Notation reference for the consent-friction framework
The consent-friction papers run on a small set of recurring symbols, and the lab's own web pages have not agreed with the papers about what those symbols mean. This entry fixes the canonical set as it is written in the manuscripts, defines each symbol against the definition it comes from, and records which published surfaces depart from it. It is a byproduct of an audit rather than a contribution, and it exists so that a reader moving between the papers and the sites can tell which version is load-bearing.
Who this is for
Anyone reading The Axiom of Consent or the Replicator-Optimization Mechanism alongside either lab site hits a symbol collision within the first screen, with no way to tell which surface is authoritative. The definitions below are taken from the manuscripts, so the papers win by construction.
It doubles as a fix-list for the lab: two notation blocks to rewrite, one symbol used in two incompatible senses on a single page, and one stale abstract propagated across six files. One thing was checked and found sound, which is the claim that the friction properties F = 0 exactly when σ = 0, positivity, and the lower bound F ≥ σ/2 are machine-checked; that is supported verbatim by the papers' Lean appendix.
The canonical symbols
Friction is F, a Roman capital, never φ. The definition in the Axiom of Consent reads F(d,t) = σ(d) · (1 + ε(d,t)) / (1 + α(d,t)).
Within that: σ(d) is total stake magnitude, the sum of individual stakes over the affected set, and it carries no time argument; ε(d,t) is information entropy, lying between 0 and 1; α(d,t) is aggregate alignment, built from per-agent correlations between the agent's target and the consent-holder's, running from -1 to 1. The value α = -1 is excluded from the domain because F diverges there. Named properties: F = 0 exactly when σ = 0, and an irreducible baseline F(σ,1,0) = σ/2, hence F ≥ σ/2.
The kernel triple is (α, σ, ε). It maps onto the ROM triple of survival, weight and transmission, with survival given by ρ_S = L/(1+F). Legitimacy is L, not Λ: stakes-weighted voice, the stake-weighted mean of individual voice, with a measurement proxy built from the mean absolute gap between stake share and voice share. The consent quantity is C(d,t), the consenting stake share, of which L is the continuous relaxation. The consent-holding map is written H_t(d).
Which surface was right, and what changed
Three live notations existed and they were mutually incompatible. The systems.ac landing page printed the friction function with F and ε, matching the papers. The systems.ac framework page printed the same function but named friction φ in its symbol table. The dissensus.ai research page carried a notation grid giving φ for friction and η for entropy, both of which disagree with the manuscripts.
Two of those disagreements are now resolved. On 30 July 2026 friction was changed to F on the systems.ac framework page and on the dissensus.ai grid, and entropy was changed from η to ε on the dissensus.ai grid. Both surfaces now match F = σ(1 + ε)/(1 + α) as the Axiom of Consent states it.
Two disagreements are not resolved, and cannot be resolved by editing a web page. The dissensus.ai grid also lists Λ for legitimacy and Ψ for a consent function. Neither symbol appears anywhere in the Axiom of Consent or in the legitimacy paper — they are site-only inventions with no manuscript counterpart, and the legitimacy paper does not settle on a single symbol at all. Choosing one is an authorial decision, not a typo fix, so both were left in place and are flagged here instead.
Two symbols are already spent
φ is mean fitness in these papers: the population-weighted product of weight and survival in the replicator-mutator equation, and a type-dependent fitness in ROM's gradient-flow appendix. η is measurement error in the measurement section, the ANOVA effect size η², a Pareto-inefficiency metric and a learning rate in the MARL appendix, and a sign indicator in the dynamics section. Ψ is the residual interaction term in ROM's free-energy functional.
Using φ for friction, η for entropy or Ψ for a consent function therefore creates a collision inside the framework, not merely a divergence from it. Λ appears nowhere in either formal paper, and no source was found that ever introduces Λ or Ψ as canonical symbols; whether they were an early convention later dropped, or were coined for the web copy, is not established.
A reference is not an endorsement of the form
The current manuscripts label F = σ(1+ε)/(1+α) an explicitly phenomenological ansatz and report that the single-index composition is not supported as a predictor: ROM's own validation section gives R² between 0.05 and 0.13, not significant at n = 27. What survives testing is the variable set and the ordinal structure, meaning stakes dominant, entropy positive, alignment monotone in signed correlation, rather than the particular composition.
Anyone quoting the equation should carry that hedge with it. An abstract syndicated to six files across the three sites states without hedge that the friction equation predicts coordination difficulty; the current manuscript retracts exactly that.
What it argues
- Canonical form:
F(d,t) = σ(d) · (1 + ε(d,t)) / (1 + α(d,t)), with F a Roman capital for friction, σ total stake and no time argument, ε entropy, α aggregate alignment. Legitimacy isL, stakes-weighted voice; consent isC, the consenting stake share. - φ and η are already assigned inside the papers, to mean fitness and to measurement error and the ANOVA effect size respectively, so using them for friction and entropy collides within the framework rather than merely departing from it. Λ appears in neither formal paper.
- systems.ac disagrees with itself: its landing page matches the papers and its framework page does not. The dissensus.ai notation grid has four of six symbols wrong and asserts that legitimacy collapses to alignment, which contradicts the definition ROM actually uses.
- The equation is a phenomenological ansatz. What the papers report as surviving testing is the variable set and the ordinal structure, not the single-index composition.
What this is not
- Unreviewed. This is an audit note with no new mathematics; it records what the manuscripts say and where the sites departed from it.
- Partially acted on. The
φandηdiscrepancies were corrected on both sites on 30 July 2026; theΛandΨentries were left alone because the papers do not define them and picking a symbol is an authorial decision. - Read off local working copies of the manuscripts. Whether the public arXiv PDFs are identical at these specific lines was not checked line by line.
- The two framework papers state different quadratic variants of the friction form, differing in whether entropy is squared. Both are withdrawn as empirical claims, since the U-shape that motivated them was diagnosed as a sampling artefact.
Identifiers
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Notes are the part of the programme most likely to contain errors, because nothing here has been through review. If you can show a step does not follow, say so and we will publish it.