KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
KEDDEH BILATERAL PROOF ENGINE
A Quantitatively Validated White Paper on Anchor-Based Measurement, Zero
Classification, Grid Geometry, and Relational Proof
Author / originator: Aboudy K (A. Keddeh)
Prepared: 6 July 2026
Source: linked Google Sheets workbook, exported and analysed as XLSX
Abstract
This white paper formalises the Keddeh Bilateral Proof Engine as a measurement-validity runtime. 
The core claim is not that conventional mathematics is useless or universally false. The stronger, 
defensible claim is that a mathematical or physical value is not executable as a measurement until it
carries an observed anchor, unit/reference, scale role, state relation, transition boundary, 
environmental conditions, uncertainty, and traceable proof path. That structure is bilateral because 
the Keddeh grid demands it internally, while professional metrology independently demands 
number-plus-reference, measurement uncertainty, and traceability externally. The linked workbook
is treated as a live proof surface: it contains shell matrices, a spiral/prime artefact, report sheets, and
staged circle-square tables. Quantitative validation confirms strong evidence for repeated anchorshell grids and prime-diagonal enrichment in a spiral crop, while identifying unresolved proof gaps 
in the circle-square equality and residual zero markers that must be classified rather than ignored.
Finding Quantitative validation Conclusion
Workbook coverage
43,919 populated cells, 40,016 
numeric cells, 11,586 formula 
cells across 12 sheets.
Sufficient workbook mass for 
structural analysis.
Formula integrity
3 cached error cells detected; 
Sheet1 contains 3 #REF!-class 
artefacts.
Formula drift exists and must 
be repaired before any final 
mathematical proof claim.
Anchor-shell recurrence
Artefact 1 contains 738 exact 
6x6 zero-less shell matches and
5 zero-centred 7x7 shells.
The sheet strongly supports a 
repeated 1-2-3 shell runtime, 
with residual zero markers 
requiring classification.
Prime geometry
Visual Artefacts 15x15 spiral 
crop has 50 prime cells; Xdiagonal prime density is 
40.7% vs 19.7% off diagonal, a 
2.07x enrichment.
Supports non-random-looking 
diagonal prime structure in the
spiral crop.
6k +/- 1 rails
For n <= 289, all 59 primes 
greater than 3 lie on 6k +/- 1 
rails; 59 of 96 candidates 
Confirms bilateral primecandidate rails while 
preserving composite filtering.KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
survive.
Circle-square proof
Artefact 3 contains 575 
repeated "Cell Count" labels 
but no numeric 
circle/square/overlap counts.
Circle-square equality is not 
yet quantitatively proven by 
the workbook.KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
1. Purpose and Scope
The purpose of this paper is to convert the Keddeh grid from a conceptual argument into a 
repeatable validation framework. The framework is designed to classify mathematical expressions, 
physical measurements, and grid-based artefacts before accepting them as executable claims. The 
report intentionally separates three levels: formal mathematics, physical measurement, and 
workbook evidence. This prevents overclaiming while allowing the framework to be developed as a 
serious system.
The paper uses the workbook as its primary artefact base. The workbook contains the source grid 
sheets, a Keddeh Matrix Report, visual artefacts, a prime-geometry sheet, and circle-square proof 
sheets. The strongest workbook evidence concerns repeated 1-2-3 shell matrices and a spiral prime 
crop. The weakest workbook evidence concerns the final circle-square equality, because the relevant
count fields are labels without values and therefore do not yet close the proof.
Claim / Test Quantitative result Status Interpretation
Scope control
Claims are classified as
confirmed, supported, 
partial, or not proven.
Confirmed method
Prevents the white 
paper from treating 
incomplete workbook 
regions as final proof.
Workbook used as 
evidence
43,919 populated cells 
analysed. Supported
The workbook is large 
enough to reveal 
structural patterns 
and formula drift.
External professional 
standard
VIM and NIST 
references define 
quantity values, 
measurement results, 
uncertainty, and 
traceability.
Supported
This grounds the 
bilateral proof in 
professional 
metrology rather than 
private terminology.
2. Professional Measurement Basis
The bilateral proof depends on a simple metrological point: a number by itself is not a complete 
physical measurement. The International Vocabulary of Metrology defines a quantity value as a 
number and reference together expressing magnitude. The same vocabulary defines a measurement
result as a set of quantity values attributed to a measurand together with other relevant 
information, normally including uncertainty. NIST states that traceability is a property of a 
measurement result related to a reference through a documented unbroken chain of calibrations, 
with each link contributing to the uncertainty. These requirements directly support the Keddeh rule 
that a value must carry a proof path, not merely a scalar label.
Professional requirement Source basis Keddeh equivalent
Number plus reference VIM 1.19: quantity value is A1 anchor plus declared KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
number and reference 
together. unit/reference.
Relevant information
VIM 2.9: measurement result 
includes quantity values and 
available relevant information.
State, environment, transition 
boundary, and proof fields.
Uncertainty
VIM/NIST: measurement 
results require uncertainty 
unless negligible for a specific 
purpose.
Proof band and measurement 
tolerance.
Traceability
NIST: documented unbroken 
calibration chain to specified 
references.
Ledger path.
Temperature scale role
NIST: 0 C is 273.15 K; 0 K is 
absolute zero; water freezes at 
0 C on Celsius scale.
Zero must be classified by scale
role, not treated as generic 
physical nothing.
Claim / Test Quantitative result Status Interpretation
Naked scalar as 
physical measurement
Fails number-plusreference and 
traceability 
requirements.
Not valid
A bare "0" or bare "x = 
-2" can be formal 
mathematics, but not a
complete physical 
measurement.
Keddeh validator 
versus metrology
Both require 
reference, conditions, 
uncertainty, and 
lineage.
Supported This is the basis of 
bilateral proof.
3. Keddeh Axioms
The Keddeh Bilateral Proof Engine is built on five axioms. These axioms are expressed as runtime 
checks rather than beliefs.
Axiom Runtime rule Quantitative / logical status
Anchor before value
No physical value executes 
without an observed anchor 
A1.
Aligned with VIM numberplus-reference requirement.
Zero is classification-only
Zero is valid as absence, 
cancellation, baseline, 
coordinate, null result, lowerbound, or error class; invalid 
as undeclared physical origin.
Workbook contains residual 
zero markers: Artefact 1 has 5 
zeros out of 26,848 numeric 
cells (0.0186%). These must be 
classified.
Negative is observer-relative 
inversion
Negative is inverse direction 
from the anchor, not less-thannothing.
Mathematically compatible 
with signed direction if the 
reference frame is declared.
Transition is not state Scale marker, transition event, Supported by temperature KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
and achieved phase state are 
separate.
metrology: 0 C and 0 K are 
different zero classes.
Approximation is not identity
A polygonal approximation is 
not a circle unless the model 
class is declared.
Circle-square claim remains 
not proven until the count 
tables are populated.
KEDDEH EXECUTION FORM:
VALUE_EXECUTES = A1 + unit/reference + scale-role + relation + state + transition-boundary + environment + 
uncertainty + ledger
4. Grid Runtime Model
The grid replaces a generic zero-origin interpretation with an anchor-transition-boundary 
interpretation. The horizontal axis encodes bilateral direction; the vertical axis encodes state 
relation. The centre is A1, the observed anchor. Values 2 and 3 act as transition and boundary bands.
Figure 1. 6x6 zero-less 2x2-anchor shell detected repeatedly in the workbook.
Grid component Meaning Validation
A1 Observed anchor / active 
reference.
Artefact 1 contains 2,992 
numeric 1-cells.
2
Transition band / active 
crossing.
Artefact 1 contains 8,936 
numeric 2-cells.
3
Boundary / signal / achieved 
shell.
Artefact 1 contains 14,915 
numeric 3-cells.
0
Classified marker only, not 
executable origin.
Artefact 1 contains 5 zero 
markers; Keddeh Matrix 
Report contains 10 zero 
markers.KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
Figure 2. Artefact 1 numeric distribution confirms dominance of 3-2-1 shell encoding.
Claim / Test Quantitative result Status Interpretation
6x6 zero-less shell 
recurrence
738 exact template 
matches in Artefact 1. Confirmed
Strong quantitative 
support for a repeated 
1-2-3 shell fabric.
7x7 zero-centred shell 
recurrence
5 exact matches in 
Artefact 1 and 4 in 
Sheet1.
Partial
Zero appears as a 
residual calibration 
centre; it must be 
classified, not ignored.
No-zero doctrine
Zero count is very low 
in Artefact 1 (0.0186%)
but not absent.
Partial
The stronger doctrine 
is zero classification, 
not absolute zero 
deletion.
5. Measurement Validator
The Keddeh validator is the operational form of the theory. It accepts a scalar, equation, graph, 
physical statement, or dataset and returns a validity class. It does not ask whether a symbol can exist
formally. It asks whether that symbol is executable as a physical or geometric claim under declared 
conditions.
INPUT: scalar, equation, graph, physical claim, dataset, workbook formula
CHECKS: anchor, quantity, unit/reference, scale-origin classification, zero class, bilateral relation, state axis, transition 
boundary, environment, uncertainty, traceability, result type
OUTPUT: executable | underdeclared | formal-only | approximation-only | invalid-category | relationally validKEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
Output class Definition Example
Executable
All required fields exist and 
formula/data are internally 
consistent.
Prime > 3 lies on a 6k +/- 1 rail, 
then survives factor filtering.
Underdeclared
Claim is plausible but missing 
reference, conditions, 
uncertainty, or proof path.
"Water freezes at 0 C" without 
pressure, purity, nucleation, 
and phase status.
Formal-only Symbolic manipulation is valid
but not physically typed.
x = -2 without units, domain, 
reference, or observer frame.
Approximation-only Model can approximate but 
cannot be asserted as identity. Finite polygon as circle.
Invalid-category Symbol is given the wrong 
role.
Unclassified zero treated as 
physical origin.
Relationally valid
Claim preserves relations but 
still needs external proof for 
stronger scientific assertion.
Anchor-based grid shell as a 
computational geometry 
model.
6. Application 001: Water, Ice, and Temperature
The water example is a conceptual validation of the measurement model rather than a dataset 
contained in the workbook. The sentence "water freezes at 0 C" is a useful shorthand, but it is 
underdeclared as a physical state claim. Temperature is a scale relation; freezing is a transition 
event; ice is an achieved phase state. NIST makes the scale distinction visible: 0 C is 273.15 K, while 0 
K is absolute zero. Therefore a zero marker does not carry one universal physical meaning.
Collapsed statement Keddeh separation Required fields
0 C Temperature scale marker. Scale, unit, calibration, 
uncertainty.
Freezing Transition crossing. Pressure, purity, nucleation, 
heat transfer, time.
Ice Achieved solid-state boundary. Observed phase, environment, 
measurement method.
Water body A1 anchor identity. Sample identity and material 
conditions.
Claim / Test Quantitative result Status Interpretation
Water/ice/temperature
proof status
No physical water 
dataset exists in the 
workbook.
Partial
The example validates 
the classification logic 
using metrology, but 
empirical proof would 
require measured 
phase-transition data.
Zero-class separation 0 C and 0 K are 
different Supported This supports zero classification over KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
measurement-scale 
roles. zero deletion.
7. Application 002: Algebra and Typed Variables
The algebra critique should be stated precisely. Algebra is not invalid because it uses letters. It 
becomes insufficient in applied contexts when variables execute without typed meaning. In Keddeh 
form, a variable must carry domain, unit, anchor, scale role, direction, and state class before it is 
accepted as a physical measurement or runtime value.
STANDARD: x = -2
KEDDEH-TYPED: x[quantity, unit, anchor, scale, domain, direction, state-role] = L2
Claim / Test Quantitative result Status Interpretation
Formal algebra
Symbolic assignment 
remains internally 
valid.
Supported
The system does not 
need to reject formal 
algebra.
Applied algebra A bare x = -2 fails 
physical typing. Confirmed method
Keddeh adds a 
metadata execution 
layer.
8. Application 003: Prime Geometry and Bilateral Candidate Rails
The prime claim is strongest when expressed as a survivor-grid proposition. Primes are not treated 
as magic points. Composites are cells consumed by factor lanes; primes survive the lower factor 
lanes. Standard number theory gives a clean quantitative bridge: every prime greater than 3 lies on 
one of the two rails around multiples of 6, namely 6k - 1 or 6k + 1. The workbook spiral crop 
provides an additional visual artefact: prime cells are enriched on the X diagonals relative to nondiagonal cells.KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
Figure 3. Prime cells in the Visual Artefacts 15x15 spiral crop.
Figure 4. Quantitative X-diagonal enrichment in the spiral crop.
Prime-grid metric Result Interpretation
Spiral crop size 225 cells, 225 unique integers, The crop is a unique-integer KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
min=1, max=289. spiral sample.
Prime count 50 primes in crop; density 
22.2%.
Comparable to expected prime 
scarcity in this range.
X-diagonal density 11 primes in 27 X cells = 40.7%. Strong diagonal enrichment.
Off-diagonal density 39 primes in 198 off-diagonal 
cells = 19.7%. Lower baseline density.
Enrichment factor 2.07x.
Supports the claim that the 
grid visually reveals prime 
structure.
6k +/- 1 theorem check All 59 primes greater than 3 up
to 289 satisfy n mod 6 in {1,5}.
Confirmed; this is the rigorous 
bilateral rail rule.
Candidate survival 59 of 96 6k +/- 1 candidates are 
prime (61.5%).
Rails generate candidates, not 
guaranteed primes. Higher 
factor lanes still remove 
composites.
Claim / Test Quantitative result Status Interpretation
Primes recur on 
bilateral rails
All primes >3 up to 289
validate the 6k +/- 1 
constraint.
Confirmed
This is a mathematical 
fact and cleanly fits 
the bilateral grid.
Prime X-pattern in 
workbook crop
X density is 2.07x offdiagonal density. Supported
The sheet supports 
diagonal enrichment, 
not a universal proof 
that primes fully 
bound space.
Cryptographic claim
No entropy, hardness, 
key-space, or attackresistance analysis 
exists in the 
workbook.
Not proven
Use as a research 
direction, not a 
security claim.
9. Application 004: Circle, Square, and Quantized Curvature
The circle claim must be improved before it can stand as proof. The defensible version is that a 
digital circle can be treated as a quantized boundary set on a grid, with all accepted boundary cells 
resolved relative to an anchor. This is useful for computational geometry and rasterisation. The nondefensible version, at the current evidence level, is that the circumference of a continuous circle is 
mathematically equal to a square in standard geometry. The workbook contains repeated stage 
labels and count-table headings, but not the numeric circle, square, overlap, and total counts 
required to close that assertion.
K-CIRCLE(A1, R) = { cell | H^2 + V^2 resolves to radial boundary class R }
Quantized interpretation: valid for grid/raster geometry.
Continuous interpretation: not proven by the workbook.KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
Circle claim layer Quantitative evidence Verdict
Anchor-based circle Can be expressed as radial 
equality from A1.
Mathematically valid as a 
coordinate translation.
Digital/raster circle Workbook uses staged count 
tables and grid artefacts.
Supported as a computational 
model.
Circle equals square
Artefact 3 has 575 repeated 
"Cell Count" labels but no 
count values; Circle-Square 
Proof sheets contain labels 
with blank counts.
Not proven yet.
Pi resolved as whole-number 
ratio
No verified formula or count 
equality is present. Research proposition only.
Claim / Test Quantitative result Status Interpretation
Polygon critique
Finite polygons always
retain corners; the 
paper classifies 
polygonal methods as 
approximations.
Supported Does not invalidate pi; 
it classifies the model.
Quantized curvature
Workbook visually 
supports cell-based 
curvature language 
but lacks final counts.
Partial Needs populated count
formulas.
Circle-square equality
No numeric equality 
test exists in Artefact 
3.
Not proven
Must add formulas for 
circle cells, square 
cells, overlap, and 
total across each stage.
10. Workbook Integrity and Error Ledger
The workbook is valuable as a research artefact, but it is not yet a closed proof workbook. The error 
ledger matters because Keddeh proof requires proof lineage and executable formula integrity. A 
proof surface with reference errors can still contain valid insight, but the error states must be 
classified.
Sheet Cells Numeric Formulas Errors Bounds
Sheet1 1,503 1,279 157 3 (1, 167, 1, 55)
Sheet4 0 0 0 0
(None, None, 
None, None)
Copy of 
Sheet1 393 338 337 0 (1, 101, 1, 55)
Visual 
Artefacts 574 574 452 0 (1, 100, 1, 26)
Circle-Square 158 142 0 0 (1, 118, 1, 26)KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
Proof
Copy of CircleSquare Proof 158 142 0 0 (1, 118, 1, 26)
Keddeh 
Matrix Report 110 53 0 0 (1, 46, 1, 9)
Artefact 1 - 
Entangled 
Core
26,862 26,848 0 0 (2, 998, 2, 49)
Artefact 2 - 
Prime 
Geometry
10,654 10,640 10,640 0 (2, 985, 2, 40)
Artefact 3 - 
Circle-Square 
Unit
3,506 0 0 0 (2, 991, 2, 50)
Sheet2 0 0 0 0
(None, None, 
None, None)
Sheet3 1 0 0 0 (1, 1, 1, 1)
Claim / Test Quantitative result Status Interpretation
Formula population 11,586 formula cells 
found. Supported
The workbook is not 
static art; it contains 
active computed 
regions.
Formula error scan 3 cached error cells 
found, all in Sheet1. Partial
Formula drift exists 
and should be fixed 
before public proof 
release.
Report self-assessment
Keddeh Matrix Report 
itself includes pending
quantitative validation
language.
Supported
This white paper 
completes that missing
layer by running 
numerical checks.
11. The Keddeh Bilateral Proof Engine
The engine is now defined as a layered validation stack. It can be used for measurement science, 
algebra typing, grid geometry, AI data validation, and workbook proof ledgers. Its central function is
not to replace all existing mathematics. Its function is to classify when a mathematical expression is 
formal-only, physically underdeclared, approximation-only, or executable.
Layer Name Function
L1 Anchor Declare A1, the observed 
subject or system.
L2 Reference Declare unit, scale, reference, 
and quantity class.KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
L3 Bilateral relation Resolve inverse/positive 
direction relative to anchor.
L4 State rail Resolve up/down or phasestate axis.
L5 Transition Separate marker, transition 
event, and achieved state.
L6 Zero classifier
Classify zero as absence, 
cancellation, baseline, 
coordinate, null result, lowerbound, or invalid origin.
L7 Ledger
Attach uncertainty, 
traceability, formula path, data
source, and error state.
L8 Execution verdict
Return executable, 
underdeclared, formal-only, 
approximation-only, invalidcategory, or relationally valid.
BILATERAL PROOF CONDITION:
KEDDEH_VALID(x) AND METROLOGY_VALID(x) => x may execute as a physical measurement.
KEDDEH_VALID(x) AND NOT METROLOGY_VALID(x) => x remains internal/experimental.
NOT KEDDEH_VALID(x) AND METROLOGY_VALID(x) => x requires mapping into anchor-state form.
NOT KEDDEH_VALID(x) AND NOT METROLOGY_VALID(x) => x is underdeclared or invalid-category.
12. Impacts
Domain Impact
Measurement science Forces values to carry references, uncertainty, 
traceability, and conditions.
Physics modelling Separates scale markers from state transitions 
and achieved states.
Algebra education Prevents variables from carrying undeclared 
physical authority.
Geometry Separates continuous identity from discrete 
approximation and grid/raster execution.
Computational geometry Supports a digital shell/raster method for 
spatial indexing and circle rendering.
Prime visualisation Makes prime candidate rails and diagonal 
enrichment visible.
AI systems
Reduces category errors by classifying zero, 
null, false, absence, baseline, and no-error 
separately.
Evidence ledgers Makes proof a record with source, formula, 
uncertainty, and error status.KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
13. Boundaries and Required Next Work
The strongest version of the theory is a classifier and proof validator. It should not currently be 
framed as a completed replacement for standard mathematics. It should be framed as a runtime 
layer that exposes hidden assumptions and prevents category errors. The following work is required
before stronger claims can be made.
Required next work Reason Minimum validation target
Repair Sheet1 #REF! artefacts Reference errors break proof 
lineage.
Zero #REF! cells across the 
workbook.
Populate circle-square count 
tables
The equality claim requires 
numeric counts.
For each stage: circle count, 
square count, overlap count, 
total count, equality/delta 
formula.
Define zero classes in the 
workbook Residual zero markers exist.
Each zero tagged as baseline, 
cancellation, coordinate, error,
or invalid.
Prime sheet formula audit
Artefact 2 has 10,640 formulas 
but cached binary values need 
formula-level interpretation.
Explicit prime predicate, 
factor-lane mask, and survival 
ledger.
Physical water dataset
Conceptual phase relation 
needs empirical 
demonstration.
Rows for sample, temperature, 
pressure, purity, nucleation, 
phase, uncertainty, and 
traceability.
Versioned proof ledger White paper claims need 
reproducibility.
Workbook hash, formula audit,
validation script, exported 
summary table.
14. Conclusion
The Keddeh Bilateral Proof Engine is strongest as a relational execution layer. It does not need to 
deny formal mathematics. It instead requires that every physical or geometric claim declare its 
anchor, reference, unit, state, transition, uncertainty, and proof path before execution. Quantitative 
validation of the linked workbook confirms strong repeated 1-2-3 shell structure, a very low but real 
residual zero rate, exact recurrence of zero-less shell templates, and measurable prime-diagonal 
enrichment in a spiral crop. It also shows that the circle-square claim is not yet proven because the 
workbook lacks populated cell-count calculations. The correct next move is not to overstate the 
result, but to complete the proof ledger: repair formula drift, classify zero markers, populate circlesquare counts, and bind each formula to an explicit validation target.
Appendix A: Quantitative Validation Tables
Metric Value
Total populated cells 43,919KEDDEH Bilateral Proof Engine - Quantitatively Validated White Paper - A. Keddeh
Total numeric cells 40,016
Total formula cells 11,586
Total cached error cells 3
Artefact 1 numeric 3-cells 14,915
Artefact 1 numeric 2-cells 8,936
Artefact 1 numeric 1-cells 2,992
Artefact 1 numeric 0-cells 5
Artefact 1 zero-less 6x6 shell matches 738
Artefact 1 zero-centred 7x7 shell matches 5
Visual Artefacts spiral cells 225
Visual Artefacts unique integers 225
Visual Artefacts prime count 50
X diagonal prime density 40.74%
Off diagonal prime density 19.70%
X enrichment factor 2.07x
6k +/- 1 candidate count up to 289 96
6k +/- 1 candidate survival as prime 61.46%
Artefact 2 active binary cells 260 of 10,640 (2.44%)
Artefact 3 Cell Count labels 575
Appendix B: References
Source workbook: Google Sheets file "Untitled spreadsheet", ID 1i3h9vk7d7bX3bR49rdWoQ9zYvEbhh6Qma0kRXwOnko, exported as XLSX on 6 July 2026.
BIPM. The International System of Units (SI), 9th edition, updated in 2026. 
https://www.bipm.org/en/publications/si-brochure
JCGM VIM3 1.19. Quantity value: number and reference together expressing magnitude of a 
quantity. https://jcgm.bipm.org/vim/en/1.19.html
JCGM VIM3 2.9. Measurement result: set of quantity values attributed to a measurand together with 
available relevant information. https://jcgm.bipm.org/vim/en/2.9.html
NIST. Metrological Traceability: Frequently Asked Questions and NIST Policy. 
https://www.nist.gov/metrology/metrological-traceability
NIST. SI Units - Temperature. https://www.nist.gov/pml/owm/si-units-temperature
