Source: https://guthlabs.ai/docs/instruments/taylor-maclaurin-series-generator/mechanism/

instrument mechanism / verified-clean

# Taylor / Maclaurin Series Generator mechanism

The inspectable method record for Taylor / Maclaurin Series Generator, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The core Taylor-polynomial method is standard after correcting the summation notation and resolving whether n denotes order or number of terms. The instrument cannot honestly claim a remainder estimate until it specifies the estimate method and its assumptions. It should label the result as an order-N Taylor polynomial, expose the convergence/domain limitations, and disclose any unsupported symbolic-function cases.

- The displayed formula is malformed: the summation index n is also used as the requested order, and the x-dependence is missing. The canonical order-N Taylor polynomial is P_N(x)=Σ_{k=0}^{N} f^(k)(a)(x-a)^k/k!. If the product requirement literally means N terms, the upper index should instead be N-1.

- The specification says 'to n terms' but the input says 'order n'; these differ by one term because order N normally includes derivatives k=0 through k=N, giving N+1 terms.

- A remainder estimate is claimed, but no remainder formula, bound assumptions, or error method is specified. A standard choice is the Lagrange bound |R_N(x)|≤M|x-a|^(N+1)/(N+1)!, requiring a valid bound M on |f^(N+1)| over the interval between a and x.

- The plotted comparison requires a stated domain and handling for points where f is undefined or where the Taylor series does not converge. The polynomial can be evaluated outside the function's domain or its convergence radius, but that comparison must be labeled honestly.

## Units and conventions

- All inputs are dimensionless mathematical real numbers unless the function explicitly defines units; x and center a must use the same units.

- For trigonometric functions, angles are assumed to be in radians.

- The test vectors interpret the input as order N, using k=0 through k=N. This interpretation must be made explicit in the product.

- For ln(x), the real-valued domain is x>0; its Taylor series about a=1 converges for |x-1|<1, with endpoint behavior requiring separate handling.

- For 1/(1-x), the Taylor series about 0 has radius of convergence 1 because of the singularity at x=1.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "function": "exp(x)",
    "center_a": "0",
    "order_N": 3,
    "evaluation_points": [
      "0.5000000000000000",
      "-1.000000000000000"
    ]
  },
  "expected_outputs": {
    "canonical_exact_symbolic_result": "P_3(x)=1+x+x^2/2+x^3/6",
    "spot_checks": [
      {
        "at": "x=0.5000000000000000",
        "input_side": 1.6487212707001282,
        "answer_side": 1.6458333333333333
      },
      {
        "at": "x=-1.000000000000000",
        "input_side": 0.3678794411714423,
        "answer_side": 0.3333333333333333
      }
    ],
    "remainder_estimate": "Not determinable from the specification because no remainder method or derivative bound is supplied."
  },
  "reference_data": {
    "interpretation": "order_N means terms k=0 through k=N",
    "exact_function_values": {
      "exp(1/2)": "e^(1/2)",
      "exp(-1)": "e^(-1)"
    },
    "exact_polynomial_values": {
      "P_3(1/2)": "79/48",
      "P_3(-1)": "1/3"
    }
  },
  "working": "Derivatives of exp(x) are all exp(x), so at a=0 every derivative equals 1. Therefore P_3(x)=1/0!+x/1!+x^2/2!+x^3/3!=1+x+x^2/2+x^3/6. At x=0.5: 1+0.5+(0.5^2)/2+(0.5^3)/6=1+0.5+0.125+0.0208333333333333=1.6458333333333333. The original value is exp(0.5)=1.6487212707001282. At x=-1: 1-1+1/2-1/6=1/3=0.3333333333333333. The original value is exp(-1)=0.3678794411714423."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The generator must reject centers where the function or required derivatives are undefined, must detect nondifferentiable branch switches, and must require an explicit angle-unit convention. Otherwise it can emit undefined coefficients or materially incorrect expansions.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/taylor-maclaurin-series-generator/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/taylor-maclaurin-series-generator/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/taylor-maclaurin-series-generator/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche8/tranche8-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`