ifc-0057

3.12 Beyond two directions: the higher-DIAL conjecture

¹1. This is a bounded conjectural extension of the two-direction architecture developed above. No later algorithm or empirical claim depends on it. ↩

The construction so far separates two local repair directions and then records a finite theory change outside their infinitesimal geometry. Once that base case is explicit, two different forms of iteration become visible. The iterated tangent endofunctor in LINCS produces a tower

\[ X,\quad TX,\quad T^2X,\quad \ldots \]

whose successive levels expose higher-order behavior of one maintained tangent structure [ Mahadevan , 2026e ] . DIAL suggests a related but different iteration. Its two sides are not merely two applications of \(T\): they are two typed classes of repair, interpreted here as assimilation and proto-accommodation, together with a law governing their interchange. Adding another tangent level and adding another repair direction must therefore be distinguished.

Tangent depth.

For an existing DIAL object \(\mathbb D\), the sequence

\[ \mathbb D,\quad T\mathbb D,\quad T^2\mathbb D,\quad \ldots \]

probes higher jets of the same two repair processes. At \(T^2\mathbb D\), for example, one may ask how the assimilatory and proto-accommodative fields accelerate, curve, or change one another. This is the direct analogue of the LINCS iteration. It does not create a third semantic mode of repair.

Repair arity.

A genuinely higher DIAL object would instead contain \(n\) independently typed infinitesimal directions. The natural syntactic candidate is

\[ \mathbb D_{\mathrm{inv}}^{(n)} := \underbrace{ \mathbb I_{\mathrm{Lie}}\otimes \cdots \otimes \mathbb I_{\mathrm{Lie}}}_{n\ \mathrm{factors}}, \qquad \mathbb D_{\mathrm{inv}}^{(1)}=\mathbb I_{\mathrm{Lie}}, \qquad \mathbb D_{\mathrm{inv}}^{(2)}=\mathbb D_{\mathrm{inv}}. \]

Associativity of the appropriate enriched tensor product would give the iterated-model reading

\[ \operatorname {Mod}\! \left( \mathbb D_{\mathrm{inv}}^{(n)},\mathcal M\right) \simeq \operatorname {Mod}\! \left( \mathbb I_{\mathrm{Lie}}, \operatorname {Mod}\! \left( \cdots \operatorname {Mod}( \mathbb I_{\mathrm{Lie}},\mathcal M)\cdots \right)\right). \]

This formula is presently a proposed presentation. It presupposes the enriched sketch doctrine, tangent stability, pullbacks, and dualizability conditions already missing in the double case.

Definition 3.10 Candidate \(n\)-fold DIAL object, proposed

For \(n\geq 1\), a candidate \(\mathrm{DIAL}^{(n)}\) object is a model of \(\mathbb D_{\mathrm{inv}}^{(n)}\) with one involution-algebroid structure in each coordinate, a DIAL interchange comparison on every two-dimensional face, and coherent comparisons among different orders of permuting the coordinates. Its smooth finite-rank realization should be an appropriate \(n\)-fold Lie-algebroid object. This is a research target, not an established internal construction.

The first new case, \(n=3\), would be cubical rather than square. Its three pairwise DIAL faces would have to satisfy braid-like coherence, and a weak or noisy realization could contain a highest-core defect invisible on every individual face. Pairwise bracket residuals would therefore not by themselves certify a triple object; a higher comparison witness would be required. The book records this possibility but does not use it in any later algorithm or experiment.

Boundary: Higher-DIAL conjecture. There should be a coherent theory of \(n\)-fold involution algebroids internal to a suitable tangent category such that: its \(n=1\) models recover involution algebroids; its \(n=2\) models recover DIAL; its strict smooth finite-rank realization agrees with the appropriate \(n\)-fold classical Lie-algebroid geometry; and its pairwise and higher comparison cells admit typed obstruction observers. Neither existence of this internal theory nor the claimed realization equivalence is established here.

The distinction between the two towers suggests a two-parameter hierarchy

\[ T^k\! \left(\mathrm{DIAL}^{(n)}\right), \qquad k\geq 0,\quad n\geq 1, \]

where \(k\) measures differential depth and \(n\) measures repair arity. A future coalgebraic construction could ask whether repeated increases in depth or arity stabilize, but no higher-DIAL endofunctor or final-coalgebra theorem is asserted in this book.

The possible semantic value of a third direction is nevertheless clear. Besides repairing a realization and revising its theory, a discovery system may need to revise the observer or admission doctrine that determines what counts as evidence. A triple object could type those three processes separately and test whether their orders agree. That interpretation is only a motivation: the third coordinate must be declared by an application rather than inferred from the integer \(3\).

. No empirical force should be attributed to higher DIAL until controlled realizations of ordinary DIAL show that two typed directions and their interchange witness add value beyond appropriate controls. A claim for \(\mathrm{DIAL}^{(3)}\) has the additional burden of exhibiting a reproducible cubical obstruction that no pairwise DIAL model can represent.