Add D4/F4 S3 orbit and covariant equivariance harness

CHANGELOG.md

@@ -1,5 +1,22 @@
 # Changelog
 
+## Unreleased — D4-TIA-S3-EQUIV v0.1.0 harness
+
+- Added the exact six-element D4/F4 triality action generated by Sakai's
+  \(w_S,w_T\) involutions and pinned a deterministic six-point D4 orbit.
+- Added six canonical \(SL_2(\mathbb Z)\) representatives for the matching
+  classes in \(SL_2(\mathbb Z)/\Gamma(2)\cong S_3\).
+- Verified that the D4/F4 and modular quotient multiplication tables agree under
+  the published generator correspondence \(g_S\leftrightarrow w_S\),
+  \(g_T\leftrightarrow w_T\).
+- Derived transformed quadratic/cubic coefficient actions from the binary-form
+  action itself and verified exact covariant equivariance for all 15 generators
+  under all six representatives: 90 symbolic identities.
+- Explicitly documented that the six modular representatives are quotient-class
+  representatives, not a literal six-element subgroup of \(SL_2(\mathbb Z)\).
+- Added a verifier, tests, roadmap integration, and no new sonification,
+  MIDI, tuning, tempo, timbre, or rendered-audio behavior.
+
 ## Unreleased — D4-TIA-COV v0.1.0 algebra engine
 
 - Added an exact sparse-polynomial engine over reduced rational arithmetic for

README.md

@@ -109,6 +109,33 @@ npm run verify:d4-tia-cov
 
 See [D4-TIA-COV v0.1.0](docs/D4_TRIALITY_COVARIANT_ENGINE.md).
 
+
+## D4-TIA-S3-EQUIV: exact triality orbit and equivariance harness
+
+The next layer is now executable as **D4-TIA-S3-EQUIV v0.1.0**. The harness
+constructs the exact six-element D4/F4 triality quotient generated by
+\(w_S,w_T\), matches it to six canonical representatives of
+\(SL_2(\mathbb Z)/\Gamma(2)\), and verifies the binary-covariant law
+
+\[
+\Psi(\alpha';u,v)=\Psi(\alpha;u',v')
+\]
+
+for all 15 exact covariants under all six representatives: **90 exact symbolic
+checks**.
+
+The implementation deliberately distinguishes quotient classes from their
+upstairs \(SL_2(\mathbb Z)\) representatives. No new MIDI, tuning, tempo,
+timbre, or rendered-audio mapping is introduced.
+
+Verify it independently:
+
+```bash
+npm run verify:d4-tia-s3
+```
+
+See [D4-TIA-S3-EQUIV v0.1.0](docs/D4_TRIALITY_S3_EQUIVARIANCE.md).
+
 ## Notation and Conventions
 
 Unless explicitly stated otherwise, scalars are in \(\mathbb C\), all vector
@@ -229,7 +256,7 @@ npm run verify
 npm run build:v3
 ```
 
-`npm run verify` validates D4-TIA-COV, D4-TIA-15, v3, and continues to validate the immutable v2 and v1
+`npm run verify` validates D4-TIA-COV, D4-TIA-S3-EQUIV, D4-TIA-15, v3, and continues to validate the immutable v2 and v1
 contracts. The build command fails closed for unsafe, existing nonempty, or
 non-`dist/` output paths and never recursively deletes a caller-selected path.
 
@@ -239,6 +266,7 @@ non-`dist/` output paths and never recursively deletes a caller-selected path.
 - [D4/F4 triality reference bridge](docs/D4_TRIALITY_REFERENCE_BRIDGE.md)
 - [D4-TIA-15 invariant-algebra sonification profile](docs/D4_TRIALITY_ALGEBRA_SONIFICATION.md)
 - [D4-TIA-COV exact covariant algebra engine](docs/D4_TRIALITY_COVARIANT_ENGINE.md)
+- [D4-TIA-S3-EQUIV orbit and equivariance harness](docs/D4_TRIALITY_S3_EQUIVARIANCE.md)
 - [Claim boundaries](docs/ETQ_V3_CLAIM_BOUNDARIES.md)
 - [v2 to v3 migration](docs/ETQ_V2_TO_V3_MIGRATION.md)
 - [Formal v3.0.1 paper and build instructions](docs/etq-303/README.md)

docs/D4_TRIALITY_S3_EQUIVARIANCE.md

@@ -0,0 +1,247 @@
+# D4-TIA-S3-EQUIV v0.1.0 — D4/F4 Orbit and Covariant Equivariance Harness
+
+**Status:** implemented research harness  
+**Harness ID:** `D4-TIA-S3-EQUIV`  
+**Version:** `0.1.0`  
+**Sonification impact:** none — this phase tests symmetry before receiver design
+
+## Purpose
+
+D4-TIA-COV made the 15 published covariants executable. This phase makes the
+relevant symmetry action executable and checks that the covariant layer actually
+commutes with the action it is supposed to respect.
+
+There are two related but distinct levels:
+
+1. the finite D4/F4 quotient
+   \[
+   W(F_4)/W(D_4)\cong S_3,
+   \]
+   generated by the exact involutions \(w_S,w_T\); and
+2. the modular quotient
+   \[
+   SL_2(\mathbb Z)/\Gamma(2)\cong S_3,
+   \]
+   with the generator correspondence
+   \(g_S\leftrightarrow w_S\) and \(g_T\leftrightarrow w_T\).
+
+The harness preserves that distinction. The six matrices used on the modular
+side are canonical representatives of quotient classes in \(SL_2(\mathbb Z)\).
+They are **not** asserted to form a literal six-element subgroup upstairs.
+
+## Exact D4/F4 triality orbit
+
+The D4/F4 side uses the same published matrices already frozen by the reference
+bridge:
+
+\[
+w_T=\operatorname{diag}(1,1,1,-1),
+\]
+
+and
+
+\[
+w_S=\frac12
+\begin{pmatrix}
+1&1&1&-1\\
+1&1&-1&1\\
+1&-1&1&1\\
+-1&1&1&1
+\end{pmatrix}.
+\]
+
+The harness verifies exactly
+
+\[
+w_S^2=I,\qquad
+w_T^2=I,\qquad
+(w_Sw_T)^3=I,
+\]
+
+constructs the six matrices
+
+\[
+e,\ S,\ T,\ ST,\ TS,\ STS,
+\]
+
+and checks closure by exact rational matrix multiplication.
+
+The fixed seed
+
+\[
+z_0=(2,4,6,10)
+\]
+
+has six distinct images:
+
+| class | orbit vector |
+|---|---|
+| \(e\) | \((2,4,6,10)\) |
+| \(S\) | \((1,5,7,9)\) |
+| \(T\) | \((2,4,6,-10)\) |
+| \(ST\) | \((11,-5,-3,-1)\) |
+| \(TS\) | \((1,5,7,-9)\) |
+| \(STS\) | \((11,-5,-3,1)\) |
+
+The orbit is a deterministic test fixture, not a physical state trajectory.
+
+## Modular quotient representatives
+
+The modular generators are
+
+\[
+g_S=
+\begin{pmatrix}
+0&-1\\
+1&0
+\end{pmatrix},
+\qquad
+g_T=
+\begin{pmatrix}
+1&1\\
+0&1
+\end{pmatrix}.
+\]
+
+The harness uses the representatives
+
+\[
+e,\ S,\ T,\ ST,\ TS,\ STS
+\]
+
+and reduces each matrix modulo 2 to identify its class in
+\(SL_2(\mathbb Z)/\Gamma(2)\).
+
+All six mod-2 classes are distinct. Their quotient multiplication table is
+computed from representative products modulo 2 and is checked to agree exactly
+with the multiplication table of the six D4/F4 matrices.
+
+This is the finite fiber-product bridge we need: matching quotient labels and
+multiplication, without conflating the upstairs representations.
+
+## Binary-form action
+
+For
+
+\[
+M=
+\begin{pmatrix}
+a&b\\
+c&d
+\end{pmatrix}
+\in SL_2(\mathbb Z),
+\]
+
+the variable action is
+
+\[
+u'=au+bv,\qquad
+v'=cu+dv.
+\]
+
+The transformed coefficients are **derived** from the source definitions
+
+\[
+f'(u,v)=f(u',v'),
+\qquad
+g'(u,v)=g(u',v'),
+\]
+
+rather than entered as a second independent table.
+
+The implementation expands those equations exactly in the sparse rational
+polynomial engine and extracts the resulting transformed
+\(\alpha_i'\) and \(\beta_j'\).
+
+## Exact covariant equivariance
+
+Sakai's covariant definition requires
+
+\[
+\Psi(\alpha';u,v)=\Psi(\alpha;u',v')
+\]
+
+under the \(SL_2(\mathbb C)\) action.
+
+For each of the six canonical \(SL_2(\mathbb Z)\) representatives, the harness:
+
+1. derives \(\alpha',\beta'\) from transformed \(f,g\);
+2. substitutes those transformed coefficients into every exact covariant;
+3. independently substitutes \(u',v'\) into the original covariant; and
+4. compares the resulting exact sparse polynomials term-for-term.
+
+With 15 covariants and six representatives, the harness performs
+
+\[
+15\times6=90
+\]
+
+exact symbolic equivariance checks.
+
+No numerical tolerance, floating-point comparison, sampling argument, or
+listener-facing receiver is involved.
+
+## Why this comes before D4-TIA-15 v2
+
+D4-TIA-15 v1 maps invariant-algebra grading metadata into symbolic MIDI fields.
+The next version is intended to sonify **evaluated covariants and their orbit
+behavior**.
+
+Before that can be done responsibly, the repository needs a tested statement of
+what is allowed to move under triality and what must remain invariant after
+projection.
+
+This harness supplies that boundary.
+
+The next phase may define a receiver \(\Phi\) only if it preserves the tested
+equivariance structure before any invariant aggregation. In other words, the
+design target is a commuting diagram, not an aesthetically chosen three-way
+permutation.
+
+## Verify
+
+Run:
+
+```bash
+npm test
+npm run verify:d4-tia-s3
+```
+
+The full root verifier also includes this harness.
+
+## Canonical fixture
+
+`examples/d4-tia-s3-equivariance.v0.1.canonical.json` freezes the exact
+six-element D4 orbit, modular quotient representatives and multiplication
+table, plus the coefficient-action and transformed-basis hashes for all six
+representatives. The companion schema binds the complete canonical summary
+with `const`, so a changed orbit, representative, quotient table, or
+equivariance commitment fails verification rather than silently redefining
+the harness.
+
+## Claim boundary
+
+Supported:
+
+- exact six-element D4/F4 triality quotient action generated by \(w_S,w_T\);
+- exact agreement with the six mod-2 modular quotient classes;
+- exact transformed coefficient actions derived from the binary forms;
+- exact covariant equivariance for all 15 constructed generators under all six
+  representatives.
+
+Not supported:
+
+- that the six representatives form a literal finite subgroup of
+  \(SL_2(\mathbb Z)\);
+- that a MIDI receiver has already been shown equivariant;
+- that orbit positions are physical states or acoustic quantities; or
+- any empirical or physical validation claim.
+
+## References
+
+- K. Sakai, *The ring of D4 triality invariants*, arXiv:2504.00546v2 (2026):
+  quotient maps and generator identification in Eqs. (2.10)–(2.16);
+  binary covariant transformation law in Definition 4.2.
+- K. Sakai, *E-strings, F4, and D4 triality*, JHEP 07 (2023) 192,
+  arXiv:2304.04878: modular-triality fiber product and the same \(w_S,w_T\)
+  convention.

docs/ROADMAP.md

@@ -65,17 +65,35 @@ audio remain outside profile identity. ETQ-101 v2 and ETQ-303 v3 are unchanged.
 This phase changes no MIDI/event mapping. It turns the literature ledger into
 executable mathematics before any further auditory interpretation.
 
+## Root D4-TIA-S3-EQUIV v0.1.0 — orbit + equivariance harness (implemented)
+
+- exact six-element \(W(F_4)/W(D_4)\cong S_3\) action generated by the
+  published \(w_S,w_T\) involutions;
+- a deterministic six-point orbit of the fixed D4 seed \((2,4,6,10)\);
+- six canonical \(SL_2(\mathbb Z)\) representatives for the corresponding
+  classes in \(SL_2(\mathbb Z)/\Gamma(2)\cong S_3\);
+- exact agreement of the D4/F4 and modular quotient multiplication tables;
+- coefficient actions derived directly from
+  \(f'(u,v)=f(u',v')\), \(g'(u,v)=g(u',v')\);
+- exact verification of
+  \(\Psi(\alpha';u,v)=\Psi(\alpha;u',v')\)
+  for all 15 constructed covariants under all six representatives; and
+- explicit separation between the finite quotient action and the upstairs
+  \(SL_2(\mathbb Z)\) representatives.
+
+The harness performs 90 exact symbolic equivariance checks and adds no new
+receiver or sonification mapping.
+
 ## Sequenced D4-TIA research follow-up
 
 The remaining D4-TIA sequence is intentionally ordered:
 
-1. **D4/F4 \(S_3\) orbit + equivariance harness** — next.
-2. **D4-TIA-15 v2** — sonify evaluated covariants/orbits while testing
-   equivariance before invariant projection.
-3. **Freeze/tag/archive** the resulting research line and then update Zenodo.
+1. **D4-TIA-15 v2** — sonify evaluated covariants/orbits while preserving the
+   tested equivariance boundary before invariant projection.
+2. **Freeze/tag/archive** the resulting research line and then update Zenodo.
 
-Do not skip directly to a new receiver mapping before the equivariance harness
-exists and is tested.
+The algebra and symmetry-action layers are now executable and tested; the next
+phase may finally define the evaluated-covariant/orbit receiver contract.
 
 The remaining numbered phases below apply only to the independent `APP/` and
 `sonification/` audio laboratories.