effectiveOutcomes_eq
plain-language theorem explainer
The effective number of outcomes per recognition event equals φ^{φ+2} ≈ 5.70. This is the perplexity (exp of Shannon entropy) of the T9 forced measure, not a hard cardinality. Anyone citing the rebuilt holographic access bound or the EventCapacityCert package needs this identity. The proof unfolds the definition and rewrites via the closed-form forced entropy plus the real power definition.
Claim. The effective outcome count of one recognition event equals $\varphi^{\varphi+2}$, where $\varphi$ is the golden ratio. Equivalently, $\mathrm{effectiveOutcomes} = \exp(\mathrm{forcedEntropy})$ with $\mathrm{forcedEntropy} = (\varphi+2)\log\varphi$.
background
This module rebuilds the holographic access bound after the deflation test killed the naive "eight states per site" orbit count. Physical per-carrier content is continuum-valued, so the right bound is forced-measurement outcome quantization: a recognition event resolves into discrete $\varphi^{-n}$-weighted outcomes from the T9 forced measure (Foundation.MeasureForcing), with $p_n = (1-\varphi^{-1})\varphi^{-n}$, mean depth $\varphi$, and partition $\varphi^2$.
The Shannon entropy of that measure is the forced entropy $(\varphi+2)\log\varphi$ nats per event. The effective outcome count is the perplexity $\exp$ of that entropy. It is a channel-capacity / average-information quantity, not a zero-error distinguishability ceiling: a forced readout can hard-distinguish infinitely many states, so $\varphi^{\varphi+2}\approx 5.70$ replaces the spurious hard ceiling of 8 without claiming injection bounds.
The identity sits on the already-proved keystone that forced entropy equals $(\varphi+2)\log\varphi$, obtained from the geometric-series entropy computation via mean rung $\varphi$ and unit total mass.
proof idea
Term-mode rewrite chain. Unfold the definition of the effective outcome count (which is $\exp$ of forced entropy). Rewrite the entropy via its closed form $(\varphi+2)\log\varphi$. Commute the product so the expression matches $\log\varphi\cdot(\varphi+2)$. Apply the real power identity $a^b = \exp(b\log a)$ for $a=\varphi>0$, yielding $\varphi^{\varphi+2}$.
why it matters
This is the central numerical identity of the rebuilt access law: the physical per-event content is $\varphi^{\varphi+2}$ effective outcomes, not the orbit count 8. It feeds directly into eventCapacityCert, which packages the entropy value, this outcome identity, the bit-rate $(\varphi+2)\log_2\varphi$, and additivity of event access over $k$ events.
In the Recognition framework it closes the capacity side of the holography story after ThetaAccessDeflation: average information per event is forced by the T9 measure (linked to the forcing chain and $\varphi$ from T5–T6), with no free alphabet. Framing remains capacity, not hard ceiling. The documented open item is the Born bridge from this entropy to a forced-measurement distribution on recognition Hilbert space; the present theorem does not touch that bridge.
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