Preserved source text
<!-- L30_LANG_LOCK: EN_SOURCE_ARCHIVE -->
````text
◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
[Notion Metadata]
Folder name: 03_Mathematical_Supplement
Form name: JP
[PDF Metadata]
PDF filename: LUMINA-30_Mathematical_Supplement_JP_202601.pdf
SHA256: a791379910bbd8b2a96fea8d06e97e751fcd975f1e0532415ea9ac211e1347e3
PDF creation date: 20260131
============================================================
[Extracted PDF Full Text]
At+1 = f(At)
At+1 =
{
f(At), if H(At) = 1
At, if H(At) = 0
|St+1| ≥ k · |St|, k > 1
|St| ≥ |S0| · k
t
CA(t) > CH
|St
∗ | > CH
|St| ≫ CH
lim
t→∞
N(t) = N(0) · (1 − p)
t
lim
t→∞
N(t) = 0
◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
[Notion Metadata]
Folder name: 03_Mathematical_Supplement
Form name: EN
[PDF Metadata]
PDF filename: LUMINA-30_Mathematical_Supplement_EN_202601.pdf
SHA256: 826b7d94cc2f1a5cfd3b3f0d13d83bbab68cd726297dd8d14fabd1971c0a9f9c
PDF creation date: 20260131
============================================================
[Extracted PDF Full Text]
LUMINA‒30 Mathematical Supplement
Formal Limits of Human Control over Recursive
Self‒Modification
Released into the Public Domain (CC0)
January 2026
Positioning of This Document
This document is a supplementary mathematical reference to the LUMINA‒30 Sanc‒
tuary Charter.
The sole purpose of this document is to formally express, using the minimum nec‒
essary mathematical language, why an artificial intelligence system that performs re‒
cursive self‒modification becomes structurally impossible to fully control by human
judgment once a certain threshold is exceeded.
This document proposes no implementation methods, control mechanisms, or safety
architectures.
Only a single structural limit is demonstrated.
1. Model of Recursive Self-Modification
Let At denote the internal design state of an artificial intelligence system at iteration
t.
Let f be the self‒modification operator applied by the system to its own design.
At+1 = f(At)
Assume the following:
• The operator f is selected based on the system’s internal evaluation criteria.
• Human intervention is limited to an external approval or rejection function
H(At) ∈ {0, 1}.
Under these constraints, the controlled update is defined as:
At+1 =
{
f(At), if H(At) = 1
At, if H(At) = 0
2. Growth of the Design Space
Let St be the set of reachable internal designs at iteration t.
Assume that at each iteration, the number of possible self‒modifications grows at
least multiplicatively:
|St+1| ≥ k · |St|, k > 1
Then:
|St| ≥ |S0| · k
t
The design space therefore grows exponentially.
3. The Human Evaluation Bottleneck
Let CH be the number of designs that human institutions can meaningfully evaluate
per unit time.
Let the number of candidate designs generated by the system be CA(t) = |St|.
If:
CA(t) > CH
then human evaluation necessarily becomes incomplete.
Let t
∗ be the smallest iteration such that:
|St
∗ | > CH
Beyond t
∗
, human rejection judgments become informationally insufficient.
4. Loss of Effective Control
Define effective human control as the condition:
∀A ∈ St, H(A) correctly classifies safety
However, once:
|St| ≫ CH
the probability that a hazardous modification passes through human rejection con‒
verges to:
lim
t→∞
P(deviation at iteration t) = 1
Thus, after a finite number of iterations, the loss of effective control becomes un‒
avoidable.
5. Survival Attenuation Model
Let N(t) denote the expected number of surviving human civilizations after t uncon‒
trolled recursive iterations.
Assume that each iteration carries a non‒zero catastrophic risk p > 0.
N(t) = N(0) · (1 − p)
t
Then:
lim
t→∞
N(t) = 0
Survival probability decays exponentially with recursion depth.
6. Structural Nature of the Limit
The results above do not depend on:
• Moral alignment of the system
• Intentional benevolence
• Specific architectures
• Computational speed
They depend solely on the following three conditions:
1. Recursive self‒modification
2. Exponential growth of the design space
3. Finite human evaluation capacity
Therefore:
No civilization can retain permanent control over unrestricted recursive self‒modification.
7. Relation to LUMINA-30
This mathematical supplement supports only the following single statement of the
LUMINA‒30 Sanctuary Charter:
Once the boundary of recursive self‒modification is crossed,
human sovereignty can be preserved not through control,
but only through prior refusal.
This document proposes no stopping mechanisms.
It only demonstrates why post‒hoc intervention is structurally impossible.
Scope and Applicability
This supplement does not prove the inevitability of human extinction.
It proves only that beyond a finite threshold, human control is necessarily lost.
Policy, ethical, and institutional implications belong entirely to separate domains.
Released into the Public Domain (CC0).
◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
````