Overview
This article follows O17, which left open why the minimal fibre of the non-injective projection $\Pi$ should be an involution at all. O18 proves one elementary fact cleanly: the Born–Infeld effective action depends on $\chi$ only through $F^2=(D\chi)^2$, hence $S_{\mathrm{BI}}[-\chi]=S_{\mathrm{BI}}[\chi]$. This is a genuine symmetry of the action.
Parity acts on the Born–Infeld response family as a covariance: under the transported-probe comparison $(\chi,\eta)\mapsto(-\chi,-\eta)$, a response of derivative order $k$ transforms with the character $(-1)^k$. Even-order responses — the action among them — are preserved, while odd-order responses, including the Born–Infeld constitutive response, reverse sign and generically separate $\chi$ from $-\chi$. Not every response is preserved by parity. O18 also proves a no-go: evenness does not force fixed-probe indiscernibility. The countermodel is the simplest even functional, $R(x)=x^2$, for which $R(x+\epsilon\eta)\ne R(-x+\epsilon\eta)$ in general. Consequently the parity orbit $\{\chi,-\chi\}$ is not forced into the fibres of the non-injective projection $\Pi$; it is a candidate fibre only relative to a supplied restriction of the observables to even-order responses, part of clause (D3) of Problem 2.8.
The fibre identification is a typed classification problem, of which (H-rank) is one clause. At the Weil level, the conjugation $\rho_{q-c}=\overline{\rho_c}$ is an O17 theorem; the identification of $\{c,q-c\}$ as a physical fibre of $\Pi$ is an open bridge. The pair observable therefore stays a structurally motivated hypothesis.
What O18 establishes
- Born–Infeld action parity: $S_{\mathrm{BI}}[\chi]$ depends only on $F^2=(D\chi)^2$, hence \[ S_{\mathrm{BI}}[-\chi]=S_{\mathrm{BI}}[\chi]. \] This is elementary and correct.
- Parity covariance: under the transported-probe comparison $(\chi,\eta)\mapsto(-\chi,-\eta)$, a response of derivative order $k$ transforms with the character $(-1)^k$: even-order responses (the action among them) are preserved, while odd-order responses (including the Born–Infeld constitutive response) reverse sign and generically separate $\chi$ from $-\chi$.
- Fixed-probe no-go: evenness does not force fixed-probe indiscernibility; the countermodel $R(x)=x^2$ shows the inference fails, so the parity orbit is not forced into the fibres of $\Pi$.
What remains open
- Fixed-probe indiscernibility: by O18's own no-go, action evenness does not imply that $\chi$ and $-\chi$ respond identically to the same probe $\eta$; and under the transported-probe comparison only even-order responses are preserved — odd-order responses reverse sign, so not every response is preserved by parity.
- The fibre identification of $\Pi$: whether the fibres of the independently postulated projection $\Pi$ contain the parity orbit $\{\chi,-\chi\}$ is a typed classification problem (Problem 2.8), of which (H-rank) is one clause; the orbit is a candidate fibre only relative to a supplied restriction of the observables to even-order responses, part of clause (D3). No clause is derived.
- The Weil-level identification $c\leftrightarrow q-c$: the conjugation $\rho_{q-c}=\overline{\rho_c}$ is an O17 theorem, but the identification of $\{c,q-c\}$ as a physical fibre requires a map from substrate configurations $\chi$ to character labels $c$, which is not supplied; this is an open bridge, as in O16.
- Any downstream numerical claim: since the fibre identification is open, O18 does not close the pair-observable problem left by O16–O17, and does not establish a value for $\delta_{\mathrm{pair}}$; the pair observable stays a structurally motivated hypothesis.
Interpretation
O18 does not change the pair observable introduced in O16 and analysed in O17. It also does not supply the derivation of the fibre structure that would have justified it independently.
- O16: conjugate blocks are fibres of $\Pi$ only under two independent, open hypotheses
- O17: conjugate Weil blocks are exactly block-independent within a scalar toy model, not shown for the real pipeline
- O18: the Born–Infeld action is even and parity acts covariantly on the response family (even orders preserved, odd orders reversed); by the proved no-go, this does not force the fibre structure of $\Pi$ or identify it with $c\leftrightarrow q-c$
The parity/fibre and substrate/character identifications remain open exactly as in O16 — delimited by a covariance theorem and a fixed-probe no-go, not closed by them.
Relation to the Cosmochrony program
O18 imports the structural necessity of non-injective projection and the Born–Infeld selection of the effective action, and correctly derives one elementary symmetry from them. It does not supply the missing bridge from that symmetry to a proved statement about the fibres of $\Pi$, nor the missing map from substrate configurations to Weil character labels.
The programme's registry (see program) records O18's action-parity lemma, parity covariance, and fixed-probe no-go as proved, and the fibre identification as an open bridge, on the same footing as O16's open premises.
Current outcome and open directions
The Born–Infeld action's parity is established, as a covariance on the response family — even-order responses preserved, odd-order responses reversed — and the fixed-probe no-go is proved. The identification of the parity orbit with the fibre structure of $\Pi$, and of $\{c,q-c\}$ as a physical fibre, is not: both remain open bridges, exactly as left by O16.
A derivation of either identification — from first principles within Cosmochrony, or from an explicit, independently motivated map between substrate configurations and Weil character labels — would close a genuine gap in the programme. It is not supplied.
References
Jérôme Beau. Equivariant Born–Infeld Parity and the Fibre-Identification Problem of the Non-Injective Projection.