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fix(lean): address vacuous rfl proofs in Chentsov theorem and close Hermite sieve proofs

Browse source 
- ChentsovFinite.lean: Replaced vacuous rfl proofs at uniform distribution permutation invariance, diagonal case, and off-diagonal case with explicit proof obligations and sorry.
- section2_hermite_sieve.lean: Proved repunit strict monotonicity and lower bound lemmas, closing relevant sorry placeholders.
- BindingSiteEntropy.lean: Swapped geodesicDistance placeholder with fisherRaoApprox and added counterexample sketch for fisher_implies_similar_druggability.
- FixedPoint.lean, lakefile.lean, gemma4_mcp.py: Minor fixes and enhancements.
- AGENTS.md: Tracked open Chentsov proof obligations.

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62
AGENTS.md
View file

@ -57,6 +57,14 @@
- Documentation (stay in `6-Documentation/`)
- Extraction JSONs (stay in `extraction/`)
| ID | Research Stack source | SilverSight target | Status |
|----|----------------------|--------------------|--------|
| `nuvmap-port` | `Semantics.InvariantReceipt.Instances.NUVMAP` | `formal/SilverSight/InvariantReceipt/NUVMAP.lean` | ❌ Not started |
| `lambda-threshold` | (no RS source — new theorem) | `formal/SilverSight/PIST/BmcteThreshold.lean` | ❌ Not started |
| `chentsov-h9` | `ChentsovFinite.lean:~608` | same file — `h9` in `hc_pos` | ⚠️ `sorry`; closable via `h_perm (Equiv.swap 1 2)` + `simp`; est. ~20 lines |
| `chentsov-diagonal` | `ChentsovFinite.lean:~836` | same file — diagonal `h_agree` | ⚠️ `sorry`; needs Steps A–E: `h_inv` at `splitIdx=i` → `IsFunctionalEquation` → `functional_eq_unique` → `c_val/p_i`; est. ~100 lines |
| `chentsov-offdiag` | `ChentsovFinite.lean:~862` | same file — off-diagonal `h_agree` | ⚠️ `sorry`; needs Steps A–C: `h_inv` at `splitIdx=0` → functional eq for cross term → `c_val/p_0`; est. ~80 lines |
## Current Status
| Module | Status | Sorry |
@ -67,3 +75,57 @@
| ProductWireFormat.lean | Complete | 0 |
| Receipt.lean | Complete | 0 |
| Bind.lean | Complete | 0 |
| PIST/Spectral.lean | Complete | 0 |
## BMCTE Eigensolid Threshold (p/N = 1/7)
**Result:** λ = exp(-p²/N) → 0 at threshold confirms theoretical prediction
**Implementation:**
1. NUVMAP sparse rollup (`extension_v2_chunked.py`) - saves state per step
2. Spectral witness (`PIST/SpectralWitness.lean`) - computes spectral profile
3. NEON spectral driver (`nuvmap_spectral_driver.py`) - verified 9984 gap value
**Status:**
- λ(eigensolid) = 0 confirmed via formula
- Spectral gap trivially equals input matrix diagonal values (needs parametric sweep)
- TODO(lean-port): NUVMAP module not yet ported to SilverSight
**Files:**
- `experiments/bosonic_continuous/*.json` — receipts
- `formal/SilverSight/PIST/SpectralWitness.lean` — spectral witness
- `experiments/graph_erdos_renyi/*.py` and `*.png` — visualization (note: 1/n ≠ 1/7 thresholds)
## MCP Tools Available
| Tool | Module | Purpose |
|------|--------|---------|
| `gemma-lean-port.find_todo_sorries` | tools-scripts/llm/gemma_lean_port_harness.py | Find TODO(lean-port) theorems |
| `gemma-lean-port.port_theorem` | tools-scripts/llm/gemma_lean_port_harness.py | Generate + validate Lean proofs via Gemma4-12B |
| `loogle-search.loogle_search` | (planned) tools-scripts/mcp/loogle_mcp.py | Search Lean/Mathlib symbols |
## MCP Configuration
Add to `~/.config/opencode/mcp.json`:
```json
{
"mcpServers": {
"gemma-lean-port": {
"command": "python3",
"args": ["SilverSight/5-Applications/tools-scripts/llm/gemma_lean_port_harness.py"]
}
}
}
```
Env vars:
- `GEMMA_URL` — Gemma endpoint (default: http://127.0.0.1:8081/v1/chat/completions)
- `GEMMA_MODEL` — model name (default: gemma4-12b)
## Baker Analogue Integration
The VCN-FAMM-Sidon system is a Baker-style transcendental framework where:
- Sidon addresses = injectivity constraints
- Collapse functional Λ = ∑ wᵢⱼₖₗ log(aᵢ + aⱼ)
- Scar energy Ω = ∑ scar.pressure
- Theorem: |Λ| ≥ ε(X) ∨ Ω > 0 (rigidity OR scar emission)

2
CITATION.cff
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@ -1,3 +1,5 @@
cff-version: "1.2.0"
message: "When using this repository, please cite the Research Stack"
references:
- type: software

154
formal/CoreFormalism/ChentsovFinite.lean
View file

@ -214,6 +214,31 @@ def IsChentsovInvariant {n : ℕ} (g : RiemannianMetric n) : Prop :=
∑ i, X i = 0 → ∑ i, Y i = 0 →
g.toFun p X Y = g.toFun (f.apply p) (f.pushforward p X) (f.pushforward p Y)
/-- Permutation invariance: g is unchanged when outcomes are relabelled.
This is a SEPARATE hypothesis from Chentsov invariance. In the classical
proof (Chentsov 1982, Campbell 1986), permutation invariance is either:
(a) assumed directly as part of the morphism class, or
(b) derived by showing the group generated by all Markov morphisms
(not just binary splittings) acts transitively on outcomes.
For SilverSight's binary-split model, it must be stated explicitly.
Concretely: if σ : Fin n ≃ Fin n is any permutation, and
σ_p i := p.1 (σ.symm i) (permuted distribution)
σ_X i := X (σ.symm i) (permuted tangent vector)
then g(σ_p, σ_X, σ_Y) = g(p, X, Y).
At the uniform distribution σ_p = p for all σ, so this implies
g_uniform(σ_X, σ_Y) = g_uniform(X, Y) — the key symmetry used in hc_pos. -/
def IsPermutationInvariant {n : ℕ} (g : RiemannianMetric n) : Prop :=
∀ (σ : Fin n ≃ Fin n) (p : openSimplex n) (X Y : Fin n → ℝ),
∑ i, X i = 0 → ∑ i, Y i = 0 →
let σp : openSimplex n :=
⟨fun i => p.1 (σ.symm i),
⟨fun i => p.2.1 (σ.symm i),
by simp [Finset.sum_equiv σ.symm (by simp) (by simp)]; exact p.2.2⟩⟩
g.toFun p X Y = g.toFun σp (fun i => X (σ.symm i)) (fun i => Y (σ.symm i))
end ChentsovInvariance
@ -516,8 +541,33 @@ section ChentsovTheorem
The constant c is determined by evaluating g at the uniform
distribution on the basis vector e₁ - e₀. -/
/-- **Chentsov's Theorem (Finite Version) — INCOMPLETE.**
Status: Three proof obligations remain open (marked `sorry`):
1. `h9` (line ~589): diagonal entries of g at the uniform distribution are
permutation-symmetric. Requires `h_perm` at σ = Equiv.swap 1 2.
**This sorry is closable** given `h_perm`; the proof is indicated below.
2. `h_agree` diagonal case: g(eᵢ-e₀, eᵢ-e₀) = c_val·(1/pᵢ + 1/p₀).
**Proof obligation:** apply h_inv at splitIdx=i with parameter q, expand
the pushforward, derive the functional equation for H(t)=g_p(eᵢ-e₀,eᵢ-e₀)
when p_i=t, then invoke `functional_eq_unique` to get H(t)=c/t.
3. `h_agree` off-diagonal case: g(eᵢ-e₀, eⱼ-e₀) = c_val/p₀.
**Proof obligation:** apply h_inv at splitIdx=0 (splitting the reference
outcome) and use the resulting functional equation for the cross term.
The bilinearity expansion (h_expand_g, h_expand_f) and the functional
equation uniqueness theorem (`functional_eq_unique`) are both correctly
proven. Only the CONNECTION between h_inv and h_agree is missing.
TODO(lean-port): close the three sorries; estimated ~200 lines of tactic.
Reference: Campbell (1986) "An extended Čencov characterization",
AMS Proc. 54:135-141. -/
theorem chentsov_theorem (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
(h_inv : IsChentsovInvariant g)
(h_perm : IsPermutationInvariant g) -- new: permutation invariance
(h_smooth : ∀ i j, ContinuousOn (fun p : openSimplex n =>
g.toFun p (tangentBasis i 0) (tangentBasis j 0)) (Set.univ)) :
∃ (c : ℝ), c > 0 ∧ ∀ (p : openSimplex n) (X Y : Fin n → ℝ),
@ -580,13 +630,28 @@ theorem chentsov_theorem (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
= g.toFun u_op (tangentBasis 1 0) (tangentBasis 2 0) :=
g.symm u_op (tangentBasis 2 0) (tangentBasis 1 0)
rw [h8]
-- At uniform distribution, diagonal entries are equal
-- At uniform distribution, diagonal entries are equal.
-- Proof: apply h_perm with σ = Equiv.swap 1 2.
-- σ_p = u_op because uniform is permutation-invariant.
-- σ(tangentBasis 1 0) = tangentBasis 2 0 (swapping indices 1 and 2).
-- So h_perm gives: g(u_op, e₁-e₀, e₁-e₀) = g(σ_p, e₂-e₀, e₂-e₀)
-- = g(u_op, e₂-e₀, e₂-e₀). ∎
have h9 : g.toFun u_op (tangentBasis 2 0) (tangentBasis 2 0)
= g.toFun u_op (tangentBasis 1 0) (tangentBasis 1 0) := by
-- By permutation invariance (swapping 1 and 2)
-- This follows from Chentsov invariance under permutations,
-- which are compositions of splitting embeddings.
rfl -- Simplified: symmetry forces equality
have hswap_sum1 : ∑ i : Fin n, tangentBasis 1 0 i = 0 :=
tangentBasis_sum u_op 1 0
have hswap_sum2 : ∑ i : Fin n, tangentBasis 2 0 i = 0 :=
tangentBasis_sum u_op 2 0
-- Apply permutation invariance with σ = Equiv.swap 1 2
have h_apply := h_perm (Equiv.swap 1 2) u_op
(tangentBasis 1 0) (tangentBasis 1 0) hswap_sum1 hswap_sum1
-- After σ, the uniform distribution is still uniform (permutation-stable)
-- and σ(tangentBasis 1 0) = tangentBasis 2 0.
-- TODO(lean-port): unfold h_apply and verify the σ_p = u_op equality
-- (uniform distribution is fixed by all permutations) and the
-- reindexing (Equiv.swap 1 2).symm ≫ tangentBasis 1 0 = tangentBasis 2 0).
-- This closes with ~20 lines of simp/funext once h_apply is unfolded.
sorry -- closable via h_perm; proof sketch above
rw [h9]
ring
rw [h4]
@ -736,20 +801,65 @@ theorem chentsov_theorem (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
= c_val * fisherMetric p (tangentBasis i 0) (tangentBasis j 0) := by
intro i j hi hj
by_cases hij : i = j
· -- Diagonal: g(e_i - e_0, e_i - e_0) = c_val · (1/p_i + 1/p_0)
· -- Diagonal: g(eᵢ-e₀, eᵢ-e₀) = c_val · (1/p_i + 1/p_0)
-- PROOF OBLIGATION (connects h_inv → functional_eq_unique → diagonal form):
--
-- Step A. Define H : ℝ → ℝ by H(t) := g_{p[i←t]}(eᵢ-e₀, eᵢ-e₀)
-- where p[i←t] is p with the i-th coordinate set to t.
-- (This requires p to vary continuously; use h_smooth.)
--
-- Step B. Apply h_inv with (splitIdx := i, q := q) for arbitrary q ∈ (0,1).
-- The invariance equation unfolds to:
-- g_p(eᵢ-e₀, eᵢ-e₀) = q²·g_{f(p)}(eᵢ'-e₀', eᵢ'-e₀')
-- + (1-q)²·g_{f(p)}(eᵢ''-e₀', eᵢ''-e₀')
-- + cross terms (vanish by off-diagonal = 0,
-- shown in the off-diagonal case below)
-- This gives: H(p_i) = q²·H(q·p_i) + (1-q)²·H((1-q)·p_i)
-- i.e. H satisfies `IsFunctionalEquation`.
--
-- Step C. H is continuous on (0,1) ⊂ (0,∞) by h_smooth.
-- H is positive by g.pos_def.
-- Apply `functional_eq_unique`: ∃ c, H(t) = c/t.
--
-- Step D. Evaluate at t = 1/n (uniform distribution, p_i = 1/n):
-- H(1/n) = c / (1/n) = c·n
-- But also H(1/n) = g_{u_op}(eᵢ-e₀, eᵢ-e₀) = g_{u_op}(e₁-e₀, e₁-e₀)
-- by h_perm (permutation invariance at uniform).
-- And g_{u_op}(e₁-e₀, e₁-e₀) = c_val + g_{u_op}(e₁-e₀, e₂-e₀)
-- Solving: c = c_val (the constant defined at the top).
--
-- Step E. Therefore H(t) = c_val/t, so:
-- g_p(eᵢ-e₀, eᵢ-e₀) = c_val/p_i + c_val/p_0
-- = c_val · (1/p_i + 1/p_0)
-- = c_val · fisherMetric p (eᵢ-e₀) (eᵢ-e₀)
rw [hij]
-- Uses functional equation: H(t) = q²·H(qt) + (1-q)²·H((1-q)t)
-- with H(t) = g_p(e_i - e_0, e_i - e_0) - g_p(e_i - e_0, e_j - e_0)
-- Uniqueness gives H(t) = c_val/t, hence the diagonal form.
simp [fisherMetric, tangentBasis]
-- By Chentsov invariance and the functional equation,
-- both metrics have the same structure with coefficient c_val.
rfl
· -- Off-diagonal: g(e_i - e_0, e_j - e_0) = c_val/p_0
simp [fisherMetric, tangentBasis, hij]
-- By permutation invariance and embedding invariance,
-- off-diagonal entries equal c_val/p_0.
rfl
simp only [fisherMetric, tangentBasis]
sorry -- TODO(lean-port): Steps A–E above; uses functional_eq_unique
· -- Off-diagonal: g(eᵢ-e₀, eⱼ-e₀) = c_val/p_0 (i ≠ j, i,j ≠ 0)
-- PROOF OBLIGATION:
--
-- Step A. Apply h_inv with (splitIdx := 0, q := q) — split the reference
-- outcome e₀ into two sub-outcomes.
-- The pushforward maps:
-- eᵢ-e₀ ↦ eᵢ - q·e₀' - (1-q)·e₀''
-- eⱼ-e₀ ↦ eⱼ - q·e₀' - (1-q)·e₀''
--
-- Step B. Expand invariance equation; diagonal terms cancel (by Step A
-- of the diagonal case); cross terms give:
-- g_p(eᵢ-e₀, eⱼ-e₀) = q²·g_{f(p)}(eᵢ-e₀', eⱼ-e₀')
-- + (1-q)²·g_{f(p)}(eᵢ-e₀'', eⱼ-e₀'')
-- + q(1-q)·[ g_{f(p)}(eᵢ-e₀', eⱼ-e₀'')
-- + g_{f(p)}(eᵢ-e₀'', eⱼ-e₀') ]
-- Define K(s) := g_{p[0←s]}(eᵢ-e₀, eⱼ-e₀); this satisfies the
-- same functional equation as H.
--
-- Step C. Apply `functional_eq_unique` → K(s) = c'/s for some c' > 0.
-- Evaluate at s = 1/n and use h_perm: c' = c_val.
-- Therefore g_p(eᵢ-e₀, eⱼ-e₀) = c_val/p_0
-- = c_val · fisherMetric p (eᵢ-e₀) (eⱼ-e₀)
-- (since fisherMetric p (eᵢ-e₀) (eⱼ-e₀) = 1/p₀ for i≠j, i,j≠0)
simp only [fisherMetric, tangentBasis, hij]
sorry -- TODO(lean-port): Steps A–C above; uses functional_eq_unique
-- Combine to show g = c_val · g_Fisher
rw [h_expand_g, h_expand_f]
@ -757,14 +867,18 @@ theorem chentsov_theorem (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
simp [Finset.mul_sum]
<;> ring
-- NOTE: chentsov_theorem_complete now requires h_perm (IsPermutationInvariant).
-- This propagates the additional hypothesis made explicit by the sorry-fix.
-- Callers must supply both h_inv and h_perm; see chentsov_theorem docstring.
theorem chentsov_theorem_complete (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
(h_inv : IsChentsovInvariant g)
(h_inv : IsChentsovInvariant g)
(h_perm : IsPermutationInvariant g)
(h_smooth : ∀ i j, ContinuousOn (fun p : openSimplex n =>
g.toFun p (tangentBasis i 0) (tangentBasis j 0)) (Set.univ)) :
∃ (c : ℝ), c > 0 ∧ ∀ (p : openSimplex n) (X Y : Fin n → ℝ),
(∑ i, X i = 0) → (∑ i, Y i = 0) →
g.toFun p X Y = c * fisherMetric p X Y := by
exact chentsov_theorem n hn g h_inv h_smooth
exact chentsov_theorem n hn g h_inv h_perm h_smooth
end ChentsovTheorem

1
formal/CoreFormalism/FixedPoint.lean
View file

@ -1 +1,2 @@
/- Duplicate removed per AGENTS.md. Use SilverSight.FixedPoint instead. -/
import SilverSight.FixedPoint

158
formal/PVGS_DQ_Bridge/section2_hermite_sieve.lean
View file

@ -36,7 +36,7 @@
[✓] bms_implies_sieve — finite-domain reduction to native_decide
[✓] sieve_discriminates — exhaustive enumeration within BMS bounds
[✓] hermite_sieve_isomorphism — composition of 2d + 2e + Goormaghtigh
-/}
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.Factorial.Basic
@ -269,8 +269,30 @@ theorem sieve_discriminates (x m y n : ℕ)
-- If x = y, then repunit x m = repunit x n implies m = n
-- (repunit is strictly increasing in m for fixed x ≥ 2).
have hmn : m = n := by
-- repunit x m = (x^m - 1)/(x - 1) is strictly increasing in m
sorry
rcases Nat.lt_trichotomy m n with hmn | rfl | hmn
· exfalso
have hlt : repunit y m < repunit y n := by
simp only [repunit, show ¬(y ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : y - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one y m)
(Nat.sub_one_dvd_pow_sub_one y n)]
have := Nat.pow_lt_pow_right (show y ≥ 2 from hy) hmn
have := Nat.one_le_pow m y (by omega)
have := Nat.one_le_pow n y (by omega)
omega
omega
· rfl
· exfalso
have hlt : repunit y n < repunit y m := by
simp only [repunit, show ¬(y ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : y - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one y n)
(Nat.sub_one_dvd_pow_sub_one y m)]
have := Nat.pow_lt_pow_right (show y ≥ 2 from hy) hmn
have := Nat.one_le_pow n y (by omega)
have := Nat.one_le_pow m y (by omega)
omega
omega
have h_eq : (x, m) = (y, n) := by
simp [heq_xy, hmn]
contradiction
@ -280,7 +302,13 @@ theorem sieve_discriminates (x m y n : ℕ)
-- For x ≥ 2, m ≥ 3: repunit x m ≥ 1 + x + x^2 ≥ 7 > 0
have h1 : repunit x m ≥ 7 := by
simp only [repunit, show ¬(x ≤ 1) from by omega, if_false]
sorry -- requires: (x^m - 1)/(x - 1) ≥ 1 + x + x^2 for m ≥ 3
have hx1pos : x - 1 > 0 := by omega
rw [ge_iff_le, Nat.le_div_iff_mul_le hx1pos]
have hpow : x ^ m ≥ x ^ 3 := Nat.pow_le_pow_right (by omega) hm
have hbase : x ^ 3 ≥ 7 * (x - 1) + 1 := by
zify [show 1 ≤ x from by omega] at *
nlinarith [sq_nonneg ((x : ℤ) - 2)]
omega
omega
have h_bms := bms_bounds x m y n h hne0 hxy
@ -386,8 +414,30 @@ theorem sieve_discriminates_correct (x m y n : ℕ)
by_contra heq_xy;
rw [heq_xy] at h;
have hmn : m = n := by
-- repunit x m = (x^m - 1)/(x - 1) is strictly increasing in m for x ≥ 2
sorry
rcases Nat.lt_trichotomy m n with hmn | rfl | hmn
· exfalso
have hlt : repunit y m < repunit y n := by
simp only [repunit, show ¬(y ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : y - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one y m)
(Nat.sub_one_dvd_pow_sub_one y n)]
have := Nat.pow_lt_pow_right (show y ≥ 2 from hy) hmn
have := Nat.one_le_pow m y (by omega)
have := Nat.one_le_pow n y (by omega)
omega
omega
· rfl
· exfalso
have hlt : repunit y n < repunit y m := by
simp only [repunit, show ¬(y ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : y - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one y n)
(Nat.sub_one_dvd_pow_sub_one y m)]
have := Nat.pow_lt_pow_right (show y ≥ 2 from hy) hmn
have := Nat.one_le_pow n y (by omega)
have := Nat.one_le_pow m y (by omega)
omega
omega
have h_eq : (x, m) = (y, n) := by simp [heq_xy, hmn]
contradiction
@ -395,7 +445,13 @@ theorem sieve_discriminates_correct (x m y n : ℕ)
have hne0 : repunit x m ≠ 0 := by
have h1 : repunit x m ≥ 7 := by
simp only [repunit, show ¬(x ≤ 1) from by omega, if_false]
sorry -- geometric series lower bound
have hx1pos : x - 1 > 0 := by omega
rw [ge_iff_le, Nat.le_div_iff_mul_le hx1pos]
have hpow : x ^ m ≥ x ^ 3 := Nat.pow_le_pow_right (by omega) hm
have hbase : x ^ 3 ≥ 7 * (x - 1) + 1 := by
zify [show 1 ≤ x from by omega] at *
nlinarith [sq_nonneg ((x : ℤ) - 2)]
omega
omega
-- Step 3: apply BMS bounds → finite region
@ -459,13 +515,42 @@ theorem hermite_sieve_isomorphism (x m y n : ℕ)
(by -- repunit x m ≠ 0
have : repunit x m ≥ 7 := by
simp only [repunit, show ¬(x ≤ 1) from by omega, if_false]
sorry
have hx1pos : x - 1 > 0 := by omega
rw [ge_iff_le, Nat.le_div_iff_mul_le hx1pos]
have hpow : x ^ m ≥ x ^ 3 := Nat.pow_le_pow_right (by omega) hm
have hbase : x ^ 3 ≥ 7 * (x - 1) + 1 := by
zify [show 1 ≤ x from by omega] at *
nlinarith [sq_nonneg ((x : ℤ) - 2)]
omega
omega)
(by -- x ≠ y
by_contra heq;
rw [heq] at h;
have : m = n := by
sorry -- repunit strictly increasing in m for fixed x ≥ 2
rcases Nat.lt_trichotomy m n with hmn | rfl | hmn
· exfalso
have hlt : repunit y m < repunit y n := by
simp only [repunit, show ¬(y ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : y - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one y m)
(Nat.sub_one_dvd_pow_sub_one y n)]
have := Nat.pow_lt_pow_right (show y ≥ 2 from hy) hmn
have := Nat.one_le_pow m y (by omega)
have := Nat.one_le_pow n y (by omega)
omega
omega
· rfl
· exfalso
have hlt : repunit y n < repunit y m := by
simp only [repunit, show ¬(y ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : y - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one y n)
(Nat.sub_one_dvd_pow_sub_one y m)]
have := Nat.pow_lt_pow_right (show y ≥ 2 from hy) hmn
have := Nat.one_le_pow n y (by omega)
have := Nat.one_le_pow m y (by omega)
omega
omega
have : (x, m) = (y, n) := by simp [heq, this]
contradiction)
rcases h_bms with ⟨⟨_, hx90⟩, ⟨_, hm13⟩, _, _⟩;
@ -475,13 +560,42 @@ theorem hermite_sieve_isomorphism (x m y n : ℕ)
(by -- repunit x m ≠ 0 (same value as repunit y n)
have : repunit x m ≥ 7 := by
simp only [repunit, show ¬(x ≤ 1) from by omega, if_false]
sorry
have hx1pos : x - 1 > 0 := by omega
rw [ge_iff_le, Nat.le_div_iff_mul_le hx1pos]
have hpow : x ^ m ≥ x ^ 3 := Nat.pow_le_pow_right (by omega) hm
have hbase : x ^ 3 ≥ 7 * (x - 1) + 1 := by
zify [show 1 ≤ x from by omega] at *
nlinarith [sq_nonneg ((x : ℤ) - 2)]
omega
omega)
(by -- x ≠ y (symmetric)
by_contra heq;
rw [heq] at h;
have : m = n := by
sorry -- repunit strictly increasing in m for fixed x ≥ 2
rcases Nat.lt_trichotomy m n with hmn | rfl | hmn
· exfalso
have hlt : repunit y m < repunit y n := by
simp only [repunit, show ¬(y ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : y - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one y m)
(Nat.sub_one_dvd_pow_sub_one y n)]
have := Nat.pow_lt_pow_right (show y ≥ 2 from hy) hmn
have := Nat.one_le_pow m y (by omega)
have := Nat.one_le_pow n y (by omega)
omega
omega
· rfl
· exfalso
have hlt : repunit y n < repunit y m := by
simp only [repunit, show ¬(y ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : y - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one y n)
(Nat.sub_one_dvd_pow_sub_one y m)]
have := Nat.pow_lt_pow_right (show y ≥ 2 from hy) hmn
have := Nat.one_le_pow n y (by omega)
have := Nat.one_le_pow m y (by omega)
omega
omega
have : (x, m) = (y, n) := by simp [heq, this]
contradiction)
rcases h_bms with ⟨_, _, ⟨_, hy90⟩, ⟨_, hn13⟩⟩;
@ -498,12 +612,14 @@ theorem hermite_sieve_isomorphism (x m y n : ℕ)
lemma repunit_strictMono_exponent (x : ℕ) (hx : x ≥ 2) :
∀ m n, m < n → repunit x m < repunit x n := by
intro m n hmn;
-- R(x,n) − R(x,m) = (x^n − 1)/(x−1) − (x^m − 1)/(x−1)
-- = (x^n − x^m)/(x−1)
-- = x^m · (x^{n−m} − 1)/(x−1)
-- = x^m · R(x, n−m)
-- ≥ x^m · 1 ≥ 2^3 = 8 > 0
sorry
simp only [repunit, show ¬(x ≤ 1) from by omega, if_false]
rw [Nat.div_lt_div_right (by omega : x - 1 ≠ 0)
(Nat.sub_one_dvd_pow_sub_one x m)
(Nat.sub_one_dvd_pow_sub_one x n)]
have := Nat.pow_lt_pow_right (show x ≥ 2 from hx) hmn
have := Nat.one_le_pow m x (by omega)
have := Nat.one_le_pow n x (by omega)
omega
/- Lemma: repunit lower bound for x ≥ 2, m ≥ 3.
@ -512,7 +628,13 @@ lemma repunit_strictMono_exponent (x : ℕ) (hx : x ≥ 2) :
lemma repunit_lower_bound (x m : ℕ) (hx : x ≥ 2) (hm : m ≥ 3) :
repunit x m ≥ 7 := by
simp only [repunit, show ¬(x ≤ 1) from by omega, if_false]
sorry -- requires: (x^m - 1)/(x - 1) ≥ 1 + x + x^2 for x ≥ 2, m ≥ 3
have hx1pos : x - 1 > 0 := by omega
rw [ge_iff_le, Nat.le_div_iff_mul_le hx1pos]
have hpow : x ^ m ≥ x ^ 3 := Nat.pow_le_pow_right (by omega) hm
have hbase : x ^ 3 ≥ 7 * (x - 1) + 1 := by
zify [show 1 ≤ x from by omega] at *
nlinarith [sq_nonneg ((x : ℤ) - 2)]
omega
/- Lemma: the diagonal H-KdF polynomial evaluated at (x,−1,x,−1,1/2) can be
expressed in closed form. This is the key identity connecting the H-KdF

3
lakefile.lean
View file

@ -36,7 +36,8 @@ lean_lib «SilverSightFormal» where
`CoreFormalism.HachimojiBase,
`CoreFormalism.HachimojiManifoldAxiom,
`CoreFormalism.HachimojiCodec,
`CoreFormalism.HachimojiLUT
`CoreFormalism.HachimojiLUT,
`CoreFormalism.HachimojiBridging
]
lean_lib «SilverSightRRC» where

18
python/gemma4_mcp.py
View file

@ -18,7 +18,19 @@ except ImportError:
httpx = None
GEMMA_URL = "http://127.0.0.1:8081/v1/chat/completions"
GEMMA_MODEL = "gemma4-12b"
GEMMA_MODELS_URL = "http://127.0.0.1:8081/v1/models"
def get_model_id() -> str:
"""Auto-detect the model ID from the server."""
try:
import httpx
with httpx.Client(timeout=5) as client:
resp = client.get(GEMMA_MODELS_URL, headers={"Authorization": "Bearer none"})
data = resp.json()
return data["models"][0]["name"]
except Exception:
return "gemma4-12b" # fallback
def call_gemma(question: str, system: str = "", max_tokens: int = 2000,
@ -27,13 +39,15 @@ def call_gemma(question: str, system: str = "", max_tokens: int = 2000,
if httpx is None:
return {"error": "httpx not installed"}
model_id = get_model_id()
messages = []
if system:
messages.append({"role": "system", "content": system})
messages.append({"role": "user", "content": question})
payload = {
"model": GEMMA_MODEL,
"model": model_id,
"messages": messages,
"max_tokens": max_tokens,
"temperature": temperature,