ULR Open Problems and Reopening Conditions

Current execution state#

M1–M8 are complete. There is therefore no mandatory ontology workstream committed to finding a ULR ontology. The open problems are not debts that leave the current verdict incomplete. They are optional follow-up programmes to be opened if a stronger candidate appears.

1. The largest conceptual problem — Is there a genuine junction residual among the four questions?#

Sharing, Identity, Role, and Formation appear connected, but each is currently explained strongly by the following theories.

The central open question is whether there is a neural-specific junction residual that does not follow from using these four component theories side by side. At least one of the following is required.

1. A theorem not implied by applying the component theories;
2. a neural-specific counterexample that separates two existing equivalences;
3. a preregistered prediction distinct from the strongest typed baseline;
4. incremental value for held-out large-model behaviour or transfer;
5. direct measurement of an obstruction to cross-time or cross-model transport.

No such item is currently established.

2. U0 — Does physical response exceed standard testing?#

In an engineered matched pair, passive-equivalent systems separated under reset/clone response. Raw port-response B3 and declared routing B4 nevertheless absorbed the result. The homogeneous tanh seed-only version also failed to produce a stable class.

The next U0 increment must not simply repeat the same endpoint at a larger scale. Informative candidates include:

• a carrier split that arises naturally during training without architecture labels;
• a gauge-descended response quotient for a recurrent Transformer, KV cache, or fast weight;
• prediction of disjoint held-out behaviour families beyond the same probe's raw standard response;
• transport of an organisation law learned only in the training environment into an unseen environment.

Termination rule: if $Q$ does not exceed B3, B4, and B*, or if the stable class does not reproduce on the preregistered split, close that candidate as weakened or absorbed.

3. U1 — Does an external relation connect to behaviour and internal mechanism?#

WordNet relations cross-fit the relation geometry of Caltech and CIFAR, but the retrieval behaviour bridge failed. The next question is whether an external taxonomy is merely an observer's comparison coordinate or predicts transportable causal structure inside the model.

The required gates are:

1. freeze the external relation before training;
2. hold out model, objective, and dataset;
3. compare with the untrained and architecture-prior baselines and B*;
4. use a held-out behavioural target rather than geometry;
5. separate an internal-carrier claim from the external baseline.

Termination rule: if the relation does not improve held-out behavioural prediction, retain Z5 as an external baseline.

4. U2 — Is there an admissible neural target for Formation?#

The formation definition is ready, but with no current carrier survivor its status is NO_ADMISSIBLE_TARGET. A new target must minimally provide the following package:

Detailed open questions include:

• How should a pointwise quotient be distinguished from the identity of a time-indexed family?
• Which invariant separates deformation within one stratum from crossing a structural boundary?
• Can transport along a noisy training trajectory be defined independently of gauge?
• Can metric co-onset serve as a secondary indicator after a structural event?
• Does intervention change the future causal role of a candidate structural type?

Start condition: a carrier must first pass the M3 reduction gate. Do not begin with defects or co-onset in the absence of a carrier.

5. U3 — Does the possibility field have an independent signature?#

The intuition that “learning changes the field of possible trajectories, while inference reads that field to produce a path” is useful, but it may currently be no more than a renaming of ordinary dynamics and predictive-state descriptions.

An adequate signature must satisfy all of the following:

• specify the state space, norm, noise coupling, and admissible trajectory grammar;
• avoid a trivial code that simply stores the full dynamics;
• provide a finite falsifier distinct from predictive state and control reachability;
• predict cross-model or cross-context transport;
• improve held-out behaviour after B*.

Without an independent signature, Z3 remains an exploratory metaphor.

6. F0 — Can UAR assembly injectivity be reformulated over the correct source?#

The former composite monomorphism was retracted. A renewed question must directly define the off-diagonal collision incidence of

for the joint quotient $Q=\Theta^{\mathrm{adm}}/G_\Sigma$. It must not transplant an exclusion from the channel source to the assembly source.

Open proof obligations:

1. fix the quotient source and map first;
2. separate diagonal and off-diagonal within the same source;
3. prove injectivity and unramifiedness separately;
4. exclude all collisions, rather than finding one good section in each fibre;
5. state the quantifier and boundary of the generic open.

This is optional mathematics and does not block M8.

7. F1 — Localisation identity#

The localisation equality between the 135-chart cover and the chart-free incidence ideal is SUPERSEDED-AS-GATE. It may be strengthened using sheaf-theoretic and localisation methods, but it is not a prerequisite for typed channel exclusion or the current ontology verdict.

The open question is how local certificates glue on overlaps and whether they agree exactly with chart-free support and radical. Without a proof, this remains an optional strengthening.

8. F2 — Channel-CERT and Assembly-CERT#

The former CERT-B/C risked mixing channel and assembly sources. Follow-up work must first separate two certificates.

• Channel-CERT: typed channel-collision boundary.
• Assembly-CERT: quotient-assembly ramification and collision boundary.

Only then can one ask whether a stronger certificate reconstructs the support or radical of the bad boundary. It is not currently marked proved or refuted.

9. Application problems for the observer theorem#

Canon 24's measurability and TV theorem is general, but applying it to an actual model requires an observer contract.

Open items:

1. Which system boundary declares the role variable in an actual Transformer?
2. What are the read, write, reset, and clone ports of a KV cache, fast weight, optimiser state, or tool memory?
3. At what finite-sample rate can the transcript law be estimated?
4. How is minimax separation for a composite role class computed?
5. When does a finite test basis guarantee full response-profile factorisation?
6. How can path-law singularity be tested for a nonstationary online policy?

Even a successful application must not promote an observer-relative result into observer-independent ontology.

10. Generalising the Set core#

The intrinsic dual organisation of Post-Canon 119–120 has been proved and audited at the Set level. Open generalisations include:

• stochastic kernels and measure-enriched categories;
• response quotients with topology and metrics;
• partial renewal and multiple persistence timescales;
• functorial transport of the joint quotient;
• finite-sample identifiability and robust perturbation;
• nontrivial realisation in naturally learned neural families.

The existence of the Set core alone does not establish a learning/inference ontology.

11. Streaming, sheaf, and constraint frontier#

Items 143–148 investigated local-to-global forcing, streaming memory, relation completion, portwise delivery, and train-only organisation in exact finite models. The next decisive question is whether there exists a uniform neural scaling law that exceeds an identity-closed baseline with the same problem family and resource ledger.

Required comparison axes:

• the same raw stream and query order;
• the same primitives and precision;
• accounting for storage ports, fanout, replay, advice, and compiler cost;
• train-only law acquisition and a sealed target;
• an approximation metric and error amplification;
• source, split, and model selection frozen before results.

A finite suite cannot exclude a tailored lookup compiler. A uniform family and scaling lower bound are therefore required.

12. Natural experiments after positive controls#

The remote-parity, Heisenberg, crossed-module, Latin-cube, and octonion designs calibrate whether the protocol can detect a planted dependency. The next step is not to make the control more elaborate, but to test the following difference:

Questions to verify:

• Was the candidate law specified independently rather than extracted post hoc from the target?
• Was the baseline given the same source access?
• Does a same-type compiler absorb the representation-specific residual?
• Are the DEV and CONFIRM datasets and manifests separate?
• Do both negative and positive controls calibrate the protocol?
• Does the learned result transport across multiple seeds, architectures, and environments?

13. The M8 reopening gate#

The ontology verdict may reopen only when a new candidate registers all of the following before seeing the result.

If any gate is added after observing the result, the result cannot be used as M8 evidence.

The maturity of post-canon evidence is also part of the gate. Items 119–142 are Git-tracked materials; item 143 is currently being revised; and items 144–159 are local, uncommitted, and untracked work products. PASS or sealed inside those documents must not be read as Canon promotion or public provenance.

14. What will not be done#

• Do not repeat the same engineered endpoint merely at larger scale.
• Do not treat an unexplained residual as a measure of ULR.
• Do not relabel an exact finite control as natural emergence.
• Do not claim efficiency on a finite benchmark while omitting a suite-tailored compiler.
• Do not declare formation from metric onset alone.
• Do not automatically call a working artefact after Canon 24 “Canon 25.”

Compressed current priorities#

The highest-information experiment is not an example that makes ULR look more plausible. It is an experiment that cleanly establishes one of the following:

1. Find a transportable, gauge-descended residual in a naturally learned neural family that remains after B*.
2. Show that such a residual is absorbed again, thereby closing the scope of Z0 typed pluralism more precisely.

Either outcome is a success for the Main research programme. ULR's current purpose is not to protect a particular conclusion, but to determine conclusively whether the junction among the four problems genuinely requires a new theory.