---
pretty_name: "MIND-QX: Order-Selective Erasure Calculus"
language:
- en
size_categories:
- n<1K
tags:
- mathematics
- theoretical-computer-science
- permutation-parity
- monotone-computation
- deletion-only-memory
- reproducible-research
- ai-agent-support
- expert-review
configs:
- config_name: triadic_protocols
  data_files:
  - split: test
    path: data/triadic_protocols.csv
---

# MIND-QX: Order-Selective Erasure Calculus

**Sharp directional-memory thresholds, delayed permutation parity, and finite focus transport.**

Research **v1.0.0** · Hugging Face release kit **v1.0.1** · 7 October 2026.
AI-assisted research draft prepared for **Maciej Nowicki**, with **Eve** as the AI drafting contributor. The original manuscript's credit line is preserved.

MIND-QX gives a finite system in which deletion-only operations encode their order in surviving directed records. A fixed monotone observer stays constant throughout every incomplete distinct-action protocol, reveals permutation parity at completion, and becomes zero under one common final reset.

The central candidate contribution is an **exact record threshold inside the specified pair-guard architecture**. Independent parity context forces both record directions, removing the sparse saving available to an isolated triad.

**[Read the manuscript](MIND_QX_Order_Selective_Erasure_Calculus.pdf)** · **[Research summary](RESEARCH_SUMMARY.md)** · **[Expert review](docs/EXPERT_REVIEW.md)** · **[AI-agent guide](docs/AI_AGENT_GUIDE.md)** · **[Novelty audit](audit/novelty_audit.md)**

## Mechanism

For each action $i$, preallocate a marker $c_i$ and selected directed records $r_{ij}$. Begin with every allocated cell present. The action is

$$
E_i(S)=S\setminus\bigl(\{c_i\}\cup\{r_{ij}:c_j\notin S\}\bigr).
$$

Guards are evaluated on the input state. Every primitive map is monotone, idempotent, and deflationary. No action creates a record. With all directions allocated, the surviving record $r_{ij}$ says that action $i$ preceded $j$.

For the declared even-parity observer, after a complete permutation $\sigma$ and one appended reset,

$$
\delta(\sigma)=n+\mathbf 1\{\sigma\text{ is even}\}.
$$

Only the chosen readout is masked before completion. Internal state already records action order. Information retained after the reset is a separate carrier.

## Sharp bounds and composition

| Protocol | Minimum directed records | Minimum total cells |
|---|---:|---:|
| One block, $n=2$ | $1$ | $3$ |
| One block, $n=3$ | $3$ | $6$ |
| One block, $n\geq4$ | $n(n-1)$ | $n^2$ |
| $m\geq2$ independent $n$-action blocks, product parity | $mn(n-1)$ | $mn^2$ |

Two separately observed sparse triads use twelve cells. Product-parity observation in the same architecture needs eighteen. The monotonicity obstruction is proved in **Theorem 8.2, Context-forced duality**. These are architecture-specific bounds; they are not lower bounds for arbitrary streaming algorithms or all memory models.

Additional results include complete-order antichains, universal upward observers, exact survivor counts, a commuting-erasure parity-histogram obstruction for $n\geq3$, and exact positive-DNF counts on the saturated reachable domain. The compact circuit and certificate DNF need not agree outside that domain.

## Run and reproduce

Python **3.10+**. Research verification uses the standard library only.

```sh
python tools/check_manifest.py
python examples/triadic_demo.py
python tools/reproduce.py --full --output runs/local-reproduction
```

The wrapper runs the unchanged original test suite in a temporary copy and saves its report and environment record to the chosen output folder. It preserves the bundled evidence and forces Python assertions to remain enabled.

**33/33 original verification groups passed in a fresh run during release preparation.** Exact enumeration covers every partial distinct-action protocol through $n=8$, all operator subset states through $n=4$, and the documented small record allocations and certificates. Larger tensor cases use labelled seeded samples.

- [Original report](results/verification_report.json)
- [Fresh reproduction report](release_validation/fresh_reproduction_report.json)
- [Claim ledger](audit/claim_ledger.json)
- [Theorem-to-check map](results/proof_test_map.csv)

Passing finite checks is evidence for the implementation. General results rest on the supplied proofs. Neither the checks nor the hashes certify historical novelty or replace expert review.

## AI-agent support

Start with [AGENTS.md](AGENTS.md) and the [review prompt](prompts/AI_REVIEW_PROMPT.md). Twelve machine-readable [review tasks](review/agent_tasks.jsonl) correspond to the mathematical claim ledger.

```sh
python tools/prepare_agent_review.py --list
python tools/prepare_agent_review.py --claim C10 --output runs/C10-review-packet.md
```

The tool prepares a text packet containing definitions, manuscript source, implementation, evidence, and a required reporting format. It can be supplied to a local or hosted research agent. It does not call an AI service or provide model weights.

Ready-to-use packets: [sharp threshold / C04](review/AGENT_PACKET_C04.md) and [context-forced duality / C10](review/AGENT_PACKET_C10.md).

Use [report_template.json](review/report_template.json) and [report_schema.json](review/report_schema.json) to return explicit proof gaps, counterexamples, tested scopes, and source equivalences. AI reports remain labelled as AI reports.

## Expert support

[EXPERT_REVIEW.md](docs/EXPERT_REVIEW.md) identifies the main proof obligations, boundary cases, and literature comparisons. [FORMALIZATION_PLAN.md](docs/FORMALIZATION_PLAN.md) breaks the claims into proof-assistant targets. [CONTRIBUTING.md](CONTRIBUTING.md) describes how to submit review evidence and corrections.

Specialists in monotone computation, finite-state systems, permutation orders, write-once memory, reliability signatures, and dynamic fault trees are invited to inspect the exact construction. No expert has been recruited, contacted, or represented as endorsing this release.

## Dataset contents

The default dataset configuration contains **six synthetic rows**, one for each action order of the sharp six-cell triad. The observation columns include the initial state, all three primitive actions, and the final reset. Its extinction vector in lexicographic order is $(4,3,3,4,4,3)$.

These rows are small exact examples for reproduction and regression checks, not a meaningful empirical machine-learning benchmark. See [DATA_CARD.md](docs/DATA_CARD.md) for the field definitions and generation procedure.

## Status and limits

| Item | Status |
|---|---|
| Source manuscript, proofs, and executable checks | Included unchanged from research v1.0.0 |
| Fresh supplied-suite pass rate | 33/33 = 100% of those test groups |
| Independent mathematical peer review | Pending; none recorded in this release |
| Proof-assistant certification | Pending |
| Historical novelty | Candidate originality; priority unconfirmed |
| Universal absence of prior art | Not established |
| Physical or RSI performance validation | Not performed |
| Public publication status | This local kit does not itself establish publication |
| License | Unspecified in source; no new reuse license granted |

The claims require distinct-label protocols, mandatory markers, the declared pair-guard rule, and fixed monotone observers. Finite recursive grouping allocates additional blocks and memory. There is no theorem of carrier-free information, physical teleportation, or unlimited improvement from fixed finite resources.

## Citation and provenance

Use [metadata/CITATION.md](metadata/CITATION.md), retaining the draft version and evidence limitations. No DOI, institutional affiliation, or journal acceptance is invented.

The [original archive](provenance/MIND_QX_OSEC_v1.0.0_research_package.zip), original README, and original manifest are preserved under `provenance/`. The current manifest covers the publishing release. [RELEASE_NOTES.md](RELEASE_NOTES.md) identifies the added support materials. [LICENSE_STATUS.md](LICENSE_STATUS.md) records the unchanged rights status.