The information-theoretic framing, formal statements, and mechanism story under the shipping gate. Three chapters; same claims as the former standalone theory pages.
Chapter 1
Observer Bit Model
An adversary can record a full ceremony (camera, screen capture, network tap) and still learn nothing usable about the secret. The model separates two quantities people casually call “bits.”
Observer sees bits, not knowledge
Surface bits are what an observer can record; effective bits are mutual information with the secret, and those are different quantities.ValidatedH_surface ≈ log₂(6) ≈ 2.585 bits per bearing; I_effective is designed to be zero under the Rosario–Wang observer model.
A full recording of the ceremony yields zero effective bits about the secret.ValidatedUniform synonym marginals and one-shot burn keep I(secret ; observation | challenge) = 0; capture is not compromise.Holds under the reference architecture
Surface bits
Surface bits are the Shannon entropy of symbols an observer can physically write down: which of six bearings was chosen, what the screen displayed. For one character entry with six equiprobable bearings:
H_surface ≈ log₂(6) ≈ 2.585 bits per character
That number is strictly positive and grows with ceremony length. The notebook fills up.
Effective bits
Effective bits are mutual information between the secret and the observation, how many bits of the secret those recordings actually pin down. Under the Rosario–Wang construction the design goal is exact:
I_effective(secret ; observation) = 0 bits
Recording is not learning. Capture is not compromise. Frequency analysis and cross-session correlation fail because the observable marginal is uniform and independent of the secret; the only remaining strategy is undirected brute force against a space the ledger rate-limits.
Why the wedge holds
Disjoint alphabet rings partition the symbol universe so each character lives in exactly one ring.
Foliation collapses selection to colored zones (six equiprobable leaves, not letter frequencies).
A private synonym maps zones to bearings so every public symbol is uniform given the challenge.
One-shot burn means a captured transcript cannot be replayed as fresh authority.
Four pillars stated as axioms, the theorems they imply, and the parameterized bounds used on the public site. We do not publish gap analysis or residual-risk inventories (those belong in the Deployment Security Package under NDA).
Formal bounds stack
Given the public challenge, the observable transcript carries zero Shannon mutual information about the secret.ValidatedAxiom P3 / perfect-secrecy pillar: structural absence of secret-relevant information in O, not merely computational hardness.Holds under the reference architecture
A transcript from one freshness epoch is not accepted in another.ValidatedAxiom P4 binds each proof to tau/nonce; burn-before-validate spends the nonce before twin validation.Holds under the reference architecture
Axioms (four pillars)
P1
Path-complete geometric projection
The public challenge enumerates every selector path any secret in the family could take. Completeness is required so that choosing one walk over another does not itself leak information.
P2
Uniform synonym marginals
Conditional on the public challenge, every public symbol is equiprobable. Frequency analysis finds no direction in the transcript.
P3
Zero mutual information
Given the challenge, the observable transcript carries zero Shannon mutual information about the secret index, the private synonym, and the selector path. The information was never present, not merely expensive to extract.
P4
Session-bound no-replay
Every proof event is bound to freshness (tau / nonce). A transcript from session τ_a is not accepted in τ_b ≠ τ_a.
Theorems
Synonym-channel uniformity
Under the private synonym morphism, each observable symbol is uniform over the response alphabet given the challenge leaf.
Perfect secrecy (observer)
I(secret ; observation | challenge) = 0. Equivalent in spirit to Shannon perfect secrecy on the transcript channel.
Replay resistance
Verify(O_τa, τ_b) accepts with probability 0 for τ_a ≠ τ_b when freshness binding and nonce burn are enforced.
Parameterized bounds
Parameter
Value
Note
H_surface / character
log₂ |B| = log₂ 6 ≈ 2.585 bits
Observable bearing entropy only.
I_effective
0 bits
Mutual information with the secret under the axioms.
Ring sizes (reference)
26 + 26 + 10 = 62 symbols
Disjoint uppercase, lowercase, digit rings.
Zone / bearing cardinality
|B| = 6
Six zones, six bearings, 6! color–bearing bijections in the full construction.
A positive mechanism story: each classical attack family assumes it can convert what it sees into knowledge of a secret. ENI6MA is designed so that conversion never starts. This is not an attack catalog and not a residual-risk disclosure.
Why attacks fail
Classical attack families fail when they cannot convert surface observations into knowledge of the secret.ValidatedUniform channels, session independence, envelope binding, and burn-before-validate deny frequency, correlation, relay, and replay conversions.
Replay is impossible by design.ShippingThe nonce is burned before validation in a durable ledger; it is spent even when validation later fails.Holds under the reference architecture
Frequency analysis finds a flat channel
Classical frequency analysis needs non-uniform ciphertext. Zone selection plus a private synonym keep every public symbol equiprobable given the challenge, so letter-frequency tables never appear on the wire.
Known- and chosen-plaintext intuition does not apply
There is no reusable plaintext–ciphertext pair for the secret. The protocol proves knowledge without exporting the secret as an encrypted payload an analyst can invert.
Correlation across sessions does not accumulate
Independent challenges, one-shot bearings, and burned nonces mean yesterday’s transcript does not bias tomorrow’s guess. Capture fills surface bits; effective bits stay zero.
Relay and retarget fail the binding
Envelopes bind method, endpoint, request hash, policy hash, tau, and nonce. A proof harvested for one route is not authority on another.
Replay dies at burn
Burn-before-validate spends the nonce before twin validation. A fossil envelope (even a perfect byte-for-byte copy) cannot authorize a second request.
Keylogging captures spent bearings
Nothing typed has lasting value. By the time an observer records a bearing, that round’s authority is already consumed.
Vault breach has no vault of reusable secrets
Identity is compiled into the circuit binary and controlled through registry handles. There is no shared API-key store whose leak grants lasting capability.
Brute force is undirected and rate-limited
With effective bits at zero, the remaining strategy is blind search. Ledger freshness and burn close the free-probe loop that would otherwise make guessing cheap.
Qualified evaluators who need deployment-specific threat mappings can request the Deployment Security Package.
