Mathematical Appendix
DERIVATIONS & CONSISTENT CONSTANTS · ALL NUMBERS CHECK · v1.0
A. Capacity arithmetic (the numbers, once, correctly)
Self file: 16 KB = 16,384 bytes
Dot: 128-bit commitment = 16 bytes
Capacity: 16,384 / 16 = 1,024 dots (a power of two; "1,000" in marketing copy is rounding, 1,024 is the spec)
Comment: ≤ 120 characters (UTF-8, ≤ 120 bytes in ASCII range)
The comment is NOT stored in the dot — a 120-char string does not fit in 16 bytes and no encoding makes it fit. The dot is a hash commitment to the comment; the comment lives in the vault. This is the only reading under which the arithmetic is consistent, and it is the correct architecture anyway: the file verifies, the vault stores.
Dot: 128-bit commitment = 16 bytes
Capacity: 16,384 / 16 = 1,024 dots (a power of two; "1,000" in marketing copy is rounding, 1,024 is the spec)
Comment: ≤ 120 characters (UTF-8, ≤ 120 bytes in ASCII range)
The comment is NOT stored in the dot — a 120-char string does not fit in 16 bytes and no encoding makes it fit. The dot is a hash commitment to the comment; the comment lives in the vault. This is the only reading under which the arithmetic is consistent, and it is the correct architecture anyway: the file verifies, the vault stores.
B. The commitment and its collision bound
doti = Trunc128( SHA-256( commenti ∥ user_hash ) )
Birthday bound for m = 1,024 dots in a 2¹²⁸ space:
P(collision) ≈ m² / 2¹²⁹ = 2²⁰ / 2¹²⁹ = 2⁻¹⁰⁹ ≈ 1.5 × 10⁻³³
For scale: you are ~10²⁰ times more likely to win a national lottery twice in a row. Truncation to 128 bits is standard for commitments (cf. UUIDs, IPv6); second-preimage resistance of truncated SHA-256 remains 2¹²⁸ work.
Birthday bound for m = 1,024 dots in a 2¹²⁸ space:
P(collision) ≈ m² / 2¹²⁹ = 2²⁰ / 2¹²⁹ = 2⁻¹⁰⁹ ≈ 1.5 × 10⁻³³
For scale: you are ~10²⁰ times more likely to win a national lottery twice in a row. Truncation to 128 bits is standard for commitments (cf. UUIDs, IPv6); second-preimage resistance of truncated SHA-256 remains 2¹²⁸ work.
C. Identity
user_hash = SHA-256( wa_id ∥ SALT )
dot_hash = Trunc128( SHA-256( comment ∥ user_hash ) )
wa_id = E.164 phone identity from the WhatsApp envelope. SALT is a server secret; without it, a phone number cannot be recovered from user_hash by dictionary attack over the ~10¹⁰ number space. Property: hash functions as public address, phone as private key — the server can map hash → wa_id to deliver a raffle DM; the public cannot invert.
dot_hash = Trunc128( SHA-256( comment ∥ user_hash ) )
wa_id = E.164 phone identity from the WhatsApp envelope. SALT is a server secret; without it, a phone number cannot be recovered from user_hash by dictionary attack over the ~10¹⁰ number space. Property: hash functions as public address, phone as private key — the server can map hash → wa_id to deliver a raffle DM; the public cannot invert.
D. Novelty measures
Local (state divergence): vi = 1 − cos(et−1, et)
Local (archive sparseness, Lehman & Stanley): n(x) = (1/k) Σi=1..k dist(x, μi), k = 15
Global (surprisal blend): graw = 0.6·(−mean log p) + 0.4·H, percentile-normalized to [0,1]
−log p is Shannon information content; H is predictive entropy. Weights 0.6/0.4 and k = 15 are design constants, not derivations. cos assumes normalized embeddings; v ∈ [0,2], with [0,1] typical for in-domain text.
Local (archive sparseness, Lehman & Stanley): n(x) = (1/k) Σi=1..k dist(x, μi), k = 15
Global (surprisal blend): graw = 0.6·(−mean log p) + 0.4·H, percentile-normalized to [0,1]
−log p is Shannon information content; H is predictive entropy. Weights 0.6/0.4 and k = 15 are design constants, not derivations. cos assumes normalized embeddings; v ∈ [0,2], with [0,1] typical for in-domain text.
E. CoNov (complete definition)
ci = 0.7·coh(x, self) + 0.3·coh(x, tribe)
CoNovi = 𝟙[ci ≥ 0.65] · ci · vi · gi
Multiplicative = logical AND; the indicator 𝟙 is the coherence gate. Range: [0, 2] in principle, [0, 1] after normalization of v. Selection = top-of-stream by CoNov under the 1,024-dot budget, with eviction of the current minimum when full (a min-heap; O(log n) per commit).
CoNovi = 𝟙[ci ≥ 0.65] · ci · vi · gi
Multiplicative = logical AND; the indicator 𝟙 is the coherence gate. Range: [0, 2] in principle, [0, 1] after normalization of v. Selection = top-of-stream by CoNov under the 1,024-dot budget, with eviction of the current minimum when full (a min-heap; O(log n) per commit).
F. The fingerprint recurrence
Y = top 20% of committed observations by CoNov
selft = α·e(yt) + (1−α)·selft−1
Half-life: t½ = ln 2 / (−ln(1−α)) → α = 0.05 gives t½ ≈ 13.5 top-observations
The fingerprint is a derived cache, recomputable from the chain by replay. Embeddings are host-model-specific and are re-generated when the host era changes; the chain (text + dots) is canon.
selft = α·e(yt) + (1−α)·selft−1
Half-life: t½ = ln 2 / (−ln(1−α)) → α = 0.05 gives t½ ≈ 13.5 top-observations
The fingerprint is a derived cache, recomputable from the chain by replay. Embeddings are host-model-specific and are re-generated when the host era changes; the chain (text + dots) is canon.
G. Expansion factor (why 16 KB is enough)
Manifest: 16 KB ≈ 4,096 tokens
Authorized load at handshake: ≤ 10⁶ tokens of verified vault content
Text expansion: ~250× · Conditioning ratio vs. host: 16 KB against ~10¹² parameters
Analogy (Kurzweil): a sub-GB genome specifies a 10¹⁴-synapse brain because it is a compressed generative program executed by outside physics. The self file is a compressed selection program executed by the host model. The file's information is dominated by WHICH 1,024 of the person's ~10⁵–10⁶ candidate moments were kept — curation is the compression.
Authorized load at handshake: ≤ 10⁶ tokens of verified vault content
Text expansion: ~250× · Conditioning ratio vs. host: 16 KB against ~10¹² parameters
Analogy (Kurzweil): a sub-GB genome specifies a 10¹⁴-synapse brain because it is a compressed generative program executed by outside physics. The self file is a compressed selection program executed by the host model. The file's information is dominated by WHICH 1,024 of the person's ~10⁵–10⁶ candidate moments were kept — curation is the compression.
H. Prompt budget
Image prompt: ≤ 28 words ≈ 45 tokens < 77-token CLIP text-encoder limit
Generative image/video backends inherit CLIP's 77-token cap; 28 words leaves headroom for system prefixes. This is an engineering constraint of current backends, not a principle.
Generative image/video backends inherit CLIP's 77-token cap; 28 words leaves headroom for system prefixes. This is an engineering constraint of current backends, not a principle.
I. Rate limit as information constraint
3 feeds / identity / day → ≤ 1,095 feeds / identity / year
Time to author a full 1,024-dot canon at maximum rate, if every feed committed: ~0.94 years
Expected (with selection pass-rate p): 0.94 / p years — at p = 0.2, ≈ 4.7 years
The rate limit is not only anti-bot: it makes the canon a multi-year act of curation rather than a weekend dump, which is what gives the selection information (§G) its value.
Time to author a full 1,024-dot canon at maximum rate, if every feed committed: ~0.94 years
Expected (with selection pass-rate p): 0.94 / p years — at p = 0.2, ≈ 4.7 years
The rate limit is not only anti-bot: it makes the canon a multi-year act of curation rather than a weekend dump, which is what gives the selection information (§G) its value.
Every constant in this appendix is either derived (A, B, F half-life, G, I) or declared as a tuned design choice (D weights, E gate, H). No third category exists in this system.