# Toroidal Mind Simulator — Source-Correlation and Drift Audit

**Audit:** `TOROIDAL-MIND-SIMULATOR-AUDIT-2026-09-01`  
**Reviewed:** `2026-09-01`  
**State:** `candidate_pending_current_human_review`  
**Publication label:** formal candidate and conformance model

This audit uses the supplied toroidal materials as a correlation test of the current Structure of Mind. It does **not** reopen seven or twelve as rival proof contests. The machine-readable authority is [TOROIDAL_MIND_SIMULATOR_AUDIT.json](TOROIDAL_MIND_SIMULATOR_AUDIT.json).

## Decision in one view

The material contributes exact and useful coordinate wisdom:

1. The normalized triad chart is an exact reversible circle coordinate system under its declared normalization.
2. Two distinguishable closed phases admit a reversible ring-torus coordinate map when `A > a > 0`.
3. A torus address is a many-to-one projection of the richer delayed history state; positional return is not complete identity-state return.
4. The Clifford, `SO(4)`, and stereographic view is a calculated reference-frame projection, not observerhood.
5. The phasor magnitude `C = |Z|` is bounded in `[0,1]` for nonnegative normalized weights, while its interpretation as Y-Theory coherence remains a candidate.

The correlation also exposes two integration errors:

- Alternative-`N` coordinate conformance was treated as if it could challenge the source-defined seven-depth Mind architecture. It cannot: the circle and torus coordinate identities are `N`-independent.
- The simulator compresses each depth to one scalar phase in one active dimension. It therefore omits the exact twelve-address recursive cell, dimensional breadth, same-depth relation, distributed convergence, and full whole-field changed return.

Seven is premise-qualified and internally derived in current Y-Theory: the declared ten-band address, adjacent inheritance, recoverable consequential relation, explicit bridge `D:=a(1/φ)=a(2/3)=7`, and recursive self-similarity together fix the complete expansion-facing depth address. Twelve is the exact `3+9=12` hierarchical address cell retained within every active depth. They are different typed structures, not alternative counts of the same object. The simulator consumes this architecture; it does not independently prove or refute it.

## Proof-scope vector

| Axis | Current result |
|---|---|
| Referent | recursive depth of each active dimension in complete Y-Theory Mind |
| Source authority | current direct-author derivation correction |
| Internal structural status | premise-qualified internal derivation established |
| Internal proof mode | ten-band address + inheritance + recoverable relation + depth bridge + recursive self-similarity |
| Primitive cardinality theorem | separate and not required by the declared positional derivation |
| Seven/twelve distinction | exactly derived |
| Simulator status | compressed candidate instantiation; not the source of proof |
| Simulator independently proves seven | no |
| Alternative-`N` coordinate relevance | does not test complete-Mind closure |
| External empirical universality | not established |
| Runtime predictive result | not validly tested for this claim |

This vector prevents three different questions from being collapsed: source-internal architectural status, a standalone primitive-cardinality theorem, and external empirical universality.

## Governed source addresses

| Artifact | SHA-256 | Authority in this audit |
|---|---|---|
| `Y_THEORY_TOROIDAL_MIND_SIMULATOR_V1.zip` | `ffce84f2567ca17d4906c5abebb1fd5f427c71f7eee88bac5fe1bcb43070d16e` | Candidate evidence; not canonical proof |
| `Y_THEORY_LIVING_SOURCE_TOROIDAL_MIND_SYNC_YLS-2026.09.01.2.zip` | `716d3d5fcd9826565e94f6c0a99fa8dbff73c5cb804c3f1be5068097478658b1` | Historical candidate bundle; not current canon |
| `pasted-text.txt` | `7c717d7d881fa9540c6bbd74b531ebde96943655ad16744774169d749cba8d80` | Candidate author explanation requiring governed review |
| `model/MATHEMATICAL_DERIVATION.md` | `f19fd3133464a8116afb566e9726aedbe836a2e6c74d4aa074b607a8860d5794` | Formal candidate derivation |
| `model/y_torus_model.py` | `3663b0b74b3f74fa598dcdc7782a2628fe2874f8aa0cb51cb675119566d50d2a` | Executable candidate model |
| `reports/VALIDATION_REPORT.md` | `a6cb448f282ce888f836857083d951585023652369e5cbae851fd20970df6ea4` | Package self-report; not independent validation |
| `reports/VALIDATION_SUMMARY.csv` | `45aac47b07b40e729e7a4eb39bf12ccc5d374c23d7aae9c57848fb5923de5f7b` | Package metric index |
| `app/index.html` | `6d64cce955d4fa54bb9fb3a7312af2b526f799379c724cb333d10974f0652dbf` | Supplied candidate browser projection |
| `Structure_of_Mind_Book.md` | `739e44020924f1ee89cfd247ed7f3b5857726a5dedb135e4ad79b9b9771ca336` | Current audited source architecture and seven/twelve typing |
| `intake/raw/YWP-20260901T164755Z-toroidal-files-are-a-drift-correlation.md` | `c2a4f168f6d0c00b2b52027e0ad33c5b22b49952a788ad020830701be7600692` | Current direct-author scope and fidelity correction |
| `corrections/AUTHOR_CORRECTION_2026-09-01_SEVEN_METRIC_CONTINUITY_DERIVATION.md` | `f0ee096823d8d022b1bbbf9b6955aa5f708ed8dd09b65c990b9d9ca4f94a4a84` | Current direct-author premise-qualified internal derivation correction |

Both current corrections are present in the living source. The external book and supplied packages are hash-addressed rather than silently copied into canon.

## Source structure that the projection must preserve

The current Structure of Mind establishes these typed relations:

- `Σ = {I, O, R}` is the triadic grammar.
- `Σ²` produces nine first self-expressions; retaining the primary roles and first self-expression gives the exact `3+9=12` hierarchical address cell.
- The one-based closed ten-band address is `a(q)=min(10,1+floor(10q))` for `0≤q≤1`.
- Equal-start adjacent inheritance gives the one-third/two-thirds envelope; arbitrary positive starts give a generalized Fibonacci recurrence, not necessarily the standard Fibonacci sequence.
- The complementary golden limit gives `1/φ`; the declared bridge `D:=a(1/φ)=a(2/3)=7` fixes the expansion-facing depth address.
- `L₁ … L₇` are therefore seven ordinal recursive depths under the full declared premise chain, and recursive self-similarity carries the same triadic operation through them.
- Each active depth can contain the twelve-address recursive cell.
- Complete Mind has dimensional breadth, represented as states such as `[x_{ℓ,k}]`, not only a single chain.
- Vertical residue carries consequence through depth; corresponding depths also relate horizontally across active dimensions.
- Distributed dimensional consequences converge at whole level and return as changed readiness, `L₁′ ≠ L₁`; convergence is not an eighth depth.
- A torus becomes an exact conditional representation when two bounded recurrent phases are distinguishable and mutually consequential. It is a downstream reversible view, not the source of seven.

## Exact coordinate results

Under the stated assumptions, the following survive the audit:

- `I + O = 1`, `P = O - I`, and `P² + R² = 1` admit the reversible chart `P = cos θ`, `R = sin θ`.
- Two closed phases `(θ, φ)` admit a ring-torus embedding with a well-defined inverse when `A > a > 0`.
- The torus address can be inverted to its two normalized triads, but it cannot recover the full delayed history state.
- The Clifford/`SO(4)`/stereographic construction is an exact calculated projection under its declared coordinate assumptions.
- For normalized nonnegative weights, `C = |Σ wℓe^{iθℓ}|` is exactly bounded in `[0,1]`.

These are coordinate or model theorems under explicit premises. They do not establish universal physical toroidality, observerhood, consciousness, or a privileged weighting law.

## Replay and counterfactual diagnostics

The supplied recursive preset was replayed with no golden ratio injected. At `N=7` it produced a measured winding ratio of `1.1932225055011505`, not `φ`. Coordinate residuals remained at machine precision.

Runs at `N = 1, 2, 6, 7, 8, 13` all preserved coordinate conformance. This is expected because coordinate conformance is `N`-independent. The non-seven runs are counterfactual implementation diagnostics; they do not vote on, test, or challenge complete-Mind closure.

Equal, Fibonacci, and reverse-Fibonacci weighting changed the phase-lock and mean-`C` results. The weighting rule is therefore a model premise rather than a neutral measurement choice. The selected `fibonacci` label names the simulator's standard weight profile; arbitrary positive starts instead define a generalized Fibonacci recurrence. The equal-start one-third/two-thirds envelope belongs to the declared source derivation, while none of these simulator replay modes establishes golden emergence.

## Defect and drift repair ledger

| ID | Result | Repair state |
|---|---|---|
| D1 | Zero delay was forced to one timestep | Repaired and tested |
| D2 | Two torus phases were presented too close to complete state | Repaired and tested with an explicit many-to-one projection boundary |
| D3 | Python/browser initial histories and initial `C` disagreed | Repaired and tested with parity fixtures |
| D4 | Replayed behavior was not byte-identical to all bulky packaged outputs | Explicitly outside the integrated runtime claim |
| D5 | Supplied checksum ledger named a stale cache file and did not bind itself | Stale ledger not imported; outer release manifest remains authoritative |
| D6 | Coordinate projection used observer language | Repaired: calculated reference-frame projection only |
| D7 | Alternative-`N` conformance was misused as a seven-depth challenge | Repaired: `N=7` is the source profile; other `N` values are counterfactual diagnostics |
| D8 | Scalar one-dimensional compression omits the full source architecture | Partially repaired by accurate labels; source-faithful multidimensional engine remains required |

## Current governed bindings

The audit binds to the live semantics of `YT-GEO-002`, `YT-NUM-004`, `YT-NUM-005`, `YT-NUM-006`, and `YT-COS-001`. Its proof-mode binding records seven as premise-qualified and internally derived through the declared positional chain while separating that result from a primitive-cardinality theorem and from external empirical universality.

The formation replay remains a separate negative runtime result: relational geometry scored `-95` milli against its best baseline and approved zero semantic relations. That run did not implement the complete source architecture and therefore does not validly test or demote seven.

No relation candidate is promoted by this audit. No historical sync installer was executed. Production deployment is not authorized by this artifact.

## Boundary and next gate

The supplied material is accepted as an exact coordinate framework and a repaired compressed laboratory. It is not accepted as:

- a new simulator proof or disproof mechanism for seven or twelve;
- a complete source-faithful Mind engine;
- evidence of universal external physical toroidality;
- evidence that the preset generates a golden attractor;
- evidence of observerhood or consciousness;
- canonical dynamics or predictive wisdom.

The next implementation gate is not another alternative-`N` contest. It is a source-compiled engine with the twelve-address cell across seven depths and active dimensions, vertical residue, horizontal same-depth relation, whole-level convergence, and changed return. Only after that fidelity gate can proposition-specific external tests be designed without confusing coordinate success with physical, biological, cognitive, or cosmological validation.