KEX-WP-02
Typed Zero and Non-Erasing Arithmetic
Separating absence, baseline, balance, stasis, missingness, and measured zero
Abstract
This paper formalises the KEX corrected-zero doctrine as a tagged-result algebra. It defines ABSENT, BASELINE, UNKNOWN, INVALID, BALANCED, MEASURED_ZERO, and STASIS; introduces non-erasing addition and difference; and specifies explicit projection into conventional scalar arithmetic. The approach preserves both operands and context when conventional cancellation yields zero. It is directly testable in data schemas, spreadsheets, runtimes, sensors, and scientific records.
1. Problem and scope
A single scalar zero can mean no item, a scale origin, a failed read, a balanced sum, a same-state distance, or a genuine measured value. Treating those meanings as interchangeable is a type error even though the numeral remains mathematically valid.
KEX therefore uses a sum type. Arithmetic operates over measured payloads; state transitions operate over S_active; compatibility projections are named functions. The host may still receive 0, but the native record retains how that 0 was produced.
The codomain of non-erasing arithmetic is larger than S_active. This avoids the false claim that state labels are closed under conventional addition.
2. Formal model
a ⊞ b = MEASURED(a+b), if a+b≠0; otherwise BALANCED(a,b,c)    (1)
project(BALANCED(a,b,c))=0,   recover(BALANCED)=(a,b,c)    (2)
difference(a,a)=STASIS(a,observer,Δt),   project(STASIS)=0    (3)
All symbols in these equations are local to this paper unless explicitly imported from Series Paper 01. A compatibility numeral such as zero may appear in ranks, matrices, encodings, distances, statistics, or projected outputs without becoming an untyped KEX active state.
3. Results and claim register
4. Architecture or state mapping
5. Application and cross-domain translation
In spreadsheets, blank, numeric zero, formula error, not-applicable, and a calibrated zero must remain distinct. In code, default-initialized zero must not pass an active-state constructor. In scientific data, the unit and instrument accompany MEASURED_ZERO.
The design resembles established tagged unions and option/result types, but adds explicit cancellation lineage and compatibility projection as first-class operations.
Cross-domain reuse is accepted only as a structure-preserving mapping. Similar words or diagrams are insufficient: the mapping must preserve identity, allowed transitions, observer/environment terms where relevant, and the authoritative proof boundary.
6. Experimental programme
7. Falsifiers and failure conditions
•	If projection is performed without recording irreversibility, the conformance claim fails.
•	If ABSENT and UNKNOWN compare equal natively, the schema fails.
•	If a measured zero lacks unit and uncertainty, it cannot support a physical claim.
8. Conclusion
Typed zero converts an expansive philosophical claim into a precise data and arithmetic contract. It preserves conventional mathematics while preventing semantic collapse in active systems.
References
[1] A. Keddeh, KEDDEH THEOREM AND BRAINK VIRTUALISED BIO-ORGANISM, Master Foundation Document v1.9, 2026. Internal source corpus.
[2] Global Alliance for Genomics and Health / hts-specs, Variant Call Format specification. https://samtools.github.io/hts-specs/
[3] KEX/BRAINK White Paper Series v1.0, KEX-WP-02, calculation receipt 31ba82c00739db6ef07e724a863dc317b885622d4eb7a1360bb7702a28b79513.
CENTRAL THESIS
The correction is not to delete zero from mathematics, but to prevent semantically different zero-like results from becoming the same authoritative state.
Series field
Value
Author and origin authority
A. Keddeh
System lineage
KEX / KEDDEH THEOREM / BRAINK / BRAINK⁶
Series
KEX/BRAINK Modular White Paper Series v1.0
Publication date
19 July 2026
Prerequisite
Series Paper 01
Next research route
Series Paper 03
STATUS DISCIPLINE
Mathematical validity, engineering performance, and physical correspondence are separate evidence lanes. This paper promotes only the claims that pass its declared gates.
PAPER BOUNDARY
The correction is not to delete zero from mathematics, but to prevent semantically different zero-like results from becoming the same authoritative state.
ID
Result
Status
Z-01
Distinct zero-like variants round-trip without collision.
FORMALISED
Z-02
Scalar equality after projection does not imply native identity.
FORMALISED
Z-03
Non-erasing arithmetic adds metadata and storage cost.
TRADE-OFF
Component/state
Function/value
Evidence or interpretation
ABSENT
tested non-presence
test rule, scope, time
BASELINE
calibration/reference
instrument, scale, version
UNKNOWN
not determined
reason and acquisition plan
INVALID
rejected
rule, evidence, failure path
BALANCED
opposing projection
operands and context
MEASURED_ZERO
instrument value
unit and uncertainty
STASIS
same active state
identity, observer, duration
Experiment
Dataset/workload
Acceptance criterion
Variant round trip
Serialize every zero-like type
Type and payload recovered exactly
Balanced pair collision
At least 100 distinct cancelling pairs
All native records remain distinct
Spreadsheet import
Blank, 0, error, NA
No silent coercion
Projection audit
Native→scalar→declared recovery
Irreversible steps reported
NON-AGGREGATE GATE
One failed mandatory falsifier blocks promotion even if other tests score highly. Results remain at the highest evidence state for which every required gate passes.
