

### histogram-rank

Type: bin_value_rank_coordinate_relation. Named check: histogram_intervals_join_sorted_rank_and_exact_counts.

257 values (17*i)%101, histogram inclusive0..100 width7, independently sorted ranks0..256. Every returned interval contains actual sorted value and requested rank; rank_start equals count below value_min and exclusive rank_end equals count<=value_max. Rank257 is absent. Value interval width and normalized rank error are distinct; no percentage claim. Histogram and ranks lose insertion order and within-bin positions. O(nlogn) sorting oracle plus bounded bin/query work.

### fptas-subsets

Type: integer_objective_feasibility_fiber_relation. Named check: fptas_objectives_join_exhaustive_subset_fibers.

n0..8 values (7*i+3)%13, weight=value, capacities0,1,7,19,40, supported tiers1,2,5,10. Enumerate all <=256 subset masks, retain attainable feasible sums and optimum. Both algorithms must return an attainable objective and objective*100 >= (100-tier)*OPT; zero optimum is therefore exact. Knapsack's absolute loss certificate also bounds actual loss and is <=tier*OPT. No selected-item witness is claimed by the API; bounded exhaustive membership proves existence here. Equal objectives leave multiple subsets compatible. Work caps2million; cost includes integer scaled DP/trimmed lists and exhaustive oracle.

### rational-numeric

Type: integer_certificate_numeric_transport. Named check: rational_certificates_join_polynomials_and_compensated_products.

Affine samples (-4,-5),(0,3),(6,15), integer x=-4..6: rational interpolation equals exact 2x+3 by cross multiplication with zero curvature certificate and equals Horner. Simpson affine integral [-4,6] equals50 within1e-12. Quantized dot fixtures a=[101,-103,108,-100], b=[100s,100s,100,-100], s=-1,+1, steps1,2,4 and tier10: exact i128 dot equals compensated sum of exactly representable integer products, error lies within certificate and certificate*100<=10*sum|products|. Additional zero-L1 fixture has all-zero certificate. This metric is not error relative to cancelling dot; affine curvature premise is grounded by the known polynomial. O(n) validation/dot and bounded quadrature; rational output alone does not reconstruct knot set or original vector pairs.

### evidence

#### Wave19 harness evidence

Eight tests pass twice in each debug/release profile in the 616-check suite. Both compiled adapter mutants fail their named assertions; exact restoration passes. Partial syntax is generated twice with identical results. Source SHA-256 894e55be473b27d0b67921ff2b44f0e2269999966a14273897cf0d198196f097.

[Native receipts](https://md.groupants.elamur.ai/free/en/mova/models/algorithm-library.groupant/#wave19-evidence) and external evidence-hashes.json bind actual logs. Finite fixture assertions do not prove all modeled relations or a shared approximation budget.