Sequence Padded Origin Shift Tool

Description

Map an index after left padding to its signed original position, preserving negative padding positions.

A sequence rule specifies an index origin, state order, and relation among terms. Advancing one step, mapping an index, and checking a local residual answer different questions. Supplied coefficients or periods define a model; they do not establish that observations follow it.

Map an index after left padding to its signed original position, preserving negative padding positions.

When to use this tool

Use Sequence Padded Origin Shift when you need this particular index map, recurrence step, count, or local algebraic term under supplied sequence conventions.

How to use the tool and how it works

Enter originalOrigin, leftPadding, position. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read newOrigin, originalPosition. Use the output convention below.

Return newOrigin=originalOrigin-leftPadding and originalPosition=position-leftPadding. Require nonnegative integer padding and position.^[1]

Input field	Meaning and units	Accepted input
originalOrigin	Original Origin	Required. Safe integer.
leftPadding	Left Padding	Required. Safe integer; minimum 0.
position	Position	Required. Safe integer; minimum 0.

Output field	Interpretation	Returned shape
newOrigin	New Origin	Required. Safe integer.
originalPosition	Original Position	Required. Safe integer.

json

{
  "input": {
    "originalOrigin": 10,
    "leftPadding": 3,
    "position": 5
  },
  "output": {
    "newOrigin": 7,
    "originalPosition": 2
  }
}

Limitations and assumptions

OriginalPosition may be negative within padding. No padded sequence, value, or source-length bound is supplied, and the coordinate convention is caller-defined.
The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.

Alternative or Complementary methods

Choose exact integer arithmetic when the index or coefficient range requires it. A complete sequence analysis needs actual terms and a consistent index and state-order convention beyond one local step.

References

Calculation and input contract: sequence/sequence-padded-origin-shift — tool.io

Similar or alternative tools

Recurrence Affine Fixed Point Shift
Calculate fixedPoint=offset/(1-multiplier) and centered=value-fixedPoint, rejecting multiplier one.
Recurrence Delay State Shift Index
Return nextIndex=index+1 for a bounded state index except the final position, which returns minus one and dropped true.