Formal security · P1–P4

Axioms, theorems, and bounds

Proofs need assumptions. These are the load-bearing ones for empty-channel authorization, so absolute product claims can be traced to a model, a reference architecture, and Claims for model scope.

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~5 min overview · ~12 min formalContinue: Quantum bounds

Overview

Four axioms (P1–P4) make empty-channel authorization falsifiable: complete challenges, flat public symbols, zero secret mutual information given the challenge, and session-bound freshness so recordings cannot authorize twice.

Formal mathematics

Why axioms

A ceremony film can look busy and still carry nothing about the secret. That claim is not magic: it rests on four pillars. Without naming them, “capture is not compromise” is a slogan. With them, every absolute-sounding line on the site can point here, then to the reference architecture, then to Claims.

Think of axioms as load-bearing walls: remove one and the theorems above it fail. P3 is the information-theoretic heart: security that does not rely on computers being too slow. P4 is the operational latch that keeps yesterday’s transcript from authorizing tomorrow.

Axioms P1–P4

Four pillars stated for the Rosario–Wang empty-channel model. Formal symbols live in the notation table.

Axiom theorem parameterized bound formal security stack — Formal bounds stack
Read bottom-up: axioms support theorems, which yield parameterized bounds. The stack is model-scoped formal security, not a computational hardness assumption alone. Takeaway: emptiness claims sit on named assumptions you can inspect.

Four Pillars dependency graph for the zero-information channel axioms — This dependency graph states the Four Pillars P1–P4 - path-complete projection, uniform private synonyms, zero mutual information, and session-bound freshness - as the axioms under which the observable channel satisfies I=0.
Follow pillar edges from path completeness and synonym uniformity into the I=0 claim, then freshness against replay. Takeaway: the zero-information condition is axiomatic, not a computational hardness assumption.

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.
$\forall j,\ \sigma_j\in\Pi(C)$
Deepening. Without path completeness, the mere fact of which leaf was reachable would bias an observer’s beliefs. The challenge must present a full menu of legal walks; see challenge geometry on How Rosario works and the Math hub’s challenge symbol. How Rosario works.
P2
Uniform synonym marginals
Conditional on the public challenge, every public symbol is equiprobable. Frequency analysis finds no direction in the transcript.
$\Pr[\upsilon=\upsilon_k\mid h_i,C]=1/H$
Deepening. Equiprobable public symbols are the flat channel: no lean in the histogram means no leak from counts. Synonym-channel uniformity is the named theorem; see A flat, unreadable channel. A flat, unreadable channel.
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.
$I(J;O\mid C)=I(\varphi;O\mid C)=I(\sigma;O\mid C)=I(M;O\mid C)=0$
Deepening. This is the information-theoretic heart: security that does not rely on computers being too slow. Hardness assumptions are not what P3 relies on. Observer perfect secrecy maps Shannon’s gold standard onto authorization transcripts. Perfect secrecy (Shannon).
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.
$\Pr[\mathsf{Verify}(O_{\tau_a},\tau_b)=1]=0\quad(\tau_a\ne\tau_b)$
Deepening. P4 is operational binding, not an entropy claim. Burn-before-validate spends the nonce before twin validation so a perfect byte copy cannot authorize twice. Product deployments earn P4 at the gate. Security (shipping).

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

Conditional on the challenge, every public response symbol is equiprobable.ValidatedAxiom P2 (synonym-channel uniformity) defeats frequency analysis on the transcript.

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

P3: zero mutual information (four-term flagship)

I(J;O\mid C)=I(\varphi;O\mid C)=I(\sigma;O\mid C)=I(M;O\mid C)=0

Read as: Once challenge C is fixed, observing transcript O gives no mutual information about secret index J, private morphism φ, selector path σ, or message M.

Arguments

$J$: : active secret index in a multi-secret family
$\varphi$: : private synonym morphism; stays off-channel
$\sigma$: : private selector path through the public scaffold
$M$: : message or content variable protected by the channel claim
$O$: : observable transcript or recorded ceremony events
$C$: : public challenge that defines the legal selector paths

Operators

$I$: : conditional mutual information
$\mid$: : conditioned on the challenge
$= 0$: : observer-effective leakage is zero under P1–P3

Significance: This is the New Dimensions flagship: structural absence of secret-relevant information in the transcript, not a hardness calendar.

Computational contrast: P3 is not a hardness assumption; it says the transcript lacks secret-relevant information rather than hiding recoverable information behind work factor.

References:Channel Zero Information-theoretic authorization New Dimensions PDF

P4: replay acceptance is zero

\Pr[\operatorname{accept\ replay}]=0

Read as: Under freshness and burn-before-validate enforcement, the probability that a replayed transcript authorizes is zero.

Arguments

$\tau$: : freshness timestamp or event-time clock tick bound into the proof
$replay$: : a captured proof event presented outside its original session binding

Operators

$\Pr$: : probability of the verifier accepting the replay event
$\operatorname{accept\ replay}$: : acceptance predicate for a copied transcript
$= 0$: : no replay acceptance when P4 is enforced

Significance: This separates transcript secrecy from lifecycle safety: a fossilized proof must not remain an authority object.

Computational contrast: Replay resistance is enforced by one-shot lifecycle and nonce burn, not by hoping a copied credential is hard to transform.

References:Session-bound no replay Burn Gate order

P2: uniform bearing marginal

P(W=w\mid C)=1/|B|

Read as: Given challenge C, each public bearing w is equally likely across the bearing set B.

Arguments

$W$: : public bearing witness emitted by the ceremony
$w$: : one particular bearing symbol
$B$: : bearing set; the reference pages use six public bearings
$C$: : public challenge conditioning the marginal

Operators

$P$: : probability of seeing a public bearing
$\mid$: : conditioned on the challenge
$1/|B|$: : uniform probability over the bearing set

Significance: Uniform marginals deny frequency analysis a compass before P3 states the stronger zero-leakage target.

Computational contrast: Flat symbol frequencies remove statistical residue; computational systems can still expose biased artifacts and rely on hardness afterward.

References:Synonym-channel uniformity

P3 in symbols: $I(J;O\mid C)=I(\varphi;O\mid C)=I(\sigma;O\mid C)=I(M;O\mid C)=0$ . P4 binds freshness τ. The private map φ never rides the channel.

Download New Dimensions

Full constructions, theorem sketches, and parameter guidance live in the UNIFIED white paper - not on this page.

Download New Dimensions PDF Math hub

Theorems

Named consequences of the pillars, still model-scoped. They are the bridge from axioms to product language.

Synonym-channel uniformity
Under the private synonym morphism, each observable symbol is uniform over the response alphabet given the challenge leaf. Frequency tables never appear as a compass on the public channel.
Perfect secrecy (observer)
I(secret ; observation | challenge) = 0. Equivalent in spirit to Shannon perfect secrecy on the transcript channel: posterior beliefs about the secret equal priors after seeing the ceremony film.
Replay resistance
Verify(O_τa, τ_b) accepts with probability 0 for τ_a ≠ τ_b when freshness binding and nonce burn are enforced. Fossils are not authority.
Decomposed-layer hiding
Path-complete SAC scaffolds leave selector-path min-entropy near n log₂ k given the public table; binding tag D is orthogonal to hiding. The scaffold shows every path; it does not mark σ.
See white paper §8 · New Dimensions PDF
Zero-information / Vernam equivalence
Under P1–P3 the observable channel matches Vernam’s mutual-information shape: I(Hidden; O | C) = 0. Mechanisms differ from OTP; the Shannon emptiness of the transcript is the shared claim.
See white paper §8 · New Dimensions PDF
Flat-amplitude Grover defense
When every morphism is equally consistent with the synonym transcript, a Grover oracle has no preferred target to amplify. Online forgery still needs separate H^{-LR/2} parameter budgeting.
See white paper §8 · New Dimensions PDF
Replay non-reusability
Freshness-bound Verify rejects O from τ_a under τ_b ≠ τ_a (ideally with probability 0; deployed systems approximate via nonce burn and TTL). Capture without reuse.
See white paper §8 · New Dimensions PDF
Forgery bound H^{-LR}
Without K and φ, online forgery succeeds with probability at most H^{-LR} classically (H^{-LR/2} under Grover-bounded search). Reference row H=6, L=7, R=10 appears under Parameterized bounds.
See white paper §8 · New Dimensions PDF

Parameterized bounds

Parameterized bounds label the reference construction on the public site: surface entropy versus effective mutual information, ring sizes, and bearing cardinality. They are not fantasy guarantees; they are the numbers product pages cite when they say “six bearings” or “effective bits at zero.” Observer bit budgets unpack $H_{\mathrm{surface}}$ versus $I_{\mathrm{effective}}$ .

Parameterized bounds for the reference construction
Parameter	Value	Note
$H_{\mathrm{surface}}$ / character	$\log_2 \lvert B\rvert = \log_2 6 \approx 2.585$ bits	Observable bearing entropy only: what can be written down.
$I_{\mathrm{effective}}$	$0$ bits	Mutual information with the secret under the axioms.
Ring sizes (reference)	26 + 26 + 10 = 62 symbols	Disjoint uppercase, lowercase, digit rings in the reference construction.
Zone / bearing cardinality	$\lvert B\rvert = 6$	Six zones, six bearings; label reference for the public site.
$H{=}6,\,L{=}7,\,R{=}10$	$H^{-LR}\approx 2^{-181}\ /\ H^{-LR/2}\approx 2^{-90}$	UNIFIED reference: classical forgery floor ≈2^{-181}; Grover-bounded ≈2^{-90}.

Four axioms P1–P4 as pillars with gold highlight on zero mutual information and a mini bounds table — Axioms · theorems · bounds · model-scoped
How to read: scan pillars P1–P4 left-to-right (path-complete, uniform marginals, zero MI, no-replay), then check the mini bounds table for H_surface versus I_effective=0. Gold on P3 marks the emptiness claim that theorems rest on. Takeaway: absolute words on this site are model-scoped to the named axioms and reference architecture.

See also (surface vs effective): What observers can measure.

Absolute mechanism claims assume the reference architecture and the axioms above. Classical attack families that assume conversion from surface to secret are denied under these pillars; the mechanism denials are on Why classical attacks fail. Affirmative claim classes live on Claims.

Product bridge

Burn-before-validate is how deployments earn P4 at the gate: the nonce is spent before twin validation, so a harvested envelope cannot authorize a second request. Shipping posture and envelope binding live on Security, not as a second formal paper, but as the operational latch on the axioms.

Overview

Why axioms

Axioms P1–P4

Path-complete geometric projection

Uniform synonym marginals

Zero mutual information

Session-bound no-replay

P3: zero mutual information (four-term flagship)

Arguments

Operators

P4: replay acceptance is zero

Arguments

Operators

P2: uniform bearing marginal

Arguments

Operators

Theorems

Synonym-channel uniformity

Perfect secrecy (observer)

Replay resistance

Decomposed-layer hiding

Zero-information / Vernam equivalence

Flat-amplitude Grover defense

Replay non-reusability

Forgery bound H^{-LR}

Parameterized bounds

Related

Product bridge