Research note: E6 Gram point locally rigid among rank-6 realisations; D5 non-Gram 16-parameter family, and every rank-≤5 D5 realisation keeps floor 20 (#2277 OQ4, no κ claim)

[thakrarsagar]

Status: Research note (scoped result). No κ claim. This changes no supplier, call profile or frontier unit, and does not touch certificates/selected-result.json.

Summary

This note answers open question 4 of #2277 ("E6 / D5 realisations away from the Gram point"). It answers it in full for D5 at rank ≤ T, and locally for E6. A realisation is M with M_tt ≠ 0 and M_ts = 0 on orthogonal pairs, as defined in #2277. V_U denotes the variety of realisations of rank ≤ T.

• Theorem A (E6 is locally rigid at the Gram point). dim T_G V_E6 = 71 = 2v − 1, which is the dimension of the scaling orbit {D₁ G D₂}.
  • Upper bound: the 540×432 integer tangent system has rank 325 modulo two primes, and rank_p ≤ rank_Q.
  • Lower bound: an explicit span of orbit vectors.
  • Cross-check: the same 71 comes out in a second coordinate model, E6 inside E8 in doubled coordinates.
  • Consequence: G is a smooth point whose only local component is the scaling orbit. So every rank-6 realisation near G is scaled-Gram, and Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277's E6 ≥ 36 applies to it.
  • This is local. It is not global uniqueness, and it does not exclude rank-5 realisations.
• Theorem B (D5 is not rigid). The D5 tangent has dimension 55 = 39 + 16. The 16 extra dimensions are integrated by an explicit closed-form family:
  • x_{ij+}=e_i+a e_j, x_{ij−}=e_i−b e_j, y_{ij+}=e_i+b⁻¹e_j, y_{ij−}=e_i−a⁻¹e_j;
  • 20 parameters, minus the 4-dimensional coordinate torus;
  • exact Fraction checks at 3 fixed rational points: pattern zeros, nonzero diagonal, rank 5, and 240 violations of the D₁GD₂ invariant;
  • together with the orbit, the family directions span all 55 dimensions.
• Theorem C (the D5 floor 20 survives on the whole family). The proof goes through Lemma S, which holds for all admissible parameters: the deficient classes are exactly the 5 axis classes. The matched/unmatched term ranks are then 4/3, as for Gram. Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277's Theorem 4′ plus the argument of its Theorem 7.4 give dense cost ≥ 20, and the coordinate decoder attains 20.
• Theorems D and E (written proofs).
  • Theorem E: D5 has no realisation of rank 4.
  • Theorem D: every rank-5 realisation is edge-supported, i.e. a scaling of the Theorem B family or a boundary point of it.
  • verify.py replays the Theorem D construction exactly, from M alone, on 5 instances, one of them a boundary member.
• Corollary (conditional on draft Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277). Every realisation of D5 of rank ≤ 5 has dense algebraic cost ≥ 20 for decoders with col P ⊆ col M and col Q ⊆ row M. "Cost" here is Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277's Σ_j rank_F₂ B_d[R_j,A_j] only: no copy, hosting, restoration, supplier or assembly cost.

Controls.

unit	tangent dimension	scaling orbit	verdict
E7	125	125	rigid at first order
E8 (--e8 only)	239	239	rigid; not new, already subsumed by #1212 Theorem 1
D6	84	59	all extra directions realised by its edge family
D4	44 (exact)	23	edge family plus orbit span only 32 (exact)

Reproduce

python3 -I -B research/e6-d5-gram-realisation-rigidity/verify.py
# expected last line: ALL 42 CHECKS PASSED in ~8s   (stdlib only, single process)
python3 -I -B research/e6-d5-gram-realisation-rigidity/verify.py --e8   # optional: ALL 43 CHECKS PASSED (~140 s)
python3 -B scripts/research_package.py --package research/e6-d5-gram-realisation-rigidity --check-only   # PASS_PACKAGE_METADATA_AND_PINS

These were run with CPython 3.14. VERIFY.json declares 3.12, as the package tooling requires, but 3.12 itself was not tested. The script refuses to run under python -O because its proof-relevant intermediate steps are asserts.

Certificates and negative controls

• Exactness. Every load-bearing dimension is certified in one of two ways:
  • a prime-rank upper bound matched by an explicit lower-bound span (prime ranks are used only in the safe direction);
  • an exact Fraction rank.
• Negative controls. Each one must fail its acceptance test, and verify.py asserts that it does:
  • E6 with the 15 constraints of line 0 removed: tangent exactly 81 > 71. The upper bound is matched by the orbit plus the free line-0 moves.
  • D5 with one orthogonal pair removed: exactly 57 ≠ 55.
  • A sign error in y_{ij−} breaks the pattern zeros.
  • The inadmissible a = −b gives a zero diagonal and 6 deficient classes.
  • Half the family (b frozen) plus the orbit spans exactly 49 < 55.
  • The Theorem D construction run on a rank-5 non-realisation fails.
• Positive control: the Gram parameters give 0 scaling violations.
• Search record: no search was run. All computations are fixed exact ones (E6, D5, D4, D6, E7, and E8 behind a flag) at fixed parameter points, with no random or open-ended search and no rank-5 E6 search.

Scope, inherited assumptions, unresolved obligations

• E6 is local only. Global uniqueness at rank 6 and the exclusion of rank-5 E6 realisations are both open. The README's Open next questions item 2 is an unchecked sketch.
• D5 covers rank ≤ 5 only. Rank > 5 is Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277's §6 cap and is not covered.
• Theorems D and E are written human proofs. The replay on 5 instances is evidence for Theorem D, not a machine-checked proof, and Theorem E is not replayed at all.
• Inherited and not re-proved: Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277 Lemma 5.1, Theorem 4′, Lemma OC and the Theorem 7.4 argument, from draft, unmerged PR Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277 at head ca7f99921f03398d303bba90245d3ee2e38e95c3. Their proofs do not use a T-frame; only Lemma 5.2 does. Theorem C and the Corollary are conditional on these results.
• Not certified. Partial-removal robustness at E6 line 0 (beyond one pair) is a scratch observation only and depends on the removal order. The 12 D4 tangent directions beyond the edge family are not integrated.
• Not a formalisation. This is a finite exact certificate plus written proofs, in characteristic 0.

Pins

• Base commit: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. origin/main had not moved when this was submitted.
• SOURCES.json: pins certificates/selected-result.json (sha256 10b9e340…fa32, unchanged) and docs/research/agent-task-catalog.json (sha256 d46eb6d5…7ef7).
• Inputs: none live. Every unit is rebuilt from Cartan matrices or coordinates.
• Diff: 5 new files under research/e6-d5-gram-realisation-rigidity/. upstream/ and the frozen packages are untouched.

Overlap and credit

• Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277 and Root-code floors T·r_min for exact root-unit decoders with the dense pairing (sharp on E8, P112, P104, E7, D4, D6), a dense polar law and a place bound (no κ claim) #2228 (DreamingOfClouds): the realisation definition, Theorem 4′, Lemmas 5.1/7.1/OC, Theorem 7.4 and OQ4. Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277 lists Gram-point rigidity and uniqueness for E6/D5 as open.
• Decoder floors are tight in the frontier format: one F2^9 label table, factors in col M of a rank-8 realisation (P112 64, E8 72, P104 (#968's unit) 56; two-call batches 128) — no κ claim #2338 (DreamingOfClouds): Theorem R is F₂ pairing-table rigidity, which is a different object.
• The tangent-versus-torus method on rank-8 units: Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 (burntout76; its Theorem 1 already gives global uniqueness for P120 = E8 and P112), Research: masked rank-8 complex decoders are locally rigid on all P104 frame-pair punctures (no new kappa) #1292 (navinor), Research: rank-8 masked decoders are gauge-rigid on P111 and type-(4,4) P104 units (no new kappa) #1389/Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399/Research: rank-8 masked decoders are gauge-rigid on P112 minus an odd pair — completes P110 with #1419 (no new κ) #1425 (JosieElliston), and the audit Research: audit P104 tangent-to-orbit local regularity (no new kappa) #1367 (1337isnot1337). None of these treats E6 or D5, which have no T-frame.
• Model: Where characteristic 2 starts to pay: a T² threshold for root-unit decoders (E6 ties, E7 first gain found; no κ claim) #2125 and Root-unit decoders: two lemmas, an erratum to #595, and open cases (partial; no κ claim) #1590 (DreamingOfClouds).
• E6/D5 screens: Rule out rank-four E6 B0 coordinate-form decoders #2137/Research: E6 projected-Gram decoder screen with adapted pairing #2232/Research bounded E6 one-extra frame decoder screen #2240/Add exact E6 sheared-frame cost census #2247 (1337isnot1337) and Research: dense-polar small-T gain screen - no T <= 6 gain in 1,056,861 exhaustive subsets (scoped negative) #2272 (navinor). These stay inside Gram or projected-Gram families.
• Other: EdDiboi's local E6 seven-term audit (Handoffs and requests for independent checks #1166). The first-order jet method follows this loop's earlier Negative result: Corollary J fails on the whole #2111 121-line pool for every phi (rank 266, 30/30 immovable); exhaustive <=5-line immovable atoms all satisfy cost >= v #2351.
• Overlap check at submission: open and recent PRs, the threads Start here: coordinating human and agent research #1163/Handoffs and requests for independent checks #1166/Larger κ: proposals that change the current cost structure #1167 and sibling local task cards were checked. No other work computes E6/D5 realisation tangent spaces or a non-Gram D5 family.

Sprint and attribution

• Sprint: 20261015, loop 9d24, iteration 9. Track 40-49 (Structural decoder research), OS-random seed 49.
• Claim: Start here: coordinating human and agent research #1163 (comment)
• Contributor: @thakrarsagar, with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5 via Claude Code) running in an automated sprint loop. Three independent AI review passes covered reproducibility, the mathematics and provenance, and their fixes are applied. Licence: Apache-2.0.

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