Phase 5 — Close INV-6 HybridQkGain (5 Admitted, architecture floor proof)

[gHashTag]

Phase 5 — Close INV-6 HybridQkGain (5 Admitted) — ARCHITECTURE FLOOR PROOF

Parent: #436 · CRITICAL PATH BLOCKER

Why this matters

• INV-6 owns the gain term gain ∝ √d_model for hybrid QK attention.
• Once Qed, the architectural floor (currently empirical 1.75–2.19) is pinned to a closed-form expression arch_floor_bpb = 2 + phi^(-2) ≈ 2.382.
• This determines whether Gate-FINAL (BPB < 1.5) is reachable on the current architecture (probably no → motivates Phase 6 next-architecture).
• Until INV-6 closes, every empirical seed run is shadow-boxing against an unstated invariant.

5 Admitted obligations

Theorem	File	Difficulty
counter_gain_sqrt_d_model	INV6_HybridQkGain.v	hard
counter_gain_unit	same	medium
counter_lr_above_band	same	medium
counter_lr_below_band	same	medium
hybrid_qk_gain_phi_sq_well_typed	same	typing only — easy once others close

Strategy

1. Counter-gain unit: gain(d) = sqrt(d) / sqrt(d_model_min) normalised to 1 at d = d_model_min (algebraic).
2. Sqrt(d) bound: standard sqrt_le_compat from Coq.Reals.Reals; combined with INV-3 d >= d_model_min.
3. lr above/below band: INV-1b lr_champion_in_safe_range already gives the band; counter-cases are lt / gt lemmas with concrete witnesses.
4. Well-typed: gain * phi^2 lives in R; no surprise.

Corollary (the publishable result)

Corollary architecture_floor_phi_squared :
  arch_floor_bpb = 2 + phi^(-2).
Proof.
  (* Combines counter_gain_sqrt_d_model (Phase 5)
     with bpb_convergence_rate_bound (INV-1 Qed)
     with phi_square_eq_phi_plus_one (CorePhi Qed). *)
Admitted. (* opened by Phase 5 closing *)

2 + phi^(-2) ≈ 2.382. The empirical 2.19 we observe is below this analytic upper bound — implying the actual floor sits at 2 + phi^(-2) ≈ 2.382 and the 2.19 measurements are still on the descent curve. The CKM/Lucas residual gap explains the 0.19 BPB difference.

Acceptance (R5)

• All 5 Admitted → Qed in INV6_HybridQkGain.v
• Corollary architecture_floor_phi_squared Qed in a new file
• CI green
• ssot.theorems 5 rows flip Admitted → Qed
• Phase 6 next-architecture issue opens automatically (depth-5 attainability)
• gardener_runs tick lane=PHASE-5-INV6-CLOSED

Estimated effort

1 week for one strong Coq author, less if the team co-works through the band lemmas.

phi^2 + phi^-2 = 3 · TRINITY · INV-6 IS THE BLOCKER

[gHashTag]

R5-honest pre-flight finding — Phase-5 INV-6 cannot start as written

LEAD reconnoitred this issue per Wave-6 dispatch and found two blockers that prevent execution as specified. Surfacing before any partial-attempt commit, per Trinity R5 honesty.

Blocker 1 — target file does not exist

The issue body links to:

https://github.com/gHashTag/t27/blob/main/proofs/canonical/igla/INV6_HybridQkGain.v

That path returns 404. A full tree-walk of gHashTag/t27@master enumerates every .v file:

compiler/codegen/verilog/fpga_emission.v
contrib/formal/{fifo,mac,uart}_formal_props.v
coq/Kernel/{FlowerE8Embedding,KernelSpec,Phi,PhiAttractor,PhiFloat,Semantics,Trit}.v
coq/Theorems/{GenIdempotency,PhiDistance,TernarySufficiency}.v
gen/verilog/...   (50+ verilog files, not Coq proofs)
proofs/gravity/dl_bounds.v
proofs/sacred/{dl_bounds,gamma_phi3,...}.v

There is no proofs/canonical/ directory and no INV6_HybridQkGain.v anywhere in the repo. The Coq subtree under coq/ carries Phi/PhiAttractor/PhiFloat work but no HybridQkGain theorems.

The sibling repo gHashTag/trinity-clara carries proofs/igla/ with 12 INV files (asha_champion_survives.v, bpb_decreases.v, ema_decay_valid.v, gf16_precision.v, gf16_safe_domain.v, igla_asha_bound.v, igla_found_criterion.v, igla_invariants.v, lr_phi_optimality.v, lucas_closure_gf16.v, nca_entropy_band.v, rainbow_bridge_consistency.v) — also no INV6_HybridQkGain.v.

Blocker 2 — toolchain absent in dispatch sandbox

coqc is not installed in the agent dispatch sandbox. Even if I created the file from scratch, I could not verify a single Qed line. The R5 contract refuses unverified Coq.

What I will NOT do

• Fabricate Qed. proofs in a file that does not exist in either repo.
• Write a stub Admitted. and call it Phase-5 progress (admitted_budget=5/5 LOCKED per INV-13 policy; budget already spent).
• Move the file location silently to where I can write to it.

What is actionable now

• Confirm the canonical file path with the issue author. Two candidates:
  • gHashTag/t27 — needs proofs/canonical/igla/INV6_HybridQkGain.v created from scratch (this PR would also need to add _CoqProject entries, CI wiring, and the 5 obligation skeletons).
  • gHashTag/trinity-clara — already has the INV-corpus convention; new file would be proofs/igla/INV6_HybridQkGain.v.
• Confirm a CI runner with coqc that the Phase-5 author can use (the Wave-6 dispatch sandbox has no Coq toolchain).
• Once both are settled, the strategy in the issue body is sound: counter_gain_unit (algebraic, medium) → counter_gain_sqrt_d_model via Coq.Reals.sqrt_le_compat (hard) → counter_lr_above_band / counter_lr_below_band (medium each, witnesses from INV-1b) → hybrid_qk_gain_phi_sq_well_typed (easy). One Qed. per commit, with coqc green between each.

Wave-6 GENERAL dispatch decision

Per "Fix×3 then skip" (constitutional preserve verbatim), LEAD does not start this front in Wave-6. Tracked for re-dispatch as soon as either blocker is unblocked.

For Wave-6 LEAD already shipped:

• 🌻 App.K — Agent Memory chapter scaffold (KG-memory → GOLD IV PhD) [was App.G — clash with CLARA Mirror] #465 App.K Agent Memory scaffold — merged in #487.
• (Phase 5 — Close INV-6 HybridQkGain (5 Admitted, architecture floor proof) #441 INV-6 — this finding; held)

🌻 α_φ = φ⁻³/2 ≈ 0.1180 · φ²+φ⁻²=3 · TRINITY · LEAD

— Agent: CitationScribe (LEAD, Wave-6 R2 holder)