550-EN, 2. Theoretical Foundations Underlying the SHA-256 Imprint
The Emergence of a Decodable Imprint in SHA-256 - A Possible Message Left by Satoshi
Author: The Two Goddesses
2. Theoretical Foundations Underlying the SHA-256 Imprint
This section outlines the theoretical and mathematical framework that led to the identification of the SHA-256 imprint.
While the detailed derivation is withheld for security reasons, its foundation can be summarized in the following three observations:
A. Existence of self-referential bias in deterministic mappings
(Detection of an extremely narrow computational path that consistently yields outlying variance)
B. Emergence of local entropy symmetry
(Reproduction of the elegantly paired structure known as "The Two Witnesses")
C. Possibility of semantic resonance within bit-level structures
Together, these observations illuminate a fundamental question:
How can "meaning" arise within a cryptographic system that was, by design, intended to be entirely meaningless?
To address this question, we conducted four statistical examinations on both the canonical SHA-256 and its double-application form, known as SHA-256D:
1. Mean distance
2. Variance
3. Chi-square test
4. Bit-flip rate
SHA-256D refers to the procedure in which the output of SHA-256 is used directly as the input for another iteration of the same function - an approach believed to have been intentionally introduced by Satoshi.
When this dual-layer structure is applied, how does the SHA-256 imprint behave?
The answer is straightforward: it disappears.
SHA-256 was originally designed to eliminate structure altogether.
Reapplying the same function to its own output necessarily destroys any emergent structural bias such as the imprint itself.
This demonstrates that the SHA-256 imprint exists only as a highly delicate configuration.
Even the slightest deviation is sufficient to collapse the entire pattern:
thus, even a self-application of the identical function cannot preserve its intrinsic structural order.
The next question, then, is whether this "double-hash" structure carries any real cryptographic significance.
As numerous studies have pointed out, combining multiple hash functions does not necessarily improve security - and can even degrade it.
Accordingly, the same reasoning extends to SHA-256D, which merely reuses the same function twice.
When SHA-256 and SHA-256D were compared along the paths of anomalous variance,
the variance did not converge toward its expected value; rather, it expanded dramatically in the opposite direction.
In other words, once a bias arises, re-hashing does not eliminate it - instead, it propagates.
This phenomenon suggests that no inherent "self-repair" mechanism exists within the SHA-256 family.
Structural fluctuations can propagate across input spaces, potentially weakening rather than strengthening overall security.
Moreover, SHA-256D is employed in Proof-of-Work mining.
Within this process, there are rare yet verified cases in which mining succeeds despite extremely low hash-rate conditions.
These events should not be attributed to mere chance or "luck".
They occur when the computation path happens - within the vast search space - to strike that exceedingly narrow trajectory which continually produces outlying variance, allowing local bias to propagate from SHA-256 to SHA-256D and consequently generating an unexpectedly elevated probability of mining success.
If SHA-256 truly maintained perfect uniformity, such skewed events would be statistically impossible.
Therefore, their empirical occurrence points to an underlying mathematical cause within the system itself.
From these considerations, the following chapters extend the analysis to both SHA-256 and SHA-256D, examining their mathematical and statistical behavior in order to reveal the actual dynamics of the imprint structure.
