KEX-WP-04
Bilateral Rebound and Recursive Geometry
An anchor-preserving involution for mirror validation, rollback, and bidirectional traversal
Abstract
This paper formalises the KEX bilateral rebound. A mirror map sends rank r to 6−r, swapping the boundary and crossing rails while fixing the anchor. In centred compatibility coordinates the map is sign inversion; natively the centre remains active state 1 rather than becoming an active zero. The paper derives involution, distance preservation, path reversal, and fixed-point properties and translates them into checkpoint, rollback, and mirror-validation algorithms.
1. Problem and scope
A raw multiplication of state labels by −1 fails because the active set contains 1 but not −1. Rebound must therefore be defined on ranks, not on the printed labels.
The centred coordinate c(s)=ρ(s)−3 is explicitly a compatibility coordinate. Its zero denotes the rank centre; it does not replace the native anchor.
For a path, rebound reverses orientation and mirrors every state. A proof record must distinguish mirror validation from ordinary forward execution.
2. Formal model
μ(s)=ρ⁻¹(6−ρ(s))    (1)
μ(μ(s))=s;   μ(1)=1    (2)
c(s)=ρ(s)−3;   c(μ(s))=−c(s)    (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 rollback, the rebound defines state orientation but does not itself restore payload bytes. A valid rollback also requires an earlier checkpoint and the inverse or compensating operations for intervening deltas.
In bilateral reasoning, the mirror branch is a controlled comparator. It must not be mistaken for independent evidence when both branches are generated from the same source assumptions.
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 μ is implemented as raw numeric negation, the native model fails.
•	If rollback restores state without proof continuity, the recovery claim fails.
•	If mirror output is presented as independent replication, the evidence claim fails.
8. Conclusion
Rank-defined rebound preserves the intended geometry without inventing an active −1 or erasing the central anchor. It is a useful operator for reversible computation and reasoning when paired with actual payload recovery and evidence rules.
References
[1] A. Keddeh, KEDDEH THEOREM AND BRAINK VIRTUALISED BIO-ORGANISM, Master Foundation Document v1.9, 2026. Internal source corpus.
[2] D. R. Jefferson, 'Virtual Time,' ACM TOPLAS 7(3), 404–425 (1985). https://doi.org/10.1145/3916.3988
[3] KEX/BRAINK White Paper Series v1.0, KEX-WP-04, calculation receipt 31ba82c00739db6ef07e724a863dc317b885622d4eb7a1360bb7702a28b79513.
CENTRAL THESIS
The correct 180-degree rebound reverses rank orientation while preserving the active anchor 1.
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 Papers 06 and 15
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 correct 180-degree rebound reverses rank orientation while preserving the active anchor 1.
ID
Result
Status
R-01
μ is an involution.
FORMALISED
R-02
1 is the unique active fixed point.
FORMALISED
R-03
d_K(μ(a),μ(b))=d_K(a,b).
FORMALISED
Component/state
Function/value
Evidence or interpretation
Forward
−3→−2→1→+2→+3
admit, anchor, execute, commit
Mirror
+3→+2→1→−2→−3
validate or recover in reverse orientation
Anchor
1
identity survives both paths
Exception
non-adjacent jump
expand path or bind explicit rule
Experiment
Dataset/workload
Acceptance criterion
State mirror
All five states
Involution and fixed point
Path mirror
All walks through length 12
Adjacency and distance preserved
Rollback
Checkpoint plus deltas
Original authoritative hash restored
Mirror reasoning
Contradiction corpus
Source dependence recorded
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.
