---
title: "algorithm-savitzky-golay — Mova code model"
description: "Mova code model of algorithms/src/savitzky_golay/mod.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/algorithms/src/savitzky_golay/mod.rs.groupant/"
---
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# algorithm-savitzky-golay

The groupant beside algorithms/src/savitzky_golay/mod.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 descriptive instance
- **source:** [`algorithms/src/savitzky_golay/mod.rs`](https://github.com/ElamurAI/mova-free/blob/main/algorithms/src/savitzky_golay/mod.rs)
- **source SHA-256:** `5888243da4b9e97a…`
- **repository folder:** [`algorithms/src/savitzky_golay/mod.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/algorithms/src/savitzky_golay/mod.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 | [approximant](#approximant) |
| **ast** | partial | `syntax.json` |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | partial | [approximant](#approximant) |
| **tests** | existing checks referenced | [tests](#tests) |

## 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 |
|---|---|---|---|
| `behavior_complement` | behavior_complement | partial source derived model | [behavior_complement](#behavior-complement) |
| `reverse_reconstruction` | reverse_reconstruction | partial source derived model | [reverse_reconstruction](#reverse-reconstruction) |
| `shifted_boundary` | shifted_boundary | partial source derived model | [shifted_boundary](#shifted-boundary) |
| `qr_composition` | qr_composition | partial source derived model | [qr_composition](#qr-composition) |
| `polynomial_invariant` | polynomial_invariant | partial source derived model | [polynomial_invariant](#polynomial-invariant) |
| `parameter_precedence` | parameter_precedence | partial source derived model | [parameter_precedence](#parameter-precedence) |

## Model text


### descriptant

#### savitzky_golay / mod.rs

Original Rust; original algorithm citation appears in module documentation. Existing FFT and Householder QR reused where appropriate.

##### Prospective epoch

#### Prospective transform models
Before implementation: DCT II uses sqrt(1/n) DC and sqrt(2/n) others, cos(pi*(j+.5)*k/n), inverse transpose; n<=2048, empty identity; O(n²) work/O(n) space. Haar full dyadic transform packs coarse value then detail levels; pair sum/difference divided sqrt(2); n<=65536 power two, no extension; empty invalid; O(n) work/space. Bluestein uses chirp exp(sign*i*pi*j²/n), symmetric inverse chirp convolution padded to power two >=2n-1, existing FFT, n<=16384; inverse divides n; empty invalid; O(n log n)/O(n). Goertzel recurrence s=x+2cos(w)s1-s2, final real=cos(w)s1-s2, imaginary=sin(w)s1 (integer-bin phase correction), n<=65536, bins<=65536, all k<n; empty input only accepts empty bins; O(n*b)/O(b). SG n<=4096, odd window<=31 and <=n, degree<=6 and <window, shifted full window at boundaries; offsets scaled by window, Vandermonde QR solves local least squares, constant coefficient predicts target; O(n*w*d²)/O(w*d+n). All finite magnitudes <=1e50, no callbacks, allocations follow standard policy except reused QR fallible allocation. Validate shapes/limits before values then compute; roundoff and rank/nonfinite failures explicit, no universal percent or residual-to-solution guarantee. Polynomial reproduction, independent direct sums, orthogonality and reconstruction are finite checks. No timing or gain claim. References: Ahmed/Natarajan/Rao (1974) DCT; Haar (1910); Bluestein (1970) linear filtering DFT; Goertzel (1958); Savitzky/Golay (1964).

##### Mathematical combiner

Source and SHA-bound partial syntax define transition F; forward/invariant approximants model F, behavior complement models excluded pairs and reconstruction models compatible antecedents. Finite independent tests constrain F. Combining these observations is not a semantic proof; no macro expansion or semantic AST. Leaf children=[]; root integrates exports/ancestor bindings.

##### Observed epoch 2026-10-30

20 new tests pass in twice debug and twice release crate runs (713 total each) with identical normalized outcomes. Five compiled production mutants fail named assertions and exact source restoration passes four tests each. Strict all-target algorithm Clippy and focused formatting pass. No timing, speed or universal accuracy claim.

### model

### behavior_complement

For validated input x let F(x) be source-defined tagged floating output. Excluded relation is (x,y) where y differs from F(x), including wrong normalization, wrong grid/layout, invalid-domain success or nonfinite success. This models excluded outputs, not only a list of tests. For output i choose start=min(saturating_sub(i,w/2),n-w), fill w rows [1,t,...,t^d] with t=(j-i)/w, solve via existing Householder QR and take coefficient zero. Uniform unit grid; shifted full windows at boundaries, no padding. n<=4096, odd 1<=w<=31 and w<=n, d<=6 and d<w, finite |x|<=1e50.

### reverse_reconstruction

A local prediction is noninjective: degree-zero fitting sees only local mean, so zero-sum changes within a window leave its prediction unchanged. With w=n and d=0 all outputs repeat global mean; [1,-1,0] and [0,0,0] are indistinguishable for w=3. w=1,d=0 is real identity. Global smoothing is not universally invertible.

### shifted_boundary

At left/right boundaries shift complete window inside data. Prediction point remains target i, not the window center; asymmetry reproduces exact polynomials in real arithmetic.

### qr_composition

Scaled Vandermonde rows have distinct t and full column rank in real arithmetic for d<w. Existing unpivoted QR can still suffer finite conditioning; its rank/allocation/nonfinite errors propagate without fabricated accuracy.

### polynomial_invariant

Degree<=d polynomial data lie in design column space; exact least squares reproduces value at t=0, including edges. No guarantee that noisy observations reveal true signal.

### parameter_precedence

Reject n cap, empty/improper/parity window, excessive degree and values before building rows. No derivatives, irregular grids, weights or callback evaluation; ties have no special branch.

### approximant

#### Forward model and invariant

For output i choose start=min(saturating_sub(i,w/2),n-w), fill w rows [1,t,...,t^d] with t=(j-i)/w, solve via existing Householder QR and take coefficient zero. Uniform unit grid; shifted full windows at boundaries, no padding. n<=4096, odd 1<=w<=31 and w<=n, d<=6 and d<w, finite |x|<=1e50.

Cost: O(n*w*(d+1)²) work and O(n+w*(d+1)) live storage. Standard allocations plus QR fallible allocations; QR errors propagate.

Successful values remain finite and input arrays are immutable. Orthogonal/DFT inverse laws require the transformed coefficients to satisfy the second call input cap; large admissible forward samples can produce coefficients above that cap, so the bounded API is not globally closed under round trips. Invalid domain checks precede computation; same error tag can cover several invalid premises. No callbacks; allocation failure is outside ordinary Result except errors inside reused QR. Transform algebra is exact over reals, implementation is rounded f64. Zero-reference relative error is undefined; cancellation precludes universal relative claims. No executable 1/2/5/10% surrogate certificate is supplied. Residuals, if present in reused QR, do not bound signal or solution error.

### tests

polynomial_reproduction checks degrees0..6 including boundary windows; published_quadratic_weights independently checks [-3,12,17,12,-3]/35; shifted_means_and_identity tests explicit means, singleton windows and limits; rejected_domains checks empty/parity/degree/caps/nonfinite.

Control: Halving fitted coefficient compiles and fails polynomial_reproduction.

Evidence: $MOVA_DATA/runs/octopus-wave26-transforms-2026-10-30. No allocator fault injection; finite arithmetic bounds and cap tests are not exhaustive f64 verification.