Appendix F. Discrete decomposition hypothesis of event rate: integer part (active sites) and residual (oscillation)

Appendix F. Discrete decomposition hypothesis: Define the following decomposition. Using the LOCK value of 9.4 (S09_04_nup_numeric). N_act can be interpreted as the discrete count of “effective events that maintain structure” inside the proton, i.e., the number of active event sites.

Define the following decomposition. Using the LOCK value of 9.4 (S09_04_nup_numeric). 3π⁴.

F.0 Purpose and scope (NON-LOCK)

Note. The canonical ν_p,can=3π⁴ is fixed geometrically by [LOCK-NU-N] in §8.0.5. This appendix decomposes the length-form value of §9.4 (292.245, the cross-check) into integer part + residual; the candidate 4(82-9)=292 reproduces the integer part only and is a separate, NON-LOCK discrete reading (its residual 0.245 is not 3π⁴'s 0.227). It is kept as illustrative, not as the source of ν_p,can. (Disambiguation: the integer 9 in 4(82-9) is internal to this NON-LOCK reading and is unrelated to the obsolete 2+2+4+1=9 shell-partition arithmetic corrected in v0.2.0 / Part II v31.) Section 9.4 derived ν_p,can≈ 292.245156s⁻¹ from only LOCK inputs (D_anch, rₚ, δ) via the closed form (S09_04_closed_form). This appendix decomposes this value into integer part + residual, and records a post-hoc hypothesis interpreting it from a discrete-structure (integer lattice/channel) viewpoint. Therefore, the contents of this appendix do not modify the main-text (LOCK/Gate) conclusions, and are not included in the PASS/FAIL decision set of the current Gate version.

F.1 Definition: integer activity and residual

Define the following decomposition.

\begin{equation} \nu_{p,\mathrm{can}} = N_{\mathrm{act}} + \varepsilon, \qquad N_{\mathrm{act}} := \lfloor \nu_{p,\mathrm{can}} \rfloor, \qquad \varepsilon := \nu_{p,\mathrm{can}} - N_{\mathrm{act}}. \end{equation}

Using the LOCK value of 9.4 ((S09_04_nup_numeric)),

\begin{equation} N_{\mathrm{act}} = 292, \qquad \varepsilon = 0.2451560251\ldots \end{equation}

we obtain.

Interpretation (post-hoc hypothesis).

N_act can be interpreted as the discrete count of “effective events that maintain structure” inside the proton, i.e., the number of active event sites. In contrast, ε can be interpreted as a residual oscillation/fluctuation component remaining due to the limits of rectification (δ) and integer lattice matching. This residual may be conceptually similar to certain terms in standard physics (e.g., Zitterbewegung), but this whitepaper does not claim an equivalence at the current stage.

F.2 Candidate discrete mapping: 82× 4 - 9× 4 = 292

The following is a candidate interpretation in which the integer part N_act=292 can be reproduced by a simple discrete-structure calculation.

Core discrete number: N_core:=82 (consistent with the core definition in 7.2)
Local rigidity minimum unit: “4-point locking” (tetrahedral) ⇒ 4 points per unit
Static input (throat) unit: N_static_q:=9 (a candidate 3-sector × 3-plug structure)

Then, if we set the total capacity points and static points as

\begin{equation} N_{\mathrm{cap}}:=4N_{\mathrm{core}}=328, \qquad N_{\mathrm{static}}:=4N_{\mathrm{static\_q}}=36 \end{equation}

as above,

\begin{equation} N_{\mathrm{cap}}-N_{\mathrm{static}}=292 \end{equation}

we get (AppF_nact_eps_numeric) as the same integer part.

Caution (definition locking required).

Equation (AppF_328_36_292) is until the meaning of “point”, “static throat”, “4-point locking”, and “9-input” is fixed by LOCK, only a post-hoc interpretation. In the current version, (AppF_328_36_292) is not incorporated into the (LOCK/Gate) conclusion, and it is recorded separately as NON-LOCK to avoid being misconstrued as a degree of freedom for numerical fitting.

F.3 Future Gate design (declaration)

To elevate this candidate interpretation to a main-text conclusion in the future, at minimum the following are required.

LOCK: explicitly fix the definition of “event sites” (meaning of points/locking/channels) and the regime (3-sector, jamming condition).
Gate: judge whether the N_act derived from LOCK matches ⌊ν_p,can⌋.
Evidence: seal reproducible files that reconstruct the core/channel structure (e.g., core82 snapshot, throat/jamming artifacts).

F.4 Deterministic computation snippet (verification)

The snippet below computes ν_p,can from the LOCK values of 9.4, and prints the integer part/residual together with the candidate calculation (AppF_328_36_292).

# verify_appendix_F.py (demonstration; NON-LOCK interpretation)
from math import pi

D_anch = 4.852620477e-12         # m  (SSOT: 2*lambda_{C,e}=2hc/(m_e c^2); jamming 4.8542 is a Gate cross-check only)
r_p    = 0.8412e-15              # m
delta  = 1.0/(pi*pi)

s_p = D_anch/(2.0*r_p)
nu  = s_p*delta

N_act = int(nu)          # floor for positive nu
eps   = nu - N_act

N_core = 82
points_per = 4
N_static_q = 9
N_cap = N_core*points_per
N_static = N_static_q*points_per
N_active_points = N_cap - N_static

print("s_p=", s_p)
print("nu_p_can=", nu)
print("N_act=", N_act)
print("eps=", eps)
print("N_active_points=", N_active_points)