HS-bit Meter
A Reportable Unit and Procedure for Measuring HS° Synchronization Cost. From Entropy Production to Tick-Canonical Reporting in Phase-Window Coordination Stacks.
1. Motivation
Phase alignment is often treated as “just a parameter” in abstract models. In real distributed systems, fine-grained phase/temporal alignment is a limiting resource. The Quantum HS◦ framework makes this precise by treating temporal (asymmetry) resource as consumable. The goal of this note is to make that consumption reportable in a simple, engineering-friendly unit (bits) and in a tick-canonical form compatible with phase-window coordination stacks.
2. Definitions: HS° consumption and HS-bit
HS° consumption: Given an alignment attempt that maps an input state ρin to an output state ρout, define the consumed HS° resource as ΔAHS := AHS(ρin) - AHS(ρout).
HS-bit: An HS-bit is one bit of HS° resource consumption: 1 HS-bit ≡ 1 bit of ΔAHS (log base 2). If logarithms are taken in base 𝑒, costs are reported in HS-nats. (Conversion: 1 nat = 1/log 2 bits.)
Remark: HS-bit is not a physical unit of time; it is an information-theoretic unit for reporting the cost of phase alignment under stated assumptions.
3. HS° Consumption Meter
HS° Consumption Meter: An HS° Consumption Meter is a procedure that outputs ΔAHS (in HS-bits or HS-nats) from one of: a pair (ρin, ρout) (state-based mode), a residual tick distribution 𝑝 (ticks mode), or proxy measurements that provide lower bounds and/or model-based estimates (bound/proxy mode).
Mode A: entropy-production mode (state-based)
Under phase diffusion (mixture of shifts, unconditioned output), HS° consumption equals entropy production: ΔAHS = S(ρout) - S(ρin).
Mode B: ticks mode (distribution-based)
In the discrete residual ticks model aligned with phase-window stacks: ΔAHS = H(p).
Mode C: bound mode (measurement-only)
If only purity is accessible (pure input case), one may report the conservative lower bound: ΔAHS ≥ -log Tr(ρ2out), with the same log base as the reported unit.
4. Interoperability bridge: windows, ticks, and success mass
Tick-canonical reporting: If an execution stack is tick-canonical (as in Conventions and Q-Address), reporting SHOULD be tick-canonical: report the tick resolution 𝑅, the window definition, and either the residual tick distribution 𝑝 or the success mass in the window.
Window success mass: Let W ⊆ {0, . . . , R - 1} be the accepted set of ticks for a phase window (wrap-safe by circular distance). Given a residual tick distribution p, define psucc := Σk ∈ W pk.
Proposition: For any distribution p, the Shannon entropy satisfies H(p) ≥ h2(psucc), where h2(x) = -x log x - (1 - x) log(1 - x) is the binary entropy (same log base).
5. Recommended reporting standard
HS-bit meter report (recommended fields): A minimal report SHOULD include: unit (HS-bits or HS-nats), model (phase diffusion and/or discrete ticks), tick resolution R, window definition, measured quantities (p, purity, or psucc), and statement of assumptions.
6. Security context
Remark: HS-bit measures alignment cost; it is not itself a security mechanism. Timeverse/Q-Address fields are public context (not secrets). Security (signatures, nonces, anti-replay policy, canonical encoding) is defined by Security Profiles.