May the force be with you: IRIS Projections, Nested Discrete Coarse‑Graining, and a Constructive Toy Mechanism for 48 → 24 → 4 Sector Splitting
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| contents | <p># May the force be with you: IRIS projections and a constructive toy mechanism for sector splitting</p> <p>## Extended abstract </p> <p>### Problem<br>How 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?</p> <p>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.</p> <p>### Setup in one line<br>We implement a nested discrete coarse-graining pipeline:</p> <p>\[<br>\mathbb{Z}_6^8 \;\xrightarrow{\text{7 seals + rigid slip}}\; \mathbb{Z}_{48}<br>\;\xrightarrow{\text{IRIS parity fiber average}}\; \mathbb{Z}_{24}<br>\;\xrightarrow{\text{wing fiber average (step +4)}}\; \mathbb{Z}_4.<br>\]</p> <p>The end product is an **effective 4-sector theory** on the Leaf, even though the Bank contains a richer discrete structure.</p> <p>---</p> <p>## 1) From eight rotors to a \(\mathbb{Z}_{48}\) bank coordinate<br>We start with eight phase-like variables (“rotors”) \(\theta_a\in S^1\), each pinned to coarse \(\mathbb{Z}_6\) steps:<br>\[<br>\theta_a \in \left\{\frac{2\pi}{6}\,n_a\right\}\quad (\text{mod } 2\pi).<br>\]</p> <p>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<br>\[<br>\Theta = \frac{1}{8}\sum_{a=1}^8 \theta_a<br>\]<br>has a minimal increment<br>\[<br>\Delta \Theta = \frac{1}{8}\frac{2\pi}{6}=\frac{2\pi}{48}.<br>\]<br>This motivates a bank-level discrete coordinate \(K\in\mathbb{Z}_{48}\) (48-step holonomy).</p> <p>**Important:** this \(\mathbb{Z}_{48}\) step is a *connectivity/slip assumption*, not a theorem of discrete dimension counting.</p> <p>---</p> <p>## 2) IRIS projection I: \(\mathbb{Z}_{48}\to\mathbb{Z}_{24}\) by parity fiber averaging<br>Write the bank label as<br>\[<br>K = 2k + p,\qquad k\in\mathbb{Z}_{24},\quad p\in\mathbb{Z}_2.<br>\]<br>The bit \(p\) is a **channel/parity fiber**. The IRIS postulate for the Leaf is:</p> <p>> Leaf observables are insensitive to \(p\). <br>> Operationally, they are obtained by averaging over \(p\) (and, for matrices, averaging over both parity legs).</p> <p>For a bank-level coupling matrix \(M_{KK'}\), the Leaf matrix is defined by:<br>\[<br>(\Pi_p M)_{kk'} \equiv \frac{1}{4}\sum_{p,p'\in\{0,1\}} M_{(2k+p)(2k'+p')}.<br>\]</p> <p>### Parity-odd bridges vanish automatically<br>If 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<br>\[<br>M_{(k,p)(k',p')} = W_{kk'} + (-1)^{p+p'}\,C_{kk'},<br>\]<br>then<br>\[<br>(\Pi_p M)_{kk'} = W_{kk'}.<br>\]<br>So the first IRIS projection kills parity-odd cross-sector bridges *by construction*.</p> <p>### Controlled parity-even leakage as a falsifiable dial<br>We also introduce a small parity-even leakage parameter \(\varepsilon\ll 1\):<br>\[<br>M_{(k,p)(k',p')} = W_{kk'} + \big((-1)^{p+p'}+\varepsilon\big)\,C_{kk'}.<br>\]<br>Then the Leaf retains a suppressed bridge term:<br>\[<br>(\Pi_p M)_{kk'} = W_{kk'} + \varepsilon\,C_{kk'}.<br>\]<br>This gives a clean signature: **Leaf cross-sector coupling scales linearly with \(\varepsilon\)**.</p> <p>---</p> <p>## 3) IRIS projection II: \(\mathbb{Z}_{24}\to\mathbb{Z}_4\) by wing fiber averaging (step +4)<br>Inside \(\mathbb{Z}_{24}\) there is a unique subgroup of order 6:<br>\[<br>\langle 4\rangle = \{0,4,8,12,16,20\}\cong \mathbb{Z}_6.<br>\]<br>This induces a canonical decomposition of the leaf index:<br>\[<br>k = s + 4w\pmod{24},\qquad s\in\mathbb{Z}_4,\quad w\in\mathbb{Z}_6.<br>\]<br>We interpret:<br>- \(s\): the **sector label** (4 sectors),<br>- \(w\): an internal **wing index** (6 “copies” inside each sector).</p> <p>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.</p> <p>For a leaf matrix \(N_{kk'}\), the sector-level matrix is<br>\[<br>(\Pi_w N)_{ss'} \equiv \frac{1}{36}\sum_{w,w'\in\{0,\dots,5\}} N_{(s+4w)(s'+4w')}.<br>\]<br>This produces a \(4\times4\) effective sector matrix.</p> <p>---</p> <p>## 4) Constructive toy model: a \(48\times 48\) coupling matrix<br>We implement the pipeline with a bank coupling matrix indexed by \((s,w,p)\) where:<br>- Each sector has an internal wing block \(W_s\in\mathbb{R}^{6\times 6}\) (symmetric, positive-ish),<br>- Cross-sector bridges are \(C_{ss'}\in\mathbb{R}^{6\times 6}\),<br>- Bridges carry the parity structure \((-1)^{p+p'}\) with leakage \(\varepsilon\).</p> <p>The bank matrix is:<br>\[<br>M^{\mathrm{bank}}_{(s,w,p)(s',w',p')}=<br>\begin{cases}<br>(W_s)_{ww'}, & s=s',\\<br>\big((-1)^{p+p'}+\varepsilon\big)(C_{ss'})_{ww'}, & s\neq s'.<br>\end{cases}<br>\]</p> <p>After parity averaging we get a \(24\times24\) leaf matrix with suppressed bridges:<br>\[<br>M_{\mathrm{leaf}} = W + \varepsilon C.<br>\]<br>After wing averaging we obtain \(M_4\in\mathbb{R}^{4\times 4}\).</p> <p>---</p> <p>## 5) Numerical results<br>### 5.1 Exact bridge removal at \(\varepsilon=0\)<br>In a representative run with \(\varepsilon=0\), the measured cross-sector coupling norms were:<br>- Bank: \(\|M_{\mathrm{bank}}\|_{\mathrm{cross}} = 4.718546240045172\)<br>- Leaf: \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} = 0\)</p> <p>So bridges exist in the Bank but vanish identically in the Leaf after the IRIS parity projection.</p> <p>The leaf diagonal remains nontrivial (not a constant “\(\mathbb{Z}_1\)” collapse), and the effective sector means (diagonal of \(M_4\)) were:<br>\[<br>\mathrm{diag}(M_4) = [0.2084,\ 0.2003,\ 0.1224,\ 0.1167].<br>\]<br>This non-degeneracy is generic in random draws, but specific “2+2” pairings are not generically robust (see below).</p> <p>### 5.2 Leakage scaling is exact (1000 trials)<br>We ran 1000 random initializations with \(\varepsilon=10^{-3}\) and checked:</p> <p>- \(\mathbb{E}[\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} / (\varepsilon\|C\|_{\mathrm{cross}})] = 0.9999999999999452 \approx 1\)<br>- \(\mathbb{E}[\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} / \|M_{\mathrm{bank}}\|_{\mathrm{cross}}] = 0.0004999997500001601 \approx \varepsilon/2\)</p> <p>This is machine-precision evidence that \(\varepsilon\) is a clean dial controlling the residual cross-sector coupling visible on the Leaf.</p> <p>### 5.3 The “2+2 hierarchy” is not generic in the i.i.d. ensemble<br>Using a strict “clear 2+2” test (gap larger than 3× within-pair spread), only 7.6% of trials passed:<br>\[<br>f_{2+2} = 0.076.<br>\]<br>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.</p> <p>---</p> <p>## 6) Interpretation and falsifiable targets<br>### What is structural<br>- **Nested projections generate 4 sectors automatically:** \(\mathbb{Z}_{24}/\langle 4\rangle \cong \mathbb{Z}_4\).<br>- **Decoupling is automatic if bridges are parity-odd:** parity fiber average kills them exactly.<br>- **A single small parameter controls residual mixing:** parity-even leakage \(\varepsilon\) produces a measurable leaf cross-coupling proportional to \(\varepsilon\).</p> <p>### What remains model-dependent<br>- How the seven constraints enforce the connectivity that yields \(\mathbb{Z}_{48}\) in a microscopic model.<br>- How to map the four sectors onto real interactions (if at all).<br>- Whether the wing index \(w\) can appear in nonlocal/topological observables while being invisible in local EM couplings.</p> <p>### A clean experimental narrative<br>If a physical realization of IRIS-like coarse visibility exists, then:<br>- at leading order the Leaf should show **strong suppression** of cross-sector bridges,<br>- any residual cross-sector “communication” should scale with a small, effectively single dial (\(\varepsilon\)),<br>- and discrete parity/winding structures may show up more cleanly in boundary/defect/topological probes than in bulk local couplings.</p> <p>---</p> <p>## Reproducibility<br>The entire mechanism can be reproduced with a few lines:<br>1) build \(M_{\mathrm{bank}}\) with parity-odd bridges and optional leakage \(\varepsilon\),<br>2) parity-average to get \(M_{\mathrm{leaf}}\),<br>3) wing-average (step +4) to get \(M_4\),<br>4) verify that \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}}\to 0\) as \(\varepsilon\to 0\), and \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}}\propto \varepsilon\) for small \(\varepsilon\). </p> <p>## v2 Addendum — Extended Abstract (Micro‑IRIS / Recursive Parity on the Sector Algebra)</p> <p>### What is new in v2<br>Version 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 \(\mathbb{Z}_{24}\!\to\!\mathbb{Z}_{4}\).</p> <p>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.</p> <p>---</p> <p>### Micro‑IRIS hypothesis (operator‑level, not state identification)<br>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 \(\mathbb{Z}_4\):<br>\[<br>I_4:\ s \mapsto s+2.<br>\]<br>We then define the micro‑IRIS projector:<br>\[<br>\Pi_{\mathrm{IRIS}}^{(\mathrm{micro})} \equiv \frac12(\mathrm{id}+I_4).<br>\]</p> <p>**Crucial interpretation:** micro‑IRIS is imposed as an **admissibility / selection rule on local couplings and operators** (Clause‑U spirit), *not* as an identification of physical states. In other words, it does **not** assert that sectors \(s=0\) and \(s=2\) (or \(s=1\) and \(s=3\)) are literally the same state; it asserts that the *operator content governing couplings* is \(I_4\)-even.</p> <p>---</p> <p>### Three-line character result (mathematically exact)<br>Let \(\chi_n\) be the characters of \(\mathbb{Z}_4\):<br>\[<br>\chi_n(s) = \exp\!\left(i\frac{\pi}{2}ns\right),\qquad n\in\{0,1,2,3\}.<br>\]<br>Then:<br>\[<br>\chi_n(s+2)=\exp(i\pi n)\chi_n(s)=(-1)^n\chi_n(s),<br>\]<br>and therefore:<br>\[<br>\Pi_{\mathrm{IRIS}}^{(\mathrm{micro})}\chi_n<br>= \frac12\bigl(1+(-1)^n\bigr)\chi_n<br>=<br>\begin{cases}<br>\chi_n,& n\ \mathrm{even}\ (n=0,2),\\<br>0,& n\ \mathrm{odd}\ (n=1,3).<br>\end{cases}<br>\]<br>This is a purely algebraic statement: **micro‑IRIS removes the \(\mathbb{Z}_2\)-odd character components of \(\mathbb{Z}_4\)**.</p> <p>---</p> <p>### Why this is “closest to the \(R^8\)” philosophy<br>In v1, the macro IRIS step is realized as a **parity fiber coarse‑graining** \(K=2k+p\) with \(p\in\mathbb{Z}_2\), implemented by averaging over the inaccessible parity fiber. The v2 hypothesis proposes that this is not a one‑shot event but a **recurrent governance rule**: whenever a canonical \(\mathbb{Z}_2\) involution is available at an effective level, admissible local operators are restricted to the even subspace under that involution.</p> <p>This reuses the same “tool” (a \(\mathbb{Z}_2\) parity projection) across scales, introducing **no new dynamical mechanism**, only a repeated admissibility principle.</p> <p>---</p> <p>### Falsifiable structural prediction (no Standard Model mapping required)<br>Consider any sector‑dependent coupling (or coupling matrix element) \(g(s)\) defined on \(s\in\mathbb{Z}_4\). Expand it in \(\mathbb{Z}_4\) characters:<br>\[<br>g(s)=\sum_{n=0}^{3}\hat g_n\,\chi_n(s).<br>\]<br>If micro‑IRIS holds as an operator‑level rule, then the \(\mathbb{Z}_2\)-odd components must be suppressed:<br>\[<br>\hat g_1 \approx 0,\qquad \hat g_3 \approx 0,<br>\]<br>up to experimental/systematic resolution. Equivalently, one should observe approximate invariance<br>\[<br>g(s+2)\approx g(s)<br>\]<br>for the relevant class of **admissible local couplings**.</p> <p>This prediction is **structural**: it does not require assigning the four sectors to specific Standard Model interactions. It only requires a measurable sector label \(s\) and a Fourier/character test on the observed coupling dependence.</p> <p>---</p> <p>### What this does **not** claim (yet)<br>- It does **not** claim that nature must realize a \(\mathbb{Z}_2\subset\mathbb{Z}_4\) “class structure”.<br>- It does **not** claim a universal 2+2 hierarchy among the four sectors.<br>- It does **not** derive micro‑IRIS from first principles; it states it as a **well‑motivated postulate** consistent with the governance/admissibility style of the \(R^8\) framework.</p> <p>The open theoretical task is to provide a derivation mechanism (e.g. “only charge‑2 composites are locally visible” or a Clause‑U admissibility argument) that explains why the Leaf coupling algebra “knows” to enforce the involution \(s\mapsto s+2\) beyond its purely algebraic existence.</p> <p>---</p> <p>### Summary<br>v2 adds a single, clean extension: **recursive IRIS** on the sector algebra. Mathematically, it is the same parity projector used at macro scale, now applied to the canonical \(\mathbb{Z}_2\) involution inside \(\mathbb{Z}_4\). Conceptually, it is an operator‑level selection rule (not state identification). Experimentally, it yields a sharp signature: **absence/suppression of the \(\mathbb{Z}_4\) odd character components** (\(n=1,3\)) in sector‑dependent couplings.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18615066 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | May the force be with you: IRIS Projections, Nested Discrete Coarse‑Graining, and a Constructive Toy Mechanism for 48 → 24 → 4 Sector Splitting Diogenes Crystalline Axiverse <p># May the force be with you: IRIS projections and a constructive toy mechanism for sector splitting</p> <p>## Extended abstract </p> <p>### Problem<br>How 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?</p> <p>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.</p> <p>### Setup in one line<br>We implement a nested discrete coarse-graining pipeline:</p> <p>\[<br>\mathbb{Z}_6^8 \;\xrightarrow{\text{7 seals + rigid slip}}\; \mathbb{Z}_{48}<br>\;\xrightarrow{\text{IRIS parity fiber average}}\; \mathbb{Z}_{24}<br>\;\xrightarrow{\text{wing fiber average (step +4)}}\; \mathbb{Z}_4.<br>\]</p> <p>The end product is an **effective 4-sector theory** on the Leaf, even though the Bank contains a richer discrete structure.</p> <p>---</p> <p>## 1) From eight rotors to a \(\mathbb{Z}_{48}\) bank coordinate<br>We start with eight phase-like variables (“rotors”) \(\theta_a\in S^1\), each pinned to coarse \(\mathbb{Z}_6\) steps:<br>\[<br>\theta_a \in \left\{\frac{2\pi}{6}\,n_a\right\}\quad (\text{mod } 2\pi).<br>\]</p> <p>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<br>\[<br>\Theta = \frac{1}{8}\sum_{a=1}^8 \theta_a<br>\]<br>has a minimal increment<br>\[<br>\Delta \Theta = \frac{1}{8}\frac{2\pi}{6}=\frac{2\pi}{48}.<br>\]<br>This motivates a bank-level discrete coordinate \(K\in\mathbb{Z}_{48}\) (48-step holonomy).</p> <p>**Important:** this \(\mathbb{Z}_{48}\) step is a *connectivity/slip assumption*, not a theorem of discrete dimension counting.</p> <p>---</p> <p>## 2) IRIS projection I: \(\mathbb{Z}_{48}\to\mathbb{Z}_{24}\) by parity fiber averaging<br>Write the bank label as<br>\[<br>K = 2k + p,\qquad k\in\mathbb{Z}_{24},\quad p\in\mathbb{Z}_2.<br>\]<br>The bit \(p\) is a **channel/parity fiber**. The IRIS postulate for the Leaf is:</p> <p>> Leaf observables are insensitive to \(p\). <br>> Operationally, they are obtained by averaging over \(p\) (and, for matrices, averaging over both parity legs).</p> <p>For a bank-level coupling matrix \(M_{KK'}\), the Leaf matrix is defined by:<br>\[<br>(\Pi_p M)_{kk'} \equiv \frac{1}{4}\sum_{p,p'\in\{0,1\}} M_{(2k+p)(2k'+p')}.<br>\]</p> <p>### Parity-odd bridges vanish automatically<br>If 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<br>\[<br>M_{(k,p)(k',p')} = W_{kk'} + (-1)^{p+p'}\,C_{kk'},<br>\]<br>then<br>\[<br>(\Pi_p M)_{kk'} = W_{kk'}.<br>\]<br>So the first IRIS projection kills parity-odd cross-sector bridges *by construction*.</p> <p>### Controlled parity-even leakage as a falsifiable dial<br>We also introduce a small parity-even leakage parameter \(\varepsilon\ll 1\):<br>\[<br>M_{(k,p)(k',p')} = W_{kk'} + \big((-1)^{p+p'}+\varepsilon\big)\,C_{kk'}.<br>\]<br>Then the Leaf retains a suppressed bridge term:<br>\[<br>(\Pi_p M)_{kk'} = W_{kk'} + \varepsilon\,C_{kk'}.<br>\]<br>This gives a clean signature: **Leaf cross-sector coupling scales linearly with \(\varepsilon\)**.</p> <p>---</p> <p>## 3) IRIS projection II: \(\mathbb{Z}_{24}\to\mathbb{Z}_4\) by wing fiber averaging (step +4)<br>Inside \(\mathbb{Z}_{24}\) there is a unique subgroup of order 6:<br>\[<br>\langle 4\rangle = \{0,4,8,12,16,20\}\cong \mathbb{Z}_6.<br>\]<br>This induces a canonical decomposition of the leaf index:<br>\[<br>k = s + 4w\pmod{24},\qquad s\in\mathbb{Z}_4,\quad w\in\mathbb{Z}_6.<br>\]<br>We interpret:<br>- \(s\): the **sector label** (4 sectors),<br>- \(w\): an internal **wing index** (6 “copies” inside each sector).</p> <p>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.</p> <p>For a leaf matrix \(N_{kk'}\), the sector-level matrix is<br>\[<br>(\Pi_w N)_{ss'} \equiv \frac{1}{36}\sum_{w,w'\in\{0,\dots,5\}} N_{(s+4w)(s'+4w')}.<br>\]<br>This produces a \(4\times4\) effective sector matrix.</p> <p>---</p> <p>## 4) Constructive toy model: a \(48\times 48\) coupling matrix<br>We implement the pipeline with a bank coupling matrix indexed by \((s,w,p)\) where:<br>- Each sector has an internal wing block \(W_s\in\mathbb{R}^{6\times 6}\) (symmetric, positive-ish),<br>- Cross-sector bridges are \(C_{ss'}\in\mathbb{R}^{6\times 6}\),<br>- Bridges carry the parity structure \((-1)^{p+p'}\) with leakage \(\varepsilon\).</p> <p>The bank matrix is:<br>\[<br>M^{\mathrm{bank}}_{(s,w,p)(s',w',p')}=<br>\begin{cases}<br>(W_s)_{ww'}, & s=s',\\<br>\big((-1)^{p+p'}+\varepsilon\big)(C_{ss'})_{ww'}, & s\neq s'.<br>\end{cases}<br>\]</p> <p>After parity averaging we get a \(24\times24\) leaf matrix with suppressed bridges:<br>\[<br>M_{\mathrm{leaf}} = W + \varepsilon C.<br>\]<br>After wing averaging we obtain \(M_4\in\mathbb{R}^{4\times 4}\).</p> <p>---</p> <p>## 5) Numerical results<br>### 5.1 Exact bridge removal at \(\varepsilon=0\)<br>In a representative run with \(\varepsilon=0\), the measured cross-sector coupling norms were:<br>- Bank: \(\|M_{\mathrm{bank}}\|_{\mathrm{cross}} = 4.718546240045172\)<br>- Leaf: \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} = 0\)</p> <p>So bridges exist in the Bank but vanish identically in the Leaf after the IRIS parity projection.</p> <p>The leaf diagonal remains nontrivial (not a constant “\(\mathbb{Z}_1\)” collapse), and the effective sector means (diagonal of \(M_4\)) were:<br>\[<br>\mathrm{diag}(M_4) = [0.2084,\ 0.2003,\ 0.1224,\ 0.1167].<br>\]<br>This non-degeneracy is generic in random draws, but specific “2+2” pairings are not generically robust (see below).</p> <p>### 5.2 Leakage scaling is exact (1000 trials)<br>We ran 1000 random initializations with \(\varepsilon=10^{-3}\) and checked:</p> <p>- \(\mathbb{E}[\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} / (\varepsilon\|C\|_{\mathrm{cross}})] = 0.9999999999999452 \approx 1\)<br>- \(\mathbb{E}[\|M_{\mathrm{leaf}}\|_{\mathrm{cross}} / \|M_{\mathrm{bank}}\|_{\mathrm{cross}}] = 0.0004999997500001601 \approx \varepsilon/2\)</p> <p>This is machine-precision evidence that \(\varepsilon\) is a clean dial controlling the residual cross-sector coupling visible on the Leaf.</p> <p>### 5.3 The “2+2 hierarchy” is not generic in the i.i.d. ensemble<br>Using a strict “clear 2+2” test (gap larger than 3× within-pair spread), only 7.6% of trials passed:<br>\[<br>f_{2+2} = 0.076.<br>\]<br>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.</p> <p>---</p> <p>## 6) Interpretation and falsifiable targets<br>### What is structural<br>- **Nested projections generate 4 sectors automatically:** \(\mathbb{Z}_{24}/\langle 4\rangle \cong \mathbb{Z}_4\).<br>- **Decoupling is automatic if bridges are parity-odd:** parity fiber average kills them exactly.<br>- **A single small parameter controls residual mixing:** parity-even leakage \(\varepsilon\) produces a measurable leaf cross-coupling proportional to \(\varepsilon\).</p> <p>### What remains model-dependent<br>- How the seven constraints enforce the connectivity that yields \(\mathbb{Z}_{48}\) in a microscopic model.<br>- How to map the four sectors onto real interactions (if at all).<br>- Whether the wing index \(w\) can appear in nonlocal/topological observables while being invisible in local EM couplings.</p> <p>### A clean experimental narrative<br>If a physical realization of IRIS-like coarse visibility exists, then:<br>- at leading order the Leaf should show **strong suppression** of cross-sector bridges,<br>- any residual cross-sector “communication” should scale with a small, effectively single dial (\(\varepsilon\)),<br>- and discrete parity/winding structures may show up more cleanly in boundary/defect/topological probes than in bulk local couplings.</p> <p>---</p> <p>## Reproducibility<br>The entire mechanism can be reproduced with a few lines:<br>1) build \(M_{\mathrm{bank}}\) with parity-odd bridges and optional leakage \(\varepsilon\),<br>2) parity-average to get \(M_{\mathrm{leaf}}\),<br>3) wing-average (step +4) to get \(M_4\),<br>4) verify that \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}}\to 0\) as \(\varepsilon\to 0\), and \(\|M_{\mathrm{leaf}}\|_{\mathrm{cross}}\propto \varepsilon\) for small \(\varepsilon\). </p> <p>## v2 Addendum — Extended Abstract (Micro‑IRIS / Recursive Parity on the Sector Algebra)</p> <p>### What is new in v2<br>Version 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 \(\mathbb{Z}_{24}\!\to\!\mathbb{Z}_{4}\).</p> <p>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.</p> <p>---</p> <p>### Micro‑IRIS hypothesis (operator‑level, not state identification)<br>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 \(\mathbb{Z}_4\):<br>\[<br>I_4:\ s \mapsto s+2.<br>\]<br>We then define the micro‑IRIS projector:<br>\[<br>\Pi_{\mathrm{IRIS}}^{(\mathrm{micro})} \equiv \frac12(\mathrm{id}+I_4).<br>\]</p> <p>**Crucial interpretation:** micro‑IRIS is imposed as an **admissibility / selection rule on local couplings and operators** (Clause‑U spirit), *not* as an identification of physical states. In other words, it does **not** assert that sectors \(s=0\) and \(s=2\) (or \(s=1\) and \(s=3\)) are literally the same state; it asserts that the *operator content governing couplings* is \(I_4\)-even.</p> <p>---</p> <p>### Three-line character result (mathematically exact)<br>Let \(\chi_n\) be the characters of \(\mathbb{Z}_4\):<br>\[<br>\chi_n(s) = \exp\!\left(i\frac{\pi}{2}ns\right),\qquad n\in\{0,1,2,3\}.<br>\]<br>Then:<br>\[<br>\chi_n(s+2)=\exp(i\pi n)\chi_n(s)=(-1)^n\chi_n(s),<br>\]<br>and therefore:<br>\[<br>\Pi_{\mathrm{IRIS}}^{(\mathrm{micro})}\chi_n<br>= \frac12\bigl(1+(-1)^n\bigr)\chi_n<br>=<br>\begin{cases}<br>\chi_n,& n\ \mathrm{even}\ (n=0,2),\\<br>0,& n\ \mathrm{odd}\ (n=1,3).<br>\end{cases}<br>\]<br>This is a purely algebraic statement: **micro‑IRIS removes the \(\mathbb{Z}_2\)-odd character components of \(\mathbb{Z}_4\)**.</p> <p>---</p> <p>### Why this is “closest to the \(R^8\)” philosophy<br>In v1, the macro IRIS step is realized as a **parity fiber coarse‑graining** \(K=2k+p\) with \(p\in\mathbb{Z}_2\), implemented by averaging over the inaccessible parity fiber. The v2 hypothesis proposes that this is not a one‑shot event but a **recurrent governance rule**: whenever a canonical \(\mathbb{Z}_2\) involution is available at an effective level, admissible local operators are restricted to the even subspace under that involution.</p> <p>This reuses the same “tool” (a \(\mathbb{Z}_2\) parity projection) across scales, introducing **no new dynamical mechanism**, only a repeated admissibility principle.</p> <p>---</p> <p>### Falsifiable structural prediction (no Standard Model mapping required)<br>Consider any sector‑dependent coupling (or coupling matrix element) \(g(s)\) defined on \(s\in\mathbb{Z}_4\). Expand it in \(\mathbb{Z}_4\) characters:<br>\[<br>g(s)=\sum_{n=0}^{3}\hat g_n\,\chi_n(s).<br>\]<br>If micro‑IRIS holds as an operator‑level rule, then the \(\mathbb{Z}_2\)-odd components must be suppressed:<br>\[<br>\hat g_1 \approx 0,\qquad \hat g_3 \approx 0,<br>\]<br>up to experimental/systematic resolution. Equivalently, one should observe approximate invariance<br>\[<br>g(s+2)\approx g(s)<br>\]<br>for the relevant class of **admissible local couplings**.</p> <p>This prediction is **structural**: it does not require assigning the four sectors to specific Standard Model interactions. It only requires a measurable sector label \(s\) and a Fourier/character test on the observed coupling dependence.</p> <p>---</p> <p>### What this does **not** claim (yet)<br>- It does **not** claim that nature must realize a \(\mathbb{Z}_2\subset\mathbb{Z}_4\) “class structure”.<br>- It does **not** claim a universal 2+2 hierarchy among the four sectors.<br>- It does **not** derive micro‑IRIS from first principles; it states it as a **well‑motivated postulate** consistent with the governance/admissibility style of the \(R^8\) framework.</p> <p>The open theoretical task is to provide a derivation mechanism (e.g. “only charge‑2 composites are locally visible” or a Clause‑U admissibility argument) that explains why the Leaf coupling algebra “knows” to enforce the involution \(s\mapsto s+2\) beyond its purely algebraic existence.</p> <p>---</p> <p>### Summary<br>v2 adds a single, clean extension: **recursive IRIS** on the sector algebra. Mathematically, it is the same parity projector used at macro scale, now applied to the canonical \(\mathbb{Z}_2\) involution inside \(\mathbb{Z}_4\). Conceptually, it is an operator‑level selection rule (not state identification). Experimentally, it yields a sharp signature: **absence/suppression of the \(\mathbb{Z}_4\) odd character components** (\(n=1,3\)) in sector‑dependent couplings.</p> |
| title | May the force be with you: IRIS Projections, Nested Discrete Coarse‑Graining, and a Constructive Toy Mechanism for 48 → 24 → 4 Sector Splitting |
| topic | Crystalline Axiverse |
| url | https://doi.org/10.5281/zenodo.18615066 |