May the force be with you: IRIS Projections, Nested Discrete Coarse‑Graining, and a Constructive Toy Mechanism for 𝑍 48 → 𝑍 24 → 𝑍 4 Sector Splitting

# May the force be with you: IRIS projections and a constructive toy mechanism for sector splitting ## Extended abstract ### ProblemHow can a *single* “bank-level” structure (one underlying force, one governance topology) **appear** at low energies as *multiple* decoupled “forces” (sectors) **without** inserting the decoupling by hand? In this work we isolate one mechanism that is *purely structural*: the low-energy (Leaf) theory does not have direct access to the full discrete channel space of the Bank. Instead, it only sees a **coarse projection** of that channel space. When cross-sector “bridge” operators live in the *projected-out* channel, they disappear from the Leaf EFT automatically. ### Setup in one lineWe implement a nested discrete coarse-graining pipeline: \[\mathbb{Z}_6^8 \;\xrightarrow{\text{7 seals + rigid slip}}\; \mathbb{Z}_{48}\;\xrightarrow{\text{IRIS parity fiber average}}\; \mathbb{Z}_{24}\;\xrightarrow{\text{wing fiber average (step +4)}}\; \mathbb{Z}_4.\] The end product is an **effective 4-sector theory** on the Leaf, even though the Bank contains a richer discrete structure. --- ## 1) From eight rotors to a \(\mathbb{Z}_{48}\) bank coordinateWe start with eight phase-like variables (“rotors”) \(\theta_a\in S^1\), each pinned to coarse \(\mathbb{Z}_6\) steps:\[\theta_a \in \left\{\frac{2\pi}{6}\,n_a\right\}\quad (\text{mod } 2\pi).\] Seven compatibility constraints (“7 seals”) restrict the allowed configurations. In the modeling regime of interest, we assume the allowed configuration graph admits **elementary slip moves** where effectively one rotor advances by one \(\mathbb{Z}_6\) step while the constraints enforce rigid locking. Then the collective mean phase\[\Theta = \frac{1}{8}\sum_{a=1}^8 \theta_a\]has a minimal increment\[\Delta \Theta = \frac{1}{8}\frac{2\pi}{6}=\frac{2\pi}{48}.\]This motivates a bank-level discrete coordinate \(K\in\mathbb{Z}_{48}\) (48-step holonomy). **Important:** this \(\mathbb{Z}_{48}\) step is a *connectivity/slip assumption*, not a theorem of discrete dimension counting. --- ## 2) IRIS projection I: \(\mathbb{Z}_{48}\to\mathbb{Z}_{24}\) by parity fiber averagingWrite the bank label as\[K = 2k + p,\qquad k\in\mathbb{Z}_{24},\quad p\in\mathbb{Z}_2.\]The bit \(p\) is a **channel/parity fiber**. The IRIS postulate for the Leaf is: > Leaf observables are insensitive to \(p\). > Operationally, they are obtained by averaging over \(p\) (and, for matrices, averaging over both parity legs). For a bank-level coupling matrix \(M_{KK'}\), the Leaf matrix is defined by:\[(\Pi_p M)_{kk'} \equiv \frac{1}{4}\sum_{p,p'\in\{0,1\}} M_{(2k+p)(2k'+p')}.\] ### Parity-odd bridges vanish automaticallyIf cross-sector “bridge” terms are **parity-odd**, i.e. they flip sign under \(p\to p+1\) on either leg, then their average is exactly zero. Concretely, if\[M_{(k,p)(k',p')} = W_{kk'} + (-1)^{p+p'}\,C_{kk'},\]then\[(\Pi_p M)_{kk'} = W_{kk'}.\]So the first IRIS projection kills parity-odd cross-sector bridges *by construction*. ### Controlled parity-even leakage as a falsifiable dialWe also introduce a small parity-even leakage parameter \(\varepsilon\ll 1\):\[M_{(k,p)(k',p')} = W_{kk'} + \big((-1)^{p+p'}+\varepsilon\big)\,C_{kk'}.\]Then the Leaf retains a suppressed bridge term:\[(\Pi_p M)_{kk'} = W_{kk'} + \varepsilon\,C_{kk'}.\]This gives a clean signature: **Leaf cross-sector coupling scales linearly with \(\varepsilon\)**. --- ## 3) IRIS projection II: \(\mathbb{Z}_{24}\to\mathbb{Z}_4\) by wing fiber averaging (step +4)Inside \(\mathbb{Z}_{24}\) there is a unique subgroup of order 6:\[\langle 4\rangle = \{0,4,8,12,16,20\}\cong \mathbb{Z}_6.\]This induces a canonical decomposition of the leaf index:\[k = s + 4w\pmod{24},\qquad s\in\mathbb{Z}_4,\quad w\in\mathbb{Z}_6.\]We interpret:- \(s\): the **sector label** (4 sectors),- \(w\): an internal **wing index** (6 “copies” inside each sector). Crucially, we choose \(w\) to be a **phantom internal index** for local couplings: it does not enter the EM axion-like probe couplings and is treated as redundant/gauge-like for operator selection. Therefore we coarse-grain it by averaging over the 6-element fiber. For a leaf matrix \(N_{kk'}\), the sector-level matrix is\[(\Pi_w N)_{ss'} \equiv \frac{1}{36}\sum_{w,w'\in\{0,\dots,5\}} N_{(s+4w)(s'+4w')}.\]This produces a \(4\times4\) effective sector matrix. --- ## 4) Constructive toy model: a \(48\times 48\) coupling matrixWe implement the pipeline with a bank coupling matrix indexed by \((s,w,p)\) where:- Each sector has an internal wing block \(W_s\in\mathbb{R}^{6\times 6}\) (symmetric, positive-ish),- Cross-sector bridges are \(C_{ss'}\in\mathbb{R}^{6\times 6}\),- Bridges carry the parity structure \((-1)^{p+p'}\) with leakage \(\varepsilon\). The bank matrix is:\[M^{\mathrm{bank}}_{(s,w,p)(s',w',p')}=\begin{cases}(W_s)_{ww'}, & s=s',\\\big((-1)^{p+p'}+\varepsilon\big)(C_{ss'})_{ww'}, & s\neq s'.\end{cases}\] After parity averaging we get a \(24\times24\) leaf matrix with suppressed bridges:\[M_{\mathrm{leaf}} = W + \varepsilon C.\]After wing averaging we obtain \(M_4\in\mathbb{R}^{4\times 4}\). --- ## 5) Numerical results### 5.1 Exact bridge removal at \(\varepsilon=0\)In a representative run with \(\varepsilon=0\), the measured cross-sector coupling norms were:- Bank: \(\|M_{\mathrm{bank}}\|_{\mathrm{cross}} = 4.718546240045172\)- Leaf: \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} = 0\) So bridges exist in the Bank but vanish identically in the Leaf after the IRIS parity projection. The leaf diagonal remains nontrivial (not a constant “\(\mathbb{Z}_1\)” collapse), and the effective sector means (diagonal of \(M_4\)) were:\[\mathrm{diag}(M_4) = [0.2084,\ 0.2003,\ 0.1224,\ 0.1167].\]This non-degeneracy is generic in random draws, but specific “2+2” pairings are not generically robust (see below). ### 5.2 Leakage scaling is exact (1000 trials)We ran 1000 random initializations with \(\varepsilon=10^{-3}\) and checked: - \(\mathbb{E}[\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} / (\varepsilon\|C\|_{\mathrm{cross}})] = 0.9999999999999452 \approx 1\)- \(\mathbb{E}[\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} / \|M_{\mathrm{bank}}\|_{\mathrm{cross}}] = 0.0004999997500001601 \approx \varepsilon/2\) This is machine-precision evidence that \(\varepsilon\) is a clean dial controlling the residual cross-sector coupling visible on the Leaf. ### 5.3 The “2+2 hierarchy” is not generic in the i.i.d. ensembleUsing a strict “clear 2+2” test (gap larger than 3× within-pair spread), only 7.6% of trials passed:\[f_{2+2} = 0.076.\]Hence, any persistent 2+2 structure should be treated as an additional structural input (e.g. a \(Z_2\subset Z_4\) class constraint), not a generic output of the nested IRIS projections. --- ## 6) Interpretation and falsifiable targets### What is structural- **Nested projections generate 4 sectors automatically:** \(\mathbb{Z}_{24}/\langle 4\rangle \cong \mathbb{Z}_4\).- **Decoupling is automatic if bridges are parity-odd:** parity fiber average kills them exactly.- **A single small parameter controls residual mixing:** parity-even leakage \(\varepsilon\) produces a measurable leaf cross-coupling proportional to \(\varepsilon\). ### What remains model-dependent- How the seven constraints enforce the connectivity that yields \(\mathbb{Z}_{48}\) in a microscopic model.- How to map the four sectors onto real interactions (if at all).- Whether the wing index \(w\) can appear in nonlocal/topological observables while being invisible in local EM couplings. ### A clean experimental narrativeIf a physical realization of IRIS-like coarse visibility exists, then:- at leading order the Leaf should show **strong suppression** of cross-sector bridges,- any residual cross-sector “communication” should scale with a small, effectively single dial (\(\varepsilon\)),- and discrete parity/winding structures may show up more cleanly in boundary/defect/topological probes than in bulk local couplings. --- ## ReproducibilityThe entire mechanism can be reproduced with a few lines:1) build \(M_{\mathrm{bank}}\) with parity-odd bridges and optional leakage \(\varepsilon\),2) parity-average to get \(M_{\mathrm{leaf}}\),3) wing-average (step +4) to get \(M_4\),4) verify that \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}}\to 0\) as \(\varepsilon\to 0\), and \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}}\propto \varepsilon\) for small \(\varepsilon\). ## v2 Addendum — Extended Abstract (Micro‑IRIS / Recursive Parity on the Sector Algebra) ### What is new in v2Version 2 introduces an **optional, scale‑recursive IRIS hypothesis** acting *inside* the Leaf: the same \(\mathbb{Z}_2\) parity‑projection principle that implements the macro coarse‑visibility \(\mathbb{Z}_{48}\!\to\!\mathbb{Z}_{24}\) is hypothesized to act again on the **sector algebra** after the nested reduction \(\mathbb{Z}_{24}\!\to\!\mathbb{Z}_{4}\). This addition is **not** used to establish the core constructive results of v1 (bridge killing, non‑\(\mathbb{Z}_1\) collapse, and the existence of four effective sectors in the toy pipeline). It is presented as a **well‑motivated postulate** that yields a clean, falsifiable structural test. --- ### Micro‑IRIS hypothesis (operator‑level, not state identification)After the nested decomposition \(k=s+4w \ (\mathrm{mod}\ 24)\) with \(s\in\mathbb{Z}_4\) (sector label) and \(w\in\mathbb{Z}_6\) (phantom wing index), we define the canonical involution on \(\mathbb{Z}_4\):\[I_4:\ s \mapsto s+2.\]We then define the micro‑IRIS projector:\[\Pi_{\mathrm{IRIS}}^{(\mathrm{micro})} \equiv \frac12(\mathrm{id}+I_4).\] **Crucial interpretation:** micro‑IRIS is imposed as an **admissibility / selection rule on local couplings and operators** (Clause‑U