Quantum mechanics mapping (completion note: lock standard-QM elements as a mapping table)

Quantum mechanics mapping: The wavefunction enters as a mapped object, not an imported axiom. Collapse becomes a counting protocol with a Gate, not a mystery. Let the lattice dimension be d∈N, and define the lattice-node set.

The wavefunction enters as a mapped object, not an imported axiom. Collapse becomes a counting protocol with a Gate, not a mystery. 2π/3.

15.1 Lattice dispersion / plane-wave solutions leftrightarrow wavefunction (state space) mapping

15.1.1 Base set and scaling (definition)

Let the lattice dimension be d∈N, and define the lattice-node set by

$\begin{equation} \mathcal{L}:=\mathbb{Z}^d \end{equation}$

Using the realization length a>0 and the realization time step Δ t>0 (their units are locked in the Realization chapter), define the real-space coordinate and the physical time corresponding to node n∈L and tick k∈Z as

\begin{equation} \mathbf{x}(\mathbf{n}) := a\,\mathbf{n},\qquad t(k):=k\,\Delta t. \end{equation}

15.1.2 Complex state field from the event log (definition)

For each node n and tick k, define the 3-sector event indicators by

\begin{equation} E_s(\mathbf{n},k)\in\{0,1\},\qquad s\in\{1,2,3\}. \end{equation}

Let the observation (aggregation) window have length M∈N and start tick k₀∈Z, and define

\begin{equation} W(k_0,M):=\{k_0,k_0+1,\dots,k_0+M-1\}. \end{equation}

Define the sector-wise event counts (log aggregates) within the window as

\begin{equation} N_s(\mathbf{n};k_0,M):=\sum_{k\in W(k_0,M)}E_s(\mathbf{n},k). \end{equation}

Fix the sector angles by

\begin{equation} \theta_s:=\frac{2\pi}{3}(s-1),\qquad s=1,2,3. \end{equation}

Define the node-wise 3-sector composite phasor by

\begin{equation} Z(\mathbf{n};k_0,M) := \sum_{s=1}^{3} N_s(\mathbf{n};k_0,M)\,e^{i\theta_s} \in\mathbb{C}. \end{equation}

Also define the total event count and the event density by

\begin{equation} N(\mathbf{n};k_0,M):=\sum_{s=1}^{3}N_s(\mathbf{n};k_0,M),\qquad \rho(\mathbf{n};k_0,M):=\frac{1}{M}\,N(\mathbf{n};k_0,M). \end{equation}

Define the phase (principal value) of the composite phasor by

\begin{equation} \varphi(\mathbf{n};k_0,M):=\arg Z(\mathbf{n};k_0,M)\in(-\pi,\pi]. \end{equation}

Then define the node-wise complex state field (complex amplitude) by

\begin{equation} \psi(\mathbf{n};k_0,M):=\sqrt{\rho(\mathbf{n};k_0,M)}\,e^{i\varphi(\mathbf{n};k_0,M)}. \end{equation}

Therefore

\begin{equation} |\psi(\mathbf{n};k_0,M)|^2=\rho(\mathbf{n};k_0,M) \end{equation}

holds identically. The definitions above are operational: they construct a state field directly from event-log aggregates and do not invoke axioms from any external theory.

15.1.3 State space (definition) and inner product (definition)

Define the complex function space

\begin{equation} \mathcal{S}:=\left\{\psi:\mathcal{L}\to\mathbb{C}\ \bigg|\ \sum_{\mathbf{n}\in\mathcal{L}}|\psi(\mathbf{n})|^2<\infty\right\} \end{equation}

as the state space. Define the inner product on S by

\begin{equation} \langle \phi,\psi\rangle := \sum_{\mathbf{n}\in\mathcal{L}}\phi(\mathbf{n})^{*}\psi(\mathbf{n}) \end{equation}

and the norm by |ψ|:=√(⟨ψ,ψ⟩). A normalized state is defined by

\begin{equation} \widehat{\psi}:=\frac{\psi}{\|\psi\|}\qquad(\psi\neq 0). \end{equation}

15.1.4 Discrete Fourier expansion and wave-number space (definition)

Define the d-dimensional Brillouin zone by

\begin{equation} \mathcal{B}:=[-\pi,\pi)^d. \end{equation}

For sufficiently fast-decaying states (or for a finite lattice with periodic boundary conditions), define the discrete Fourier transform by

\begin{equation} \widetilde{\psi}(\boldsymbol{\kappa}) := \sum_{\mathbf{n}\in\mathcal{L}} \psi(\mathbf{n})\,e^{-i\boldsymbol{\kappa}\cdot\mathbf{n}}, \qquad \boldsymbol{\kappa}\in\mathcal{B}. \end{equation}

Here boldsymbolκ is the dimensionless lattice wave number. Define the real-space wave vector (with dimension L⁻¹) by

\begin{equation} \mathbf{k}:=\frac{1}{a}\boldsymbol{\kappa}. \end{equation}

15.1.5 One-tick evolution operator, dispersion, and plane waves (definition)

Assume a regime in which tick evolution is linear and translation-invariant (i.e., the same rule under the same closure stack). Define the one-tick evolution operator by

\begin{equation} \psi(\cdot,k+1)=\mathcal{U}\,\psi(\cdot,k). \end{equation}

Translation invariance means that for every m∈L, with the translation operator T_m defined by

\begin{equation} (\mathcal{T}_{\mathbf{m}}\psi)(\mathbf{n}) := \psi(\mathbf{n}+\mathbf{m}), \end{equation}

one has

\begin{equation} \mathcal{U}\,\mathcal{T}_{\mathbf{m}}=\mathcal{T}_{\mathbf{m}}\,\mathcal{U} \qquad(\forall\,\mathbf{m}\in\mathcal{L}). \end{equation}

Under this condition, define the plane waves (lattice harmonic modes) by

\begin{equation} \phi_{\boldsymbol{\kappa}}(\mathbf{n}) := e^{i\boldsymbol{\kappa}\cdot\mathbf{n}}, \qquad \boldsymbol{\kappa}\in\mathcal{B}. \end{equation}

Then φ_boldsymbolκ can be an eigenmode of U. Define its eigenvalue in phase form by

\begin{equation} \mathcal{U}\,\phi_{\boldsymbol{\kappa}} = e^{-i\Omega(\boldsymbol{\kappa})}\,\phi_{\boldsymbol{\kappa}}, \qquad \Omega(\boldsymbol{\kappa})\in(-\pi,\pi]. \end{equation}

Define the lattice dispersion (angular frequency) by

\begin{equation} \omega(\boldsymbol{\kappa}) := \frac{1}{\Delta t}\,\Omega(\boldsymbol{\kappa}). \end{equation}

Therefore, a single-mode plane-wave solution can be written as

\begin{equation} \psi_{\boldsymbol{\kappa}}(\mathbf{n},k) = A\, e^{i\boldsymbol{\kappa}\cdot\mathbf{n}} e^{-i\Omega(\boldsymbol{\kappa})k} = A\, \exp\!\Big(i\boldsymbol{\kappa}\cdot\mathbf{n}-i\omega(\boldsymbol{\kappa})t(k)\Big). \end{equation}

This follows directly from the definitions (S15_01_Udef)–(S15_01_dispersion_phase).

15.1.6 Mapping table to external text (standard QM) (target-text lock)

In this section, “standard QM” is treated as an external target text and is not used as a derivation basis. The following table fixes only a notation mapping to the standard-QM notation (target text).

VP/lattice–event system (definition)	Standard-QM target text (term)	Mapping rule (locked)
Node n∈L	position variable x	x=an
Tick k∈Z	time variable t	t=kΔ t
Complex state field ψ(n;k₀,M)	wavefunction ψ(x,t)	ψ(x(n),t(k))leftrightarrow ψ(n,k)
Dimensionless wave number boldsymbolκ	wave vector k	k=boldsymbolκ/a
Eigenphase Ω(boldsymbolκ)	angular frequency ω	ω=Ω/Δ t
Mode exp(iboldsymbolκ·n-iΩ k)	plane wave exp(ik·x-iω t)	(S15_01_xt), (S15_01_kmap), (S15_01_omega)

15.1.7 Declared scope: allowed vs forbidden uses (locked)

Allowed uses (internal computations in this document)

The complex state field ψ defined by (S15_01_psi_def) and the identity (S15_01_abs2_rho).
The Fourier expansion (S15_01_FT) and the eigenphase (dispersion) definition (S15_01_dispersion_phase).
The plane-wave mode representation (S15_01_plane_solution) (mode basis in a translation-invariant regime).

Forbidden uses (may not be used as derivation grounds; target-text only)

Using the axioms of standard QM (wavefunction axiom, operator axiom, projection postulate, etc.) as justification.
Adopting specific continuum equations from standard QM (target text) in place of this document's internal closures/axioms.

15.2 Rewriting the Born rule, measurement, and collapse as event frequency and gates (link to observation protocols)

15.2.1 Partition of an observation module (definition)

Let the lattice region referenced by the observation module be a finite set Λ⊂L, and define the set of output labels by

\begin{equation} \mathcal{M}:=\{1,2,\dots,M_{\mathrm{out}}\}. \end{equation}

Define a partition map that splits Λ into mutually disjoint subregions by

\begin{equation} \Pi:\Lambda\to\mathcal{M}, \qquad \Omega_m:=\Pi^{-1}(m), \qquad \Omega_m\cap\Omega_{m'}=\varnothing\ (m\neq m'),\quad \bigcup_{m\in\mathcal{M}}\Omega_m=\Lambda. \end{equation}

The family Ωₘ defines the outcome channels (output bins).

15.2.2 Event counts and outcome frequencies (definition)

For an observation window W(k₀,M), define the event count of outcome channel m by

\begin{equation} C_m(k_0,M) := \sum_{\mathbf{n}\in\Omega_m}\sum_{k\in W(k_0,M)}\sum_{s=1}^{3}E_s(\mathbf{n},k) = \sum_{\mathbf{n}\in\Omega_m} N(\mathbf{n};k_0,M). \end{equation}

The total event count is C_tot:=Σ_m∈MCₘ. Define the outcome frequency (normalized event frequency) by

\begin{equation} P_m(k_0,M) := \frac{C_m(k_0,M)}{C_{\mathrm{tot}}(k_0,M)} \qquad (C_{\mathrm{tot}}>0). \end{equation}

Thus Pₘ is an outcome frequency computed directly from the event log.

15.2.3 Equivalent expression in terms of the complex state field (definition)

From the definition (S15_01_psi_def) and identity (S15_01_abs2_rho) in §15.1, for each channel one has

\begin{equation} C_m(k_0,M) = M\sum_{\mathbf{n}\in\Omega_m}\rho(\mathbf{n};k_0,M) = M\sum_{\mathbf{n}\in\Omega_m}|\psi(\mathbf{n};k_0,M)|^2. \end{equation}

Therefore (S15_02_Pm) can be rewritten exactly as

\begin{equation} P_m(k_0,M) = \frac{\sum_{\mathbf{n}\in\Omega_m}|\psi(\mathbf{n};k_0,M)|^2}{\sum_{\mathbf{n}\in\Lambda}|\psi(\mathbf{n};k_0,M)|^2} \qquad \left(\sum_{\mathbf{n}\in\Lambda}|\psi(\mathbf{n};k_0,M)|^2>0\right). \end{equation}

Here (S15_02_Pm_bornform) is not derived from the foundations of standard QM. It is an identity that holds equivalently by construction from the event-log definition and (S15_01_psi_def).

15.2.4 Gate decision (condition for a measurement event; definition)

A measurement is an event that creates a log entry “some m was selected.” To formalize this, define an outcome-selection rule (gate) as

$\begin{equation} m^\star := \operatorname*{arg\,max}_{m\in\mathcal{M}} \ \mathcal{G}_m\big(C_m(k_0,M);\Theta_m\big). \end{equation}$

Here Gₘ(·;Θₘ) is the gate function for channel m, and Θₘ is a pre-registered set of threshold/decision parameters. Define the condition for producing a measurement log by

$\begin{equation} \mathrm{PASS}_{\mathrm{meas}} :\Longleftrightarrow \left[ \exists\,m^\star\in\mathcal{M}\ \text{s.t.}\ \mathcal{G}_{m^\star}\big(C_{m^\star}(k_0,M);\Theta_{m^\star}\big)=1 \right]. \end{equation}$

If PASS_meas does not hold, the protocol does not generate a measurement log (no outcome selection).

15.2.5 Operational definition of collapse (conditional renormalization) (definition)

When the measurement outcome m^* is fixed in the log (i.e., when (S15_02_PASS_meas) holds), define the following mask operator (channel filter) on the state field ψ(·;k₀,M) from the same window:

\begin{equation} (\mathcal{P}_{m}\psi)(\mathbf{n}) := \begin{cases} \psi(\mathbf{n}), & \mathbf{n}\in\Omega_m,\\ 0, & \mathbf{n}\in\Lambda\setminus\Omega_m, \end{cases} \qquad \mathbf{n}\in\Lambda. \end{equation}

Then define the post-measurement state by conditional renormalization:

$\begin{equation} \widehat{\psi}_{\mathrm{post}} := \frac{\mathcal{P}_{m^\star}\widehat{\psi}_{\mathrm{pre}}}{\left\|\mathcal{P}_{m^\star}\widehat{\psi}_{\mathrm{pre}}\right\|} \qquad \left(\left\|\mathcal{P}_{m^\star}\widehat{\psi}_{\mathrm{pre}}\right\|>0\right). \end{equation}$

Here ψ_pre is the normalized state immediately before measurement (computed under the same protocol). Equation (S15_02_collapse) is a convention for re-expressing the state after the event log is fixed, and cannot be applied without a gate pass.

15.2.6 Mapping table to the standard-QM target text (target-text lock)

In this subsection, standard-QM terms (Born rule, measurement, collapse) are treated as target text. The following table fixes only the notation mapping based on (S15_02_Pm_bornform) and (S15_02_collapse).

VP/event–gate system (definition)	Standard-QM target text (term)	Mapping rule (locked)
Outcome frequency Pₘ	probability pₘ	pₘleftrightarrow Pₘ
Σ_(Ωₘ)\|ψ\|²/Σ_(Λ)\|ψ\|²	“\|ψ\|² rule”	(S15_02_Pm_bornform)
Gate pass PASS_meas	condition for measurement to occur	(S15_02_PASS_meas)
Conditional renormalization (S15_02_collapse)	projection (collapse) notation	notation mapping only

15.2.7 Declared scope: allowed vs forbidden uses (locked)

Allowed uses (internal computations in this document)

Log-based computation of event counts Cₘ and outcome frequencies Pₘ ((S15_02_Cm)–(S15_02_Pm)).
The identity (S15_02_Pm_bornform) (the ψ-form).
Conditional renormalization (S15_02_collapse) after a gate pass.

Forbidden uses (may not be used as derivation grounds; target-text only)

Adopting the standard-QM “measurement postulate” in place of the gate definition in this document.
Back-injecting the standard-QM “probability interpretation” as the justification for the event-frequency definition.

15.3 Operators, commutation relations, uncertainty, and the spin/statistics link (3-sector): separating theorem / hypothesis / validation

15.3.1 Operational definition of observables (theorem)

Define a linear operator O:S→S on the state space S as an observable operator. For a normalized state ψ, define its expectation and variance by

\begin{equation} \langle \mathcal{O}\rangle_{\widehat{\psi}} := \langle \widehat{\psi},\mathcal{O}\widehat{\psi}\rangle, \qquad (\Delta \mathcal{O})^2_{\widehat{\psi}} := \left\langle \widehat{\psi},(\mathcal{O}-\langle \mathcal{O}\rangle_{\widehat{\psi}})^2\widehat{\psi}\right\rangle. \end{equation}

Equation (S15_03_mean_var) is a purely operator-algebraic definition determined by the inner product (S15_01_inner).

15.3.2 Basic operators: position, translation, and finite differences (theorem)

For coordinate component j∈1,…,d, define the position (multiplication) operator X_j by

\begin{equation} (\mathcal{X}_j\psi)(\mathbf{n}) := x_j(\mathbf{n})\,\psi(\mathbf{n}) = a\,n_j\,\psi(\mathbf{n}). \end{equation}

For the unit lattice vector e_j, define the translation (shift) operator T_j by

\begin{equation} (\mathcal{T}_j\psi)(\mathbf{n}) := \psi(\mathbf{n}+\mathbf{e}_j), \qquad (\mathcal{T}_j^{-1}\psi)(\mathbf{n}) := \psi(\mathbf{n}-\mathbf{e}_j). \end{equation}

The commutation relation (exact identity) follows by direct computation:

\begin{align} (\mathcal{X}_j\mathcal{T}_j\psi)(\mathbf{n}) &= a\,n_j\,\psi(\mathbf{n}+\mathbf{e}_j), \notag\\ (\mathcal{T}_j\mathcal{X}_j\psi)(\mathbf{n}) &= a\,(n_j+1)\,\psi(\mathbf{n}+\mathbf{e}_j), \notag\\ \Rightarrow\quad \big([\mathcal{X}_j,\mathcal{T}_j]\psi\big)(\mathbf{n}) &= (\mathcal{X}_j\mathcal{T}_j-\mathcal{T}_j\mathcal{X}_j)\psi(\mathbf{n}) = -a\,\psi(\mathbf{n}+\mathbf{e}_j) = -a\,(\mathcal{T}_j\psi)(\mathbf{n}). \end{align}

Hence

\begin{equation} [\mathcal{X}_j,\mathcal{T}_j]=-a\,\mathcal{T}_j, \qquad [\mathcal{X}_j,\mathcal{T}_j^{-1}]=+a\,\mathcal{T}_j^{-1} \end{equation}

holds exactly.

15.3.3 Momentum-like operator and commutation relation (theorem + hypothesis)

(Theorem) Symmetric-difference generator

Define the symmetric-difference operator D_j by

\begin{equation} \mathcal{D}_j := \frac{1}{2a}\big(\mathcal{T}_j-\mathcal{T}_j^{-1}\big). \end{equation}

(Hypothesis H-Pmap) Mapping to the “momentum” notation in the standard-QM target text

Define the constant

\begin{equation} \hbar_{\mathrm{map}}:=\frac{h}{2\pi} \end{equation}

as a mapping constant (derived from the realization input h). Define the momentum-like operator P_j by

\begin{equation} \mathcal{P}_j := \frac{\hbar_{\mathrm{map}}}{i}\,\mathcal{D}_j = \frac{\hbar_{\mathrm{map}}}{2ia}\big(\mathcal{T}_j-\mathcal{T}_j^{-1}\big). \end{equation}

By (S15_03_comm_XT2), P_j satisfies the exact commutation relation

\begin{align} [\mathcal{X}_j,\mathcal{P}_j] &= \frac{\hbar_{\mathrm{map}}}{2ia}\Big([\mathcal{X}_j,\mathcal{T}_j]-[\mathcal{X}_j,\mathcal{T}_j^{-1}]\Big) \notag\\ &= \frac{\hbar_{\mathrm{map}}}{2ia}\Big(-a\mathcal{T}_j-a\mathcal{T}_j^{-1}\Big) = \frac{i\hbar_{\mathrm{map}}}{2}\big(\mathcal{T}_j+\mathcal{T}_j^{-1}\big). \end{align}

For the plane-wave mode (S15_01_planewave),

\begin{equation} \mathcal{P}_j\,\phi_{\boldsymbol{\kappa}} = \left(\frac{\hbar_{\mathrm{map}}}{a}\sin\kappa_j\right)\phi_{\boldsymbol{\kappa}} \end{equation}

holds. Therefore, in the small-κ_j regime (slow-variation regime) where sinκ_j≃ κ_j,

\begin{equation} \mathcal{P}_j\,\phi_{\boldsymbol{\kappa}} \simeq \left(\hbar_{\mathrm{map}}\,\frac{\kappa_j}{a}\right)\phi_{\boldsymbol{\kappa}} = \left(\hbar_{\mathrm{map}}\,k_j\right)\phi_{\boldsymbol{\kappa}} \qquad(\kappa_j\to 0). \end{equation}

We map this approximation to the standard-QM target-text notation p=ħ k only as a translation rule, and it must not be used as a derivation basis.

15.3.4 Uncertainty inequality (theorem; derivation as a mathematical theorem)

Derive an operator uncertainty inequality using the Cauchy–Schwarz inequality in an inner-product space:

\begin{equation} |\langle u,v\rangle|^2\le \langle u,u\rangle\,\langle v,v\rangle. \end{equation}

For a normalized state ψ and two operators A,B, define

\begin{equation} \delta\mathcal{A}:=\mathcal{A}-\langle\mathcal{A}\rangle_{\widehat{\psi}},\qquad \delta\mathcal{B}:=\mathcal{B}-\langle\mathcal{B}\rangle_{\widehat{\psi}}. \end{equation}

Let

\begin{equation} u:=\delta\mathcal{A}\,\widehat{\psi},\qquad v:=\delta\mathcal{B}\,\widehat{\psi}. \end{equation}

Then

\begin{equation} \langle u,u\rangle = \langle \widehat{\psi},(\delta\mathcal{A})^2\widehat{\psi}\rangle = (\Delta\mathcal{A})^2_{\widehat{\psi}}, \qquad \langle v,v\rangle = (\Delta\mathcal{B})^2_{\widehat{\psi}}. \end{equation}

By (S15_03_CS),

\begin{equation} (\Delta\mathcal{A})^2_{\widehat{\psi}}\,(\Delta\mathcal{B})^2_{\widehat{\psi}} \ge |\langle u,v\rangle|^2 = \left|\left\langle \widehat{\psi},\delta\mathcal{A}\,\delta\mathcal{B}\,\widehat{\psi}\right\rangle\right|^2. \end{equation}

Let z:=⟨ ψ,δAδBψ⟩∈C and decompose

\begin{equation} z = \frac{1}{2}\langle \widehat{\psi},\{\delta\mathcal{A},\delta\mathcal{B}\}\widehat{\psi}\rangle + \frac{1}{2}\langle \widehat{\psi},[\delta\mathcal{A},\delta\mathcal{B}]\widehat{\psi}\rangle. \end{equation}

Here X,Y:=XY+YX is the anticommutator and [X,Y]:=XY-YX is the commutator. Therefore

\begin{equation} |z|^2 = \left(\Re z\right)^2+\left(\Im z\right)^2 \ge \left(\Im z\right)^2. \end{equation}

Also

\begin{equation} \Im z = \frac{1}{2i}\left(z-z^{*}\right) = \frac{1}{2i}\left\langle \widehat{\psi},\left(\delta\mathcal{A}\delta\mathcal{B}-\delta\mathcal{B}\delta\mathcal{A}\right)\widehat{\psi}\right\rangle = \frac{1}{2i}\langle \widehat{\psi},[\mathcal{A},\mathcal{B}]\widehat{\psi}\rangle. \end{equation}

Combining (S15_03_RS0)–(S15_03_Imz) yields

\begin{equation} \Delta\mathcal{A}\,\Delta\mathcal{B} \ge \frac{1}{2}\left|\left\langle \widehat{\psi},\frac{1}{i}[\mathcal{A},\mathcal{B}]\,\widehat{\psi}\right\rangle\right| = \frac{1}{2}\left|\langle [\mathcal{A},\mathcal{B}]\rangle_{\widehat{\psi}}\right|. \end{equation}

Equation (S15_03_uncertainty) is a mathematical theorem derived from the inner product and operator definitions.

15.3.5 Spin-like label from 3-sector structure (hypothesis) and statistics (hypothesis)

(Theorem) 3-sector probability vector

For node n and window W(k₀,M), define the sector fractions by

\begin{equation} p_s(\mathbf{n};k_0,M):=\frac{N_s(\mathbf{n};k_0,M)}{N(\mathbf{n};k_0,M)} \qquad(N(\mathbf{n};k_0,M)>0), \qquad \sum_{s=1}^{3}p_s=1. \end{equation}

Define the normalized sector phasor by

\begin{equation} u(\mathbf{n};k_0,M):=\sum_{s=1}^{3}p_s(\mathbf{n};k_0,M)\,e^{i\theta_s} \end{equation}

and its phase by

\begin{equation} \Phi(\mathbf{n};k_0,M):=\arg u(\mathbf{n};k_0,M)\in(-\pi,\pi]. \end{equation}

Thus Φ is an angular variable computed from the 3-sector occupancy fractions.

(Hypothesis H-S1) Definition of a two-valued spin-like label

Consider two consecutive windows (k₀,M) and (k₀+M,M) and define the phase increment by

\begin{equation} \Delta\Phi(\mathbf{n};k_0,M) := \mathrm{Wrap}_{(-\pi,\pi]}\!\Big(\Phi(\mathbf{n};k_0+M,M)-\Phi(\mathbf{n};k_0,M)\Big). \end{equation}

Here Wrap_((-π,π]) reduces a value into the interval (-π,π]. Define the spin-like label by

\begin{equation} \sigma(\mathbf{n};k_0,M) := \mathrm{sgn}\big(\Delta\Phi(\mathbf{n};k_0,M)\big) \in\{-1,0,+1\}. \end{equation}

Treat σ=0 as indeterminate (gate failure). Map only σ=±1 to the standard-QM target-text “two-valued spin projection” notation, and classify σ=0 as a state with no target-text counterpart.

(Lemma T-S1) Two-valuedness is forced in the cyclic-shift regime (added v0.5.0).

Suppose the 3-sector occupancies evolve, between consecutive windows, by a cyclic shift: pₛ(·;k₀+M,M)=p_(s-r)(·;k₀,M) with r∈+1,-1 (one sector step per window — the synchronized-rotation regime of §14.0, where the sign is the CW/CCW rotation sense, §14.3.2). With θₛ=2π(s-1)/3, the phasor obeys u(k₀+M)=Σₛ p_(s-r)e^(iθₛ)=e^(2π i r/3)u(k₀), hence ΔΦ=Wrap_((-π,π])(2π r/3)=± 2π/3 exactly: σ=r∈+1,-1 and σ=0 cannot occur. Spatial parity reverses the rotation sense (θₛ↦-θₛ, so Φ↦-Φ and σ↦-σ): the label set is a single parity orbit +1,-1, a forced Z₂. Consequence for the grading: the two-valuedness of the label is a theorem conditional on the cyclic-shift premise ([F]{} given the premise); what remains hypothesis (H-S1 proper, [H]{}) is (i) that physical one-quantum states are in this regime, and (ii) the identification of σ with the target-text spin-tfrac12 projection (magnitude ħ/2, the SU(2) double cover, the statistics link), which this lemma does not address. The logic-gaps audit graded the entry “hypothesis”; the corrected grading is two-valuedness: conditional theorem; spin identification: hypothesis.

(Hypothesis H-STAT1) Exclusive-occupancy rule and mapping to statistics

Define an exclusive-occupancy regime as a regime in which two objects with the same label cannot stably co-exist simultaneously on the same node (or the same slot in a contact graph). Declare the occupancy rule as

\begin{equation} \mathcal{N}(\mathbf{n})\in\{0,1\}\quad(\text{exclusive-occupancy regime}), \qquad \mathcal{N}(\mathbf{n})\in\{0,1,2,\dots\}\quad(\text{non-exclusive regime}). \end{equation}

Assign the standard-QM target-text terms “Fermi/Bose statistics” to the regime classification above only as a label mapping.

15.3.6 Validation items (judged only by gates; fixed list)

(V-OP) Operator-level agreement items

(V-OP1) Under the same protocol, whether the mode eigenvalues of P_j satisfy (S15_03_P_eig) (Fourier-mode fit Gate).
(V-OP2) In the slow-variation regime, whether the expectation of (S15_03_comm_XP_exact) converges to ⟨ tfraciħ_map2(T_j+T_j⁻¹)⟩≃ iħ_map (regime Gate).

(V-S) Spin-like label items

(V-S1) Within the same partition module, whether the event frequencies of the two channels σ=±1 match the predicted frequencies computed from (S15_02_Pm) (two-channel Gate).
(V-S2) Whether the occurrence rate of σ=0 is below a pre-registered upper bound (indeterminacy upper-bound Gate).

(V-STAT) Statistics-regime items

(V-STAT1) In the exclusive-occupancy regime, whether the simultaneous-occupancy violation log count is 0 (exclusion-violation Gate).
(V-STAT2) In the non-exclusive regime, whether the multi-occupancy distribution satisfies the pre-registered sampling protocol (sampling Gate).

15.4 Standard Model boundary: share (observational numbers) vs ban (gauge dynamics), and scope declaration

15.4.1 Form of the boundary declaration (definition)

In this section, “Standard Model” and “standard QM” are treated as external target texts. The role of this chapter is to provide mapping tables that translate target-text terms/symbols into this document's operational definitions (LOCK, Gate, event log). It provides no justification and no derivational support. Accordingly, fix the following two categories:

$\begin{equation} \mathcal{C}_{\mathrm{share}}:\ \text{share (observational numbers / reporting units)}, \qquad \mathcal{C}_{\mathrm{ban}}:\ \text{ban (target-text dynamics / axioms / structure)}. \end{equation}$

15.4.2 Shared items Cₛhare (observational numbers and reporting units; fixed list)

The following items are shared only as “reported quantities” or “reporting units.” Sharing means unit conversion and table-based correspondence; it does not mean adopting the target text's theoretical structure.

(SH1) The choice of reporting units for mass, length, time, energy, etc. (SI or the document's realization units).
(SH2) The numerical values themselves when given observationally (e.g., mass ratios, length ratios, frequency ratios).
(SH3) Labels of reported quantities referred to by terms such as “charge,” “spin,” and “probability.”

15.4.3 Banned items C_ban (gauge dynamics and axiom structure; fixed list)

The following items, even if present in the target text, are not permitted in the derivation chain of this document.

(BN1) Target-text dynamical generators such as gauge fields, gauge symmetry, Lagrangians, and action integrals.
(BN2) Internal target-text construction principles such as renormalization, perturbation expansions, and vacuum structure.
(BN3) Treating the axioms of standard QM (state axiom, measurement axiom, operator axiom) as grounds.
(BN4) Adopting target-text continuum equations (e.g., a specific wave equation) in place of this document's closures.

15.4.4 Mapping table to the standard-QM/Standard-Model target text (including banned scope; fixed)

This document (operational definitions / outputs)	Target text (term)	Handling (share / ban)
Event log Eₛ, counts Nₛ, frequencies Pₘ	probability p	share: report item; ban as justification
Complex state field ψ (event density + phase construction)	wavefunction ψ	share: notation mapping; ban as axiom
Gate PASS and conditional renormalization	measurement/collapse	share: notation mapping; ban as axiom
Operators X,T,P and commutators	operators/commutators	share: mapping table; ban target-text axioms
3-sector phase-increment label σ	spin (projection)	share: label mapping; ban dynamics structure
Exclusive/non-exclusive occupancy regimes	statistics (Fermi/Bose)	share: label mapping; ban target-text principle

15.4.5 Effective scope (regime declaration; locked)

The mapping of this chapter is defined only in the following regimes:

(R1) A regime where an event log exists (C_tot>0) and a state field ψ can be constructed by (S15_01_psi_def).
(R2) A regime where the observation-module partition Π, the gate functions Gₘ, and the thresholds Θₘ are pre-registered.
(R3) If an operator mapping is claimed (in particular, the target-text mapping of P_j), a slow-variation regime (small |boldsymbolκ|) must be separately certified by a Gate.

If these regime conditions do not hold, no conclusions may be produced using the target-text mapping tables.

15.5 Blackbody radiation: the energy quantum reinterpreted as a lattice-stiffness threshold

This section does not derive energy quantization from a deeper principle; it reinterprets the quantum as the cost of crossing a rigidity-shell barrier, ε_b(ν)=h_VPν (the Planck hypothesis stated in lattice language, with h_VP calibrated to data; grade CALIB, §16.4). Given that reinterpretation, the spectrum shape follows from standard Bose–Einstein statistics over the rigidity-shell modes and the 3D mode count; the contribution here is conceptual (the quantum as lattice stiffness), not a derivation of h or of quantization. (Planck's original paper: M. Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Annalen der Physik 309(3), 553–563 (1901). DOI: 10.1002/andp.19013090310.)

15.5.1 Geometry of mode count (statement): g(ν)∝ ν²

In a 3D cavity, the number of wave modes is given by the volume of a spherical shell in k-space. As a function of frequency ν, the mode density per unit volume scales geometrically as

\begin{equation} g(\nu)\,d\nu \propto \nu^{2}\,d\nu \end{equation}

(continuum geometry). Including the usual electromagnetic prefactors gives g(ν)=8πν²/c_ref³, but the key point here is that the ν² scaling comes from pure geometry.

15.5.2 Rigidity-shell barrier and the action constant h_VP (definition)

In VP theory, “softness” is not a property of the particle itself but the result of structural events (§3.2.1). For a high-frequency mode to exist, the lattice rigidity shell must be periodically deformed. Under linear elastic response, the strain rate ε is proportional to frequency. Hence the minimum barrier energy ε_b required to penetrate the shell (or to cross the critical deformation once) satisfies

\begin{equation} \epsilon_b(\nu) \propto \dot{\varepsilon} \propto \nu. \end{equation}

Fix the proportionality constant in action (energy×time) units and define

\begin{equation} \boxed{\epsilon_b(\nu) := h_{\mathrm{VP}}\,\nu} \end{equation}

Here h_VP is not an arbitrary “unknown constant”; it is the minimum action (structural resistance coefficient) required for the VP shell to allow one cycle of deformation.

15.5.3 Integer event count n and the emergence of (e^x-1)⁻¹ (derivation)

The key point is that “excitation” is not continuous but is tallied as the number of barrier-crossing events. In a mode of frequency ν, one barrier crossing costs ε_b(ν)=h_VPν; therefore the accumulated energy after n crossing events is

\begin{equation} E_n(\nu) = n\,h_{\mathrm{VP}}\,\nu,\qquad n\in\{0,1,2,\dots\}. \end{equation}

This is not “postulating E=n hν” but follows from the fact that n is an event count and thus must be an integer (discreteness).

In thermal equilibrium at temperature T, the Boltzmann weight is

\begin{equation} P(n\mid \nu) \propto \exp\!\left(-\frac{E_n(\nu)}{k_B T}\right) =\exp\!\left(-\frac{n h_{\mathrm{VP}}\nu}{k_B T}\right) \end{equation}

and the partition function closes as a geometric series:

\begin{equation} Z(\nu)=\sum_{n=0}^{\infty} e^{-n\beta h_{\mathrm{VP}}\nu}=\frac{1}{1-e^{-\beta h_{\mathrm{VP}}\nu}},\qquad \beta:=\frac{1}{k_B T}. \end{equation}

Therefore the mean energy per mode is

\begin{equation} \langle E\rangle(\nu) = -\frac{\partial}{\partial \beta}\ln Z =\frac{h_{\mathrm{VP}}\nu}{e^{h_{\mathrm{VP}}\nu/(k_B T)}-1}. \end{equation}

This term produces the characteristic “-1” of the Planck spectrum.

15.5.4 Planck spectrum (conclusion): geometry × stiffness filter

The energy density per unit volume is u(ν)=g(ν)⟨ E⟩(ν), hence

\begin{equation} u(\nu) \propto \underbrace{\nu^{2}}_{\text{geometric mode count}} \times \underbrace{\frac{h_{\mathrm{VP}}\nu}{e^{h_{\mathrm{VP}}\nu/(k_B T)}-1}}_{\text{rigidity-shell filter (discrete events)}} \;\;\Rightarrow\;\; u(\nu)\propto \frac{\nu^{3}}{e^{h_{\mathrm{VP}}\nu/(k_B T)}-1}. \end{equation}

Thus the high-frequency cutoff is not a “mystery of light”; it occurs because the rigidity shell structurally blocks (filters) high-frequency deformation.

Interpretive note (NON-LOCK)

If one loosely says “light is gas-like,” this should be read as a statement about excitation statistics on top of a jammed background (State 4), not as a claim that the vacuum medium itself is unjammed. A concise regime dictionary for this distinction is provided in Appendix L (see L.5).

Blackbody radiation by rigidity-shell filtering (schematic).
At low frequency the geometric growth of modes (ν²) domin — Blackbody radiation by rigidity-shell filtering (schematic). At low frequency the geometric growth of modes (ν²) dominates, whereas at high frequency the rigidity-shell barrier exceeds thermal excitation and a cutoff occurs.

15.5.5 Gate: conditions for using the blackbody regime (definition)

This derivation may be used as a “Planck spectrum derivation” only when the following conditions hold:

\begin{equation} \mathrm{PASS}_{\mathrm{BB}} :\Longleftrightarrow \text{(thermal equilibrium)}\wedge\text{(optically thick cavity)}\wedge\text{(linear response)}\wedge\text{(integer event count $n$ holds)}. \end{equation}

If conditions are not satisfied, this section is set to INCONCLUSIVE.

Open — What This Framework Does Not Claim

Stated plainly. §12 (electron one-second) carries an unresolved O(1) factor. Other open items are flagged in place within their parent sections in Part I—αₑₘ (§14.5, non-evidence) and the absolute magnitude of gravity (§17.4, a proven obstruction = the hierarchy problem).