khipu-computational-toolkit

Phase 2: Summation Patterns

Generated: 2026-03-08
Database: K-CAT SQLite database (built from KFG source data)
Detector: src/analysis/kfg_summation_detector.py
Status: ✅ Complete

Research Question

What fraction of khipus embed arithmetic summation relationships — cords whose numeric values sum to other cords? What pattern types appear, and how often are they combined?

Methodology

Pattern Types

The KFG Summation Detector implements nine structural relationship types derived from the Ascher & Ascher fieldmark vocabulary:

Code	Pattern	Description
`pendant_pendant_sum`	Pendant → Pendant	One pendant’s value = sum of adjacent pendants
`colored_pendant_sum`	Color group	Pendants of the same color sum to another pendant
`indexed_pendant_sum`	Indexed pendant	A designated “total” pendant = sum of its group
`subsidiary_pendant_sum`	Subsidiary → Pendant	Subsidiary cord values sum to their parent pendant
`indexed_subsidiary_sum`	Indexed subsidiary	A subsidiary acts as a color-group total
`pendant_sub_neighbor`	Pendant–subsidiary	A pendant equals the sum of a neighbor’s subsidiaries
`group_group_sum`	Group → Group	One group of pendants sums to another group
`group_sum_bands`	Group sum bands	Khipu split at midpoint; left-half group totals equal right-half
`ascher_decreasing_group`	Decreasing pattern	Groups form a decreasing arithmetic sequence

Detection Logic

For each khipu, the detector:

Loads all cords with their values, hierarchy levels, and colors from the K-CAT database
Enumerates candidate relationships for each pattern type
Checks whether the arithmetic identity holds exactly (tolerance = 0, i.e., exact integer match)
A khipu is scored as has_summation = True if at least one relationship of any type matches

Tolerance = 0 means the numeric equality must hold exactly, with no rounding. Cords with value = 0 (null placeholder) are excluded from summation candidates.

Per-Pattern Criteria

pendant_pendant_sum: contiguous window of pendants; minimum 2 non-zero summands; exact sum match.
indexed_pendant_sum: designated total pendant value ≥ 7 (KFG significance threshold); window of pendants in same sub-group.
colored_pendant_sum: pendants of identical color code sum to another pendant; minimum 2 summands.
subsidiary_pendant_sum: subsidiary cord value ≥ 11; multiples of 10 when value < 100 excluded (coincidental match filter); minimum 2 non-zero pendants in summing window.
indexed_subsidiary_sum: subsidiary acts as color-group total; value ≥ 5; multiples of 10 (< 100) and multiples of 100 (< 1000) excluded; grouped by same level + same color; deduplicated by (sum_cord_id, frozenset(summand_ids)).
pendant_sub_neighbor: pendant value = sum of an adjacent pendant’s subsidiaries; significance threshold > 1 occurrence per khipu (single occurrence deemed accidental by KFG).
group_group_sum: one group’s total = sum of other groups; group sum threshold ≥ 21; multiples of 10 (unless ≥ 100) excluded.
group_sum_bands: khipu split at midpoint; left-half group totals equal right-half group totals.
ascher_decreasing_group: groups form a monotonically decreasing sequence of totals.

Handedness Tracking

For each summation relationship, the detector records handedness — whether the sum cord appears to the left or right of its summand window in the pendant sequence:

Left-handed: Sum cord’s cord_index < min(summand_cord_index)
Right-handed: Sum cord’s cord_index > max(summand_cord_index)
Undefined: For patterns where position is not linear (e.g., colored_pendant_sum, group_group_sum)

Dual Sum Detection

The detector identifies dual sums — cords whose value matches multiple distinct summand windows. This is computed by grouping relationships by sum_cord_id and checking for multiple unique summand_window_hashes.

Figure-8 Knot Proximity Analysis

Figure-8 knots (E, EE in knot_clusters.knot_type) do not encode numeric value. For each summation relationship, the detector checks whether a figure-8 knot appears on or adjacent to the sum cord and summands, using structural proximity flags (has_left_exact, has_right_exact, has_left_close, has_right_close).

K-CAT Summation Results

Corpus-Wide Coverage

Metric	Count	Rate
Khipus tested	709	—
With any summation pattern	515	72.6%
Without any detected pattern	194	27.4%

By Pattern Type

Pattern Type	Khipus	Rate	Relationships
`pendant_pendant_sum`	410	57.8%	7,018
`colored_pendant_sum`	276	38.9%	3,526
`indexed_pendant_sum`	204	28.8%	1,835
`pendant_sub_neighbor`	178	25.1%	341
`subsidiary_pendant_sum`	145	20.5%	1,034
`ascher_decreasing_group`	144	20.3%	280
`group_sum_bands`	104	14.7%	175
`group_group_sum`	102	14.4%	262
`indexed_subsidiary_sum`	54	7.6%	203

pendant_pendant_sum is the single most common pattern (57.8%). Color-based grouping (colored_pendant_sum, 38.9%) is the second most prevalent.

Handedness Analysis

Pendant-pendant sum handedness (410 khipus with PPS patterns, 7,018 relationships):

Direction	Count	Rate
Left-handed	3,204	45.7%
Right-handed	3,814	54.3%
Total relationships	7,018	—

The corpus-wide handedness ratio is +0.09 (slight right bias).

Dual Sum Detection

Dual sum prevalence (pendant_pendant_sum only):

Metric	Count
PPS sum cords with dual decompositions	1,240 (21.5% of unique sum cords)

A sum cord has a dual decomposition when its value can be matched by multiple distinct summand windows. 21.5% of unique PPS sum cords have at least two valid windows, reflecting the combinatorial nature of contiguous-window enumeration.

Figure-8 Knot Proximity Analysis

For PPS relationships, figure-8 structural proximity indicators:

Metric	Count	Rate
PPS relationships with any figure-8 indicator	3,270	46.6% of 7,018

Figure-8 location distribution (PPS relationships with figure-8 indicators):

Location	Count	% of figure-8 PPS
right_exact	1,362	41.7%
left_exact	1,337	40.9%
left_close	1,126	34.4%
right_close	1,094	33.5%

Note: A single PPS relationship can have multiple figure-8 location flags, so percentages sum to more than 100%. The KFG author notes: “8knot markers were probably optional (like parentheses for example), and that maybe why correlation is so bad.”

Data Quality Notes

Tolerance 0 is strict. Exact integer arithmetic is required. Khipus with partially decoded cord values may fail a match even though a genuine summation structure exists — this biases toward under-detection.
value = 0 exclusion. Cords with value = 0 (null placeholder) are excluded as candidate summing terms. Khipus with many undecoded cords therefore have fewer candidates.
colored_pendant_sum and compound color codes. The K-CAT database stores compound color codes (e.g., MB:W, KB-DB) as single strings. The detector extracts the dominant color component before grouping.
194 khipus with no detected pattern. These include objects with predominantly undecoded values, as well as any khipus structured by conventions not yet modeled.

How to Re-run

# Corpus sweep (replicates the numbers above)
from src.analysis.kfg_summation_detector import KFGSummationDetector
import sqlite3, pandas as pd

DB = 'data/kfg/khipu_database.db'
detector = KFGSummationDetector(DB)

conn = sqlite3.connect(DB)
khipu_ids = pd.read_sql('SELECT kfg_id FROM khipu_metadata', conn)['kfg_id'].tolist()
conn.close()

for kid in khipu_ids:
    summary = detector.summarize(kid)   # tolerance=0 by default
    # summary['has_summation'], summary['pattern_stats'], etc.

Limitations

The detector tests arithmetic identity only. It has no model of intent: a coincidental three-cord sum (e.g., 1 + 2 = 3) passes the same test as any other arithmetic match.
The corpus sweep uses tolerance = 0. A small tolerance (1–2 units) may be appropriate when cord values are subject to transcription uncertainty; such analysis is left for future work.
Pattern type taxonomy follows Ascher & Ascher (1978, 1981).

See Citations and Acknowledgments in the project README for primary sources, data attribution, and toolkit provenance.

Corpus sweep run against K-CAT SQLite database. Re-run with KFGSummationDetector.summarize() on the current database to refresh these figures.

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