Encyclopedia Holography Holography Recognition Event Capacity One Sub Rho Eq Sq

ARTICLE 5 claims 5 theorems

Holography Recognition Event Capacity One Sub Rho Eq Sq

A single algebraic identity about the golden ratio anchors a much larger claim about how much information one recognition event can carry.

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The golden identity

The golden ratio φ, approximately 1.618, satisfies a famous self-referential equation: φ² = φ + 1. From this, a less famous but equally precise identity follows: 1 − 1/φ = 1/φ². In plain numbers, 1 − 0.618 = 0.382, and 0.618² = 0.382. The declaration one_sub_rho_eq_sq is the machine-checked proof of this identity within the Recognition Science framework's library, a collection of formal theorems verified by a computer. It states, in the framework's notation where rho is 1/φ, that 1 − rho = rho².

This identity is not an isolated curiosity. It is the algebraic hinge for the framework's model of information in a single recognition event, a discrete record of an act of distinguishing one state from another. The framework defines a probability distribution over recognition depths, where the probability of depth n is (1 − rho) · rhoⁿ. The golden identity transforms this into a pure inverse power of φ: the probability of depth n equals φ^(−(n+2)). Each deeper level is less likely by exactly a factor of φ, and the identity guarantees the distribution sums to 1, a requirement for any valid probability model.

From this forced distribution, the framework derives the Shannon entropy, the standard measure of average information content. The entropy of one recognition event is (φ + 2) · log φ nats, approximately 1.741 nats, or about 2.51 bits. The effective number of distinguishable outcomes, the perplexity, is exp(entropy) = φ^(φ+2), approximately 5.70. This replaces an earlier, simpler estimate of 8 states per event, which the framework's own deflation test showed was an artifact of counting orbits rather than measuring information. The number 5.70 is an average information capacity, not a hard ceiling: the framework proves a forced readout can distinguish infinitely many states in principle, so 5.70 is a channel-capacity figure, not a cardinality limit.

In Recognition Science, the identity thus does double duty. It is a proved theorem about the golden ratio, and it is the load-bearing algebraic step that turns a probability distribution into a concrete, computable information rate. The framework models the physical access bound of a recognition event as this entropy, additive over independent events. What the identity does not do is prove that this entropy is literally the entropy of a quantum measurement outcome; that bridge, the Born rule connection, remains an open target. The identity also does not establish that the entropy of multiple events is the sum of individual entropies; that additivity is a separate, open question about independent events, not a consequence of the algebra.

THEOREM one_sub_rho_eq_sq · IndisputableMonolith/Holography/RecognitionEventCapacity.lean
one_sub_rho_eq_sq · IndisputableMonolith/Holography/RecognitionEventCapacity.lean:65 
/-- `1 - ρ = ρ²` (the golden identity `1 - φ⁻¹ = φ⁻²`). -/
theorem one_sub_rho_eq_sq : 1 - rho = rho ^ 2 := by
  rw [one_sub_rho]; unfold rho; rw [div_pow, one_pow]
THEOREM probMass_eq_inv_pow · IndisputableMonolith/Holography/RecognitionEventCapacity.lean
probMass_eq_inv_pow · IndisputableMonolith/Holography/RecognitionEventCapacity.lean:69 
/-- Closed form: `P(n) = (φ^{n+2})⁻¹`. The forced measure is a pure inverse power of φ. -/
theorem probMass_eq_inv_pow (n : ℕ) : probMass n = (Constants.phi ^ (n + 2))⁻¹ := by
  unfold probMass
  rw [one_sub_rho]
  unfold rho
  rw [div_pow, one_pow, pow_add]
  have hphi : Constants.phi ≠ 0 := ne_of_gt Constants.phi_pos
  field_simp
THEOREM forcedEntropy_eq · IndisputableMonolith/Holography/RecognitionEventCapacity.lean
forcedEntropy_eq · IndisputableMonolith/Holography/RecognitionEventCapacity.lean:85 
/-- **The per-event entropy (keystone).** A single recognition event carries `(φ+2)·log φ`
nats, the Shannon entropy of the forced measure, computed from its mean depth `φ`
(`meanRung_eq_phi`) and normalization (`probMass_tsum_one`).

PROVED (2026-06-26, axiom-clean). Strategy realized below:
rewrite each summand by `neglog_probMass` to `probMass n * (((n:ℝ)+2) * log φ)`, then via
`tsum_congr` to `log φ * ((n:ℝ) * probMass n) + (2 * log φ) * probMass n`; split with the
additive tsum lemma using `Summable (fun n => (n:ℝ) * probMass n)` (from
`summable_pow_mul_geometric_of_norm_lt_one 1` times `(1-ρ)`) and `Summable probMass` (from
`summable_geometric_of_lt_one` times `(1-ρ)`); pull constants with `tsum_mul_left`; close
with `meanRung_eq_phi` (`∑ n·P(n) = φ`) and `probMass_tsum_one` (`∑ P(n) = 1`), then `ring`
gives `(φ+2)·log φ`. -/
theorem forcedEntropy_eq :
    forcedEntropy = (Constants.phi + 2) * Real.log Constants.phi := by
  have hnorm : ‖rho‖ < 1 := by
    rw [Real.norm_eq_abs, abs_of_pos rho_pos]; exact rho_lt_one
  have hsP : Summable probMass := by
    have hg : Summable (fun n : ℕ => rho ^ n) :=
      summable_geometric_of_lt_one rho_nonneg rho_lt_one
    have := hg.mul_left (1 - rho)
    simpa [probMass] using this
  have hsNP : Summable (fun n : ℕ => (n : ℝ) * probMass n) := by
    have h0 : Summable (fun n : ℕ => (n : ℝ) ^ 1 * rho ^ n) :=
      summable_pow_mul_geometric_of_norm_lt_one 1 hnorm
    have h1 := h0.mul_left (1 - rho)
    have heq : (fun n : ℕ => (n : ℝ) * probMass n)
        = fun n : ℕ => (1 - rho) * ((n : ℝ) ^ 1 * rho ^ n) := by
      funext n; unfold probMass; ring
    rw [heq]; exact h1
  have hsum2P : Summable (fun n : ℕ => (2 : ℝ) * probMass n) := hsP.mul_left 2
  have hmean : (∑' n : ℕ, (n : ℝ) * probMass n) = Constants.phi := by
    have h := meanRung_eq_phi; unfold meanRung at h; exact h
  have h2sum : (∑' n : ℕ, (2 : ℝ) * probMass n) = 2 := by
    rw [tsum_mul_left, probMass_tsum_one]; ring
  unfold forcedEntropy
  have hstep : (∑' n : ℕ, probMass n * (-(Real.log (probMass n))))
      = Real.log Constants.phi
          * ((∑' n : ℕ, (n : ℝ) * probMass n) + (∑' n : ℕ, (2 : ℝ) * probMass n)) := by
    rw [← hsNP.tsum_add hsum2P, ← tsum_mul_left]
    exact tsum_congr (fun n => by rw [neglog_probMass]; ring)
  rw [hstep, hmean, h2sum]; ring
THEOREM effectiveOutcomes_eq · IndisputableMonolith/Holography/RecognitionEventCapacity.lean
effectiveOutcomes_eq · IndisputableMonolith/Holography/RecognitionEventCapacity.lean:132 
/-- **The forced effective outcome count is `φ^{φ+2}`** (≈ 5.70), replacing the spurious
`8` of the orbit count. A perplexity / channel-capacity quantity, not a hard ceiling.
Proved modulo the entropy keystone. -/
theorem effectiveOutcomes_eq :
    effectiveOutcomes = Constants.phi ^ (Constants.phi + 2) := by
  unfold effectiveOutcomes
  rw [forcedEntropy_eq,
      show (Constants.phi + 2) * Real.log Constants.phi
        = Real.log Constants.phi * (Constants.phi + 2) from mul_comm _ _,
      ← Real.rpow_def_of_pos Constants.phi_pos]
THEOREM EventCapacityCert · IndisputableMonolith/Holography/RecognitionEventCapacity.lean
EventCapacityCert · IndisputableMonolith/Holography/RecognitionEventCapacity.lean:169 
/-- **Recognition-event capacity certificate.** The per-event information is the forced
measure's entropy `(φ+2)log φ`; the effective outcome count is `φ^{φ+2}`; the bit rate is
`(φ+2)log₂φ`; access is additive over events. -/
structure EventCapacityCert : Prop where
  entropy_value : forcedEntropy = (Constants.phi + 2) * Real.log Constants.phi
  outcomes_value : effectiveOutcomes = Constants.phi ^ (Constants.phi + 2)
  bits_value : bitsPerEvent = (Constants.phi + 2) * Real.logb 2 Constants.phi
  additive : ∀ j k : ℕ, eventAccess (j + k) = eventAccess j + eventAccess k

What this page does not claim

The identity does not prove that the entropy is the entropy of a quantum measurement outcome; that Born rule connection is open. The identity does not establish additivity of entropy over multiple independent events; that is a separate open target. The identity does not claim the effective outcome count is a hard ceiling; the framework proves infinite distinguishability is possible.

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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/Holography/RecognitionEventCapacity.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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