# Admissibility Paths Overview ## Purpose This folder compares the main admissibility paths currently available to the `j3_oc` branch. Its purpose is not to declare a winner too early. Its purpose is to: - separate the genuinely different admissibility strategies - make their strengths and risks explicit - compare them against the real burdens of the branch - leave a clear record of why one path may later be preferred This is a comparison pack, but it now also records the current working investigation strategy. Related notes: - [0b - ambient-to-observable reduction.md](../0b%20-%20ambient-to-observable%20reduction.md) - [0c - falsification and decision criteria.md](../0c%20-%20falsification%20and%20decision%20criteria.md) - [2a - statics.md](../2a%20-%20statics.md) - [2d - consistency.md](../2d%20-%20consistency.md) --- ## Candidate paths The current comparison pack treats four paths as the main serious candidates: 1. `Hermitian-first` 2. `Hermitian + rank/orbit` 3. `Stabilizer-sensitive` 4. `Observable-viability first` These are not four different theory programs. They are four different ways to define the first serious admissibility cut on $$ X \in J3(C \otimes O). $$ --- ## Evaluation criteria The paths should be compared against the actual burdens of the branch, not only by elegance. The main tests are: 1. `Discipline` Does the path give a real admissibility rule rather than vague cleanup? 2. `u` Does the path improve the status of `u`, even if only up to orbit or stabilizer class? 3. `SU(3)` Does the path help distinguish ambient stabilizer from reduced internal symmetry and possible color? 4. `Reduction` Does the path support a credible later route to a reduced `Spin(2,3)`-type image? 5. `Observable viability` Does the path help later observable extraction rather than leaving it fully unstructured? 6. `Ad hoc risk` Does the path risk reverse-engineering the answer? 7. `Comparative value against J3(O)` Does the path help explain why the complexified branch earns something over the real branch? --- ## Quick matrix | Path | Strength | Main risk | Provisional role | |---|---|---|---| | Hermitian-first | safest baseline control | too weak by itself | likely stage one, not whole answer | | Hermitian + rank/orbit | more selective and geometric | may still not sharpen `u` or `SU(3)` enough | strongest middle candidate | | Stabilizer-sensitive | directly targets branch burdens | highest ad hoc danger | strong but risky refinement stage | | Observable-viability first | ties ambient branch to physics early | may start too far downstream | useful check, weak as first cut | --- ## Current working strategy The current working strategy is: - treat `Hermitian + rank/orbit` as the active strong path - keep `Hermitian-first` alive as a weaker fallback, not as an equal competitor - return to the weaker path only if the stronger path hits a real roadblock and that fallback clarifies where the difficulty actually lies - keep `Stabilizer-sensitive` alive in parallel as a high-upside refinement track - treat `Observable-viability first` mainly as a pressure test rather than a primary backbone This means the branch is no longer merely comparing paths in the abstract. It is running: 1. one main strong path 2. one weaker fallback path 3. one parallel high-upside refinement path That is the most disciplined way to explore multiple admissibility options without multiplying the whole branch. --- ## Provisional reading The current best non-overclaiming picture is: - `Hermitian-first` is the safest weak baseline - `Hermitian + rank/orbit` is the strongest candidate for the main admissibility backbone and should now be treated as the active strong path - `Stabilizer-sensitive` should remain alive in parallel as a refinement track unless it can later justify becoming part of the main backbone - `Observable-viability first` is more useful as a screening or comparison criterion than as the very first cut That is not a final victory claim. It is the current investigative posture. The first explicit test candidate inside the strong path is now: - Hermitian rank-`1` / projective-like admissibility This is a provisional working choice meant to make the strong path falsifiable, not a declaration that rank `1` is already the correct physical sector. The nearest comparison case inside that same strong path is now: - Hermitian rank-`2` or a corresponding non-minimal orbit sector So the current strong-path investigation is no longer only: - "is rank `1` good?" It is now: - "is rank `1` informative enough to keep, or is rank `2` the first admissible sector that actually sharpens the branch burdens?" --- ## What would count as a winner A path should count as the current preferred admissibility path only if it does at least one of the following better than the others: 1. improves the `u` story 2. improves the `SU(3)` story 3. sharpens the reduction map 4. helps explain why `J3(C \otimes O)` earns more than `J3(O)` If no path clearly does that, the right conclusion is: - keep multiple paths alive - treat one as the active strong path - keep one weaker fallback available - do not pretend the branch decision is settled --- ## Current decision matrix inside the strong path The active strong path now contains its own internal comparison: | Candidate inside `Hermitian + rank/orbit` | Present role | Reason | |---|---|---| | rank `1` | primary test case | strongest first cut, lowest ad hoc risk, best falsifiability | | rank `2` | nearest escalation test | richer internal structure if rank `1` proves too thin | The current working verdict is: - keep rank `1` as the main live candidate - keep rank `2` close at hand as the first serious escalation - do not promote rank `2` merely because it carries more room This means the branch now has a clearer nested strategy: 1. main strong path = `Hermitian + rank/orbit` 2. internal primary test = rank `1` 3. internal escalation test = rank `2` 4. parallel refinement track = stabilizer-sensitive admissibility The stabilizer-sensitive track is now concrete enough to have its own ordered probes: 1. `u`-compatibility probe 2. residual-internal-symmetry probe 3. joint `u` / `SU(3)` probe That is helpful because it keeps the refinement track from sounding stronger than it is. --- ## File map - [1a - Hermitian-first path.md](1a%20-%20Hermitian-first%20path.md) - [1b - Hermitian plus rank-orbit path.md](1b%20-%20Hermitian%20plus%20rank-orbit%20path.md) - [1c - Stabilizer-sensitive path.md](1c%20-%20Stabilizer-sensitive%20path.md) - [1d - Observable-viability path.md](1d%20-%20Observable-viability%20path.md) --- ## Working bottom line This pack exists so the branch can investigate several admissibility paths in parallel without multiplying the whole framework. That is the right level of branching for now.