nlPUnit_continuous
plain-language theorem explainer
The unit-normalized cubic reparametrization of ledger imbalance is a continuous real-valued map on ledger states. Gravity and gap-closure arguments cite it when packaging a continuous, swap-odd, balance-vanishing, unit-normalized momentum that still fails additivity. The proof is a short continuity chase: imbalance is continuous, the cubic polynomial is continuous, and scaling by one-half preserves continuity.
Claim. The map $z \mapsto \mathrm{nlP}(\mathrm{imbalance}(z))/2$ on ledger states is continuous, where $\mathrm{imbalance}(z)=z_1-z_2$ and $\mathrm{nlP}(m)=m+m^3$.
background
This module sits in the Seven Gaps gravity line and attacks unconditional momentum additivity from the recognition composition law alone. The consumer already shows that additivity under consolidation, continuity, balance-vanishing, and unit normalization force EnergyEqualsCost; the remaining question is whether the composition law selects additivity.
Ledger states carry a net imbalance $z \mapsto z_1-z_2$, defined by subtraction with no cost. The nonlinear momentum coordinate is $\mathrm{nlP}(m)=m+m^3$; the unit-normalized reparametrization is that cubic applied to imbalance and divided by two. The ambient cost on the chart is $J$-cost, which satisfies the recognition composition law independently of any momentum observable.
The module's no-go exhibits two continuous, balance-vanishing, unit-normalized momenta that disagree on additivity: plain imbalance (additive) and absolute imbalance (not additive). The cubic unit map is a further swap-odd witness in the same spirit.
proof idea
Term-mode continuity composition. First, imbalance is continuous as the difference of the two continuous projections on the product carrier. Cubing and addition preserve continuity, so $z \mapsto \mathrm{imbalance}(z)+(\mathrm{imbalance}(z))^3$ is continuous. Division by the constant $2$ is continuous. A final congruence identifies that composite with the definition of the unit-normalized cubic map.
why it matters
Feeds the swap-odd package that bundles continuity, swap-oddness, balance-vanishing, unit normalization, and explicit failure of additivity for this cubic unit map. That package is one of the constructive witnesses inside the module verdict: composition-law independence of momentum additivity, together with the ambient cost satisfying RCL.
In the Recognition framework this closes a gap-5 premise attack: debit-credit parity and the composition law do not substitute for the kinetic condition when forcing additive momentum and thence EnergyEqualsCost. It is local to the stated LedgerState chart with ambient $J$-cost, not a quantified class theorem over all cost systems. Downstream the verdict records that outcome (b) landed as the no-go headline and (c) as the constructive corollary.
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