Encyclopedia Foundation Foundation Pair Kernel Periodic3 Symbol Torus Quantized Wave Ne Zero Of Ne Zero
ARTICLE 3 claims 3 theorems
Foundation Pair Kernel Periodic3 Symbol Torus Quantized Wave Ne Zero Of Ne Zero
On a finite three-dimensional torus, every non-constant pattern of vibration has a non-zero frequency signature, a fact that makes the discrete world's wave mechanics well-defined.
The nonzero mode theorem
A finite three-dimensional torus is a box where opposite faces are glued together, so that moving past one edge brings you back on the other side. In the Recognition Science framework, such a torus is a discrete model of space, a finite grid of sites rather than a continuous expanse. Each site is identified by three coordinates, and the whole grid wraps around in every direction. This is the classical setting for studying waves on a compact space without boundaries.
The declaration symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode is a theorem in the framework's machine-checked library of formal theorems. It states that if you take any non-zero mode on this torus, meaning any pattern of vibration that is not completely flat and uniform, then its associated frequency symbol is also non-zero. The symbol is a complex number that encodes how the wave's spatial pattern relates to its temporal evolution, a kind of fingerprint for the mode.
In plain terms, the theorem rules out a degenerate case: a non-trivial spatial pattern cannot have a zero frequency signature. If it did, the mathematics of solving for the wave's response would break down, as division by this symbol would be impossible. The theorem ensures that for every non-constant mode, the machinery of spectral analysis, which decomposes complicated patterns into simple building blocks, remains well-defined and invertible.
In Recognition Science, this is a technical but necessary step. The framework models physical structure as a discrete ledger of recognition events, and this torus is a carrier for its periodic wave mechanics. The theorem belongs to a chain that builds up a consistent picture of how waves behave on this finite, periodic space. It is a structural guarantee, not a physical prediction: it does not say which modes exist or how they evolve, only that the non-zero ones have the algebraic property needed for the theory to proceed.
What this theorem does not claim is equally important. It does not identify this finite periodic torus with the infinite three-dimensional space of everyday physics, nor with the finite open box used in other parts of the framework. It asserts no convergence result as the grid size grows without bound. The shift-eigenmode equations, which would tie these modes to specific physical wave solutions, remain hypotheses, not theorems. The declaration is a precise, local result about the algebra of a finite grid, and it earns its place by making the spectral method on this torus sound.
THEOREM symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.lean
/-- A nonzero quantized torus frequency has a nonzero denominator for the
cubic-lattice symbol. The `N = 1` case is included because its only mode
is the zero mode. -/
theorem symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode
{N : ℕ} [NeZero N] (m : Fin 3 → Fin N) (hm : m ≠ 0) :
symbol (torusQuantizedWave N m) ≠ 0 := by
intro hsym
have hsum :
(∑ i : Fin 3, (1 - Real.cos (torusQuantizedWave N m i))) = 0 := by
unfold symbol at hsym
linarith
have hnn :
∀ i ∈ (Finset.univ : Finset (Fin 3)),
(0 : ℝ) ≤ 1 - Real.cos (torusQuantizedWave N m i) := by
intro i _
have hcos := Real.cos_le_one (torusQuantizedWave N m i)
linarith
have hcoord_zero : ∀ i : Fin 3, m i = 0 := by
intro i
have hzero :=
(Finset.sum_eq_zero_iff_of_nonneg hnn).1 hsum i (Finset.mem_univ i)
have hcos : Real.cos (torusQuantizedWave N m i) = 1 := by
linarith
have hNpos_nat : 0 < N := Nat.pos_of_ne_zero (NeZero.ne N)
have hNpos : 0 < (N : ℝ) := Nat.cast_pos.mpr hNpos_nat
have h2pi_pos : 0 < (2 * Real.pi : ℝ) := by positivity
have hm_lt : ((m i).val : ℝ) < (N : ℝ) := by
exact_mod_cast (m i).isLt
have hlt : torusQuantizedWave N m i < 2 * Real.pi := by
change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) < 2 * Real.pi
rw [div_lt_iff₀ hNpos]
exact mul_lt_mul_of_pos_left hm_lt h2pi_pos
have hnonneg : 0 ≤ torusQuantizedWave N m i := by
change 0 ≤ 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ)
positivity
have hgt : -(2 * Real.pi) < torusQuantizedWave N m i := by
linarith
have hqzero : torusQuantizedWave N m i = 0 :=
(Real.cos_eq_one_iff_of_lt_of_lt hgt hlt).1 hcos
have hval_real : ((m i).val : ℝ) = 0 := by
change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) = 0 at hqzero
have hden : (N : ℝ) ≠ 0 := ne_of_gt hNpos
have hnum : 2 * Real.pi * ((m i).val : ℝ) = 0 := by
exact ((div_eq_zero_iff).1 hqzero).resolve_right hden
exact (mul_eq_zero.mp hnum).resolve_left (ne_of_gt h2pi_pos)
apply Fin.ext
exact Nat.cast_eq_zero.mp hval_real
apply hm
funext i
exact hcoord_zero i
THEOREM symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.lean
/-- A nonzero quantized torus frequency has a nonzero denominator for the
cubic-lattice symbol. The `N = 1` case is included because its only mode
is the zero mode. -/
theorem symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode
{N : ℕ} [NeZero N] (m : Fin 3 → Fin N) (hm : m ≠ 0) :
symbol (torusQuantizedWave N m) ≠ 0 := by
intro hsym
have hsum :
(∑ i : Fin 3, (1 - Real.cos (torusQuantizedWave N m i))) = 0 := by
unfold symbol at hsym
linarith
have hnn :
∀ i ∈ (Finset.univ : Finset (Fin 3)),
(0 : ℝ) ≤ 1 - Real.cos (torusQuantizedWave N m i) := by
intro i _
have hcos := Real.cos_le_one (torusQuantizedWave N m i)
linarith
have hcoord_zero : ∀ i : Fin 3, m i = 0 := by
intro i
have hzero :=
(Finset.sum_eq_zero_iff_of_nonneg hnn).1 hsum i (Finset.mem_univ i)
have hcos : Real.cos (torusQuantizedWave N m i) = 1 := by
linarith
have hNpos_nat : 0 < N := Nat.pos_of_ne_zero (NeZero.ne N)
have hNpos : 0 < (N : ℝ) := Nat.cast_pos.mpr hNpos_nat
have h2pi_pos : 0 < (2 * Real.pi : ℝ) := by positivity
have hm_lt : ((m i).val : ℝ) < (N : ℝ) := by
exact_mod_cast (m i).isLt
have hlt : torusQuantizedWave N m i < 2 * Real.pi := by
change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) < 2 * Real.pi
rw [div_lt_iff₀ hNpos]
exact mul_lt_mul_of_pos_left hm_lt h2pi_pos
have hnonneg : 0 ≤ torusQuantizedWave N m i := by
change 0 ≤ 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ)
positivity
have hgt : -(2 * Real.pi) < torusQuantizedWave N m i := by
linarith
have hqzero : torusQuantizedWave N m i = 0 :=
(Real.cos_eq_one_iff_of_lt_of_lt hgt hlt).1 hcos
have hval_real : ((m i).val : ℝ) = 0 := by
change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) = 0 at hqzero
have hden : (N : ℝ) ≠ 0 := ne_of_gt hNpos
have hnum : 2 * Real.pi * ((m i).val : ℝ) = 0 := by
exact ((div_eq_zero_iff).1 hqzero).resolve_right hden
exact (mul_eq_zero.mp hnum).resolve_left (ne_of_gt h2pi_pos)
apply Fin.ext
exact Nat.cast_eq_zero.mp hval_real
apply hm
funext i
exact hcoord_zero i
THEOREM symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.lean
/-- A nonzero quantized torus frequency has a nonzero denominator for the
cubic-lattice symbol. The `N = 1` case is included because its only mode
is the zero mode. -/
theorem symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode
{N : ℕ} [NeZero N] (m : Fin 3 → Fin N) (hm : m ≠ 0) :
symbol (torusQuantizedWave N m) ≠ 0 := by
intro hsym
have hsum :
(∑ i : Fin 3, (1 - Real.cos (torusQuantizedWave N m i))) = 0 := by
unfold symbol at hsym
linarith
have hnn :
∀ i ∈ (Finset.univ : Finset (Fin 3)),
(0 : ℝ) ≤ 1 - Real.cos (torusQuantizedWave N m i) := by
intro i _
have hcos := Real.cos_le_one (torusQuantizedWave N m i)
linarith
have hcoord_zero : ∀ i : Fin 3, m i = 0 := by
intro i
have hzero :=
(Finset.sum_eq_zero_iff_of_nonneg hnn).1 hsum i (Finset.mem_univ i)
have hcos : Real.cos (torusQuantizedWave N m i) = 1 := by
linarith
have hNpos_nat : 0 < N := Nat.pos_of_ne_zero (NeZero.ne N)
have hNpos : 0 < (N : ℝ) := Nat.cast_pos.mpr hNpos_nat
have h2pi_pos : 0 < (2 * Real.pi : ℝ) := by positivity
have hm_lt : ((m i).val : ℝ) < (N : ℝ) := by
exact_mod_cast (m i).isLt
have hlt : torusQuantizedWave N m i < 2 * Real.pi := by
change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) < 2 * Real.pi
rw [div_lt_iff₀ hNpos]
exact mul_lt_mul_of_pos_left hm_lt h2pi_pos
have hnonneg : 0 ≤ torusQuantizedWave N m i := by
change 0 ≤ 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ)
positivity
have hgt : -(2 * Real.pi) < torusQuantizedWave N m i := by
linarith
have hqzero : torusQuantizedWave N m i = 0 :=
(Real.cos_eq_one_iff_of_lt_of_lt hgt hlt).1 hcos
have hval_real : ((m i).val : ℝ) = 0 := by
change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) = 0 at hqzero
have hden : (N : ℝ) ≠ 0 := ne_of_gt hNpos
have hnum : 2 * Real.pi * ((m i).val : ℝ) = 0 := by
exact ((div_eq_zero_iff).1 hqzero).resolve_right hden
exact (mul_eq_zero.mp hnum).resolve_left (ne_of_gt h2pi_pos)
apply Fin.ext
exact Nat.cast_eq_zero.mp hval_real
apply hm
funext i
exact hcoord_zero i
What this page does not claim
The theorem does not prove that the torus is a physical model of space. The theorem does not establish any convergence or approximation result for large grid sizes. The theorem does not assert that the shift-eigenmode equations hold.
Verify this page
Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:
$ lake env lean IndisputableMonolith/Foundation/PairKernelPeriodic3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)
A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.
Derived articles
This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:
- What physical interpretation, if any, does the framework assign to the modes on this finite torus?
- How does the spectral method on this torus connect to the infinite-lattice Fourier analysis used elsewhere in the framework?
- What would it mean for the framework if a non-zero mode did have a zero symbol?
- What role does the torus carrier play in the broader Recognition Science derivation of three-dimensional space?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
THEOREM symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.lean
/-- A nonzero quantized torus frequency has a nonzero denominator for the cubic-lattice symbol. The `N = 1` case is included because its only mode is the zero mode. -/ theorem symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode {N : ℕ} [NeZero N] (m : Fin 3 → Fin N) (hm : m ≠ 0) : symbol (torusQuantizedWave N m) ≠ 0 := by intro hsym have hsum : (∑ i : Fin 3, (1 - Real.cos (torusQuantizedWave N m i))) = 0 := by unfold symbol at hsym linarith have hnn : ∀ i ∈ (Finset.univ : Finset (Fin 3)), (0 : ℝ) ≤ 1 - Real.cos (torusQuantizedWave N m i) := by intro i _ have hcos := Real.cos_le_one (torusQuantizedWave N m i) linarith have hcoord_zero : ∀ i : Fin 3, m i = 0 := by intro i have hzero := (Finset.sum_eq_zero_iff_of_nonneg hnn).1 hsum i (Finset.mem_univ i) have hcos : Real.cos (torusQuantizedWave N m i) = 1 := by linarith have hNpos_nat : 0 < N := Nat.pos_of_ne_zero (NeZero.ne N) have hNpos : 0 < (N : ℝ) := Nat.cast_pos.mpr hNpos_nat have h2pi_pos : 0 < (2 * Real.pi : ℝ) := by positivity have hm_lt : ((m i).val : ℝ) < (N : ℝ) := by exact_mod_cast (m i).isLt have hlt : torusQuantizedWave N m i < 2 * Real.pi := by change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) < 2 * Real.pi rw [div_lt_iff₀ hNpos] exact mul_lt_mul_of_pos_left hm_lt h2pi_pos have hnonneg : 0 ≤ torusQuantizedWave N m i := by change 0 ≤ 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) positivity have hgt : -(2 * Real.pi) < torusQuantizedWave N m i := by linarith have hqzero : torusQuantizedWave N m i = 0 := (Real.cos_eq_one_iff_of_lt_of_lt hgt hlt).1 hcos have hval_real : ((m i).val : ℝ) = 0 := by change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) = 0 at hqzero have hden : (N : ℝ) ≠ 0 := ne_of_gt hNpos have hnum : 2 * Real.pi * ((m i).val : ℝ) = 0 := by exact ((div_eq_zero_iff).1 hqzero).resolve_right hden exact (mul_eq_zero.mp hnum).resolve_left (ne_of_gt h2pi_pos) apply Fin.ext exact Nat.cast_eq_zero.mp hval_real apply hm funext i exact hcoord_zero iThe declaration symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode is a theorem in the framework's machine-checked library of formal theorems. symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.leanTHEOREM symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.lean
/-- A nonzero quantized torus frequency has a nonzero denominator for the cubic-lattice symbol. The `N = 1` case is included because its only mode is the zero mode. -/ theorem symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode {N : ℕ} [NeZero N] (m : Fin 3 → Fin N) (hm : m ≠ 0) : symbol (torusQuantizedWave N m) ≠ 0 := by intro hsym have hsum : (∑ i : Fin 3, (1 - Real.cos (torusQuantizedWave N m i))) = 0 := by unfold symbol at hsym linarith have hnn : ∀ i ∈ (Finset.univ : Finset (Fin 3)), (0 : ℝ) ≤ 1 - Real.cos (torusQuantizedWave N m i) := by intro i _ have hcos := Real.cos_le_one (torusQuantizedWave N m i) linarith have hcoord_zero : ∀ i : Fin 3, m i = 0 := by intro i have hzero := (Finset.sum_eq_zero_iff_of_nonneg hnn).1 hsum i (Finset.mem_univ i) have hcos : Real.cos (torusQuantizedWave N m i) = 1 := by linarith have hNpos_nat : 0 < N := Nat.pos_of_ne_zero (NeZero.ne N) have hNpos : 0 < (N : ℝ) := Nat.cast_pos.mpr hNpos_nat have h2pi_pos : 0 < (2 * Real.pi : ℝ) := by positivity have hm_lt : ((m i).val : ℝ) < (N : ℝ) := by exact_mod_cast (m i).isLt have hlt : torusQuantizedWave N m i < 2 * Real.pi := by change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) < 2 * Real.pi rw [div_lt_iff₀ hNpos] exact mul_lt_mul_of_pos_left hm_lt h2pi_pos have hnonneg : 0 ≤ torusQuantizedWave N m i := by change 0 ≤ 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) positivity have hgt : -(2 * Real.pi) < torusQuantizedWave N m i := by linarith have hqzero : torusQuantizedWave N m i = 0 := (Real.cos_eq_one_iff_of_lt_of_lt hgt hlt).1 hcos have hval_real : ((m i).val : ℝ) = 0 := by change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) = 0 at hqzero have hden : (N : ℝ) ≠ 0 := ne_of_gt hNpos have hnum : 2 * Real.pi * ((m i).val : ℝ) = 0 := by exact ((div_eq_zero_iff).1 hqzero).resolve_right hden exact (mul_eq_zero.mp hnum).resolve_left (ne_of_gt h2pi_pos) apply Fin.ext exact Nat.cast_eq_zero.mp hval_real apply hm funext i exact hcoord_zero iIt states that if you take any non-zero mode on this torus, meaning any pattern of vibration that is not completely flat and uniform, then its associated frequency symbol is also non-zero. symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.leanTHEOREM symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.lean
/-- A nonzero quantized torus frequency has a nonzero denominator for the cubic-lattice symbol. The `N = 1` case is included because its only mode is the zero mode. -/ theorem symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode {N : ℕ} [NeZero N] (m : Fin 3 → Fin N) (hm : m ≠ 0) : symbol (torusQuantizedWave N m) ≠ 0 := by intro hsym have hsum : (∑ i : Fin 3, (1 - Real.cos (torusQuantizedWave N m i))) = 0 := by unfold symbol at hsym linarith have hnn : ∀ i ∈ (Finset.univ : Finset (Fin 3)), (0 : ℝ) ≤ 1 - Real.cos (torusQuantizedWave N m i) := by intro i _ have hcos := Real.cos_le_one (torusQuantizedWave N m i) linarith have hcoord_zero : ∀ i : Fin 3, m i = 0 := by intro i have hzero := (Finset.sum_eq_zero_iff_of_nonneg hnn).1 hsum i (Finset.mem_univ i) have hcos : Real.cos (torusQuantizedWave N m i) = 1 := by linarith have hNpos_nat : 0 < N := Nat.pos_of_ne_zero (NeZero.ne N) have hNpos : 0 < (N : ℝ) := Nat.cast_pos.mpr hNpos_nat have h2pi_pos : 0 < (2 * Real.pi : ℝ) := by positivity have hm_lt : ((m i).val : ℝ) < (N : ℝ) := by exact_mod_cast (m i).isLt have hlt : torusQuantizedWave N m i < 2 * Real.pi := by change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) < 2 * Real.pi rw [div_lt_iff₀ hNpos] exact mul_lt_mul_of_pos_left hm_lt h2pi_pos have hnonneg : 0 ≤ torusQuantizedWave N m i := by change 0 ≤ 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) positivity have hgt : -(2 * Real.pi) < torusQuantizedWave N m i := by linarith have hqzero : torusQuantizedWave N m i = 0 := (Real.cos_eq_one_iff_of_lt_of_lt hgt hlt).1 hcos have hval_real : ((m i).val : ℝ) = 0 := by change 2 * Real.pi * ((m i).val : ℝ) / (N : ℝ) = 0 at hqzero have hden : (N : ℝ) ≠ 0 := ne_of_gt hNpos have hnum : 2 * Real.pi * ((m i).val : ℝ) = 0 := by exact ((div_eq_zero_iff).1 hqzero).resolve_right hden exact (mul_eq_zero.mp hnum).resolve_left (ne_of_gt h2pi_pos) apply Fin.ext exact Nat.cast_eq_zero.mp hval_real apply hm funext i exact hcoord_zero iIt does not identify this finite periodic torus with the infinite three-dimensional space of everyday physics, nor with the finite open box used in other parts of the framework. symbol_torusQuantizedWave_ne_zero_of_ne_zero_mode · IndisputableMonolith/Foundation/PairKernelPeriodic3.lean