UNNS SUBSTRATE RESEARCH PROGRAM · ESC-I · ECHO–ADMISSIBILITY CORRESPONDENCE CHAMBER Iunns.tech
RECURSIVE PROJECTION · STRUCTURAL ADMISSIBILITY
Recursive Projection and Structural Admissibility
From Complex Echoes to Predictive Boundaries
UNNS RESEARCH COLLECTIVE · SYNTHESIS MANUSCRIPT · 13 OCTOBER 2026
Real 2×2 recursion → exact classification in ℚ → echo at finite horizon → frozen, hashed rules → projected recursion Π∘T → independent re-check
Spectrum alone does not fix the regime. For xn+1 = Π(Txn) the projection, its thresholds and the starting state decide the outcome. ESC-I tests a structure-aware rule for this against exact arithmetic, within a stated domain; it restates classical theory with certificates and claims no law beyond it.
EXACT ARITHMETIC · ℚRULES FROZEN + HASHED BEFORE COMPARISON0 CONTRADICTIONS · 4,197 NONLINEAR CASES819 VERDICTS FINITE-HORIZON ONLY1 RULE FALSIFIED, REFINED, RE-CONFIRMEDCOMPLEX ALGEBRA FROM ADMISSIBILITY ALONE · OPENno formal proof-checking run
01
From the complex-number hypothesis to structural admissibility
ESC-I is a constructive continuation of the early UNNS complex-number programme, not a departure from it. The four ideas of the original manuscripts remain the starting point; the audit turned them into statements with explicit hypotheses.
WHAT ESC-I ADDS
Four items, each a result of ESC-I itself. The method (preregistration, hashing, falsification-first) is inherited from the wider UNNS programme.
02
Exact classification and the echo
Each real 2×2 recursion is classified exactly (Fractions; Jury criterion, Jordan and eigenvector tests, root-of-unity criterion τ ∈ {0, ±1}). The echo reads the same orbit at finite horizon N = and tolerance . Fifteen designed examples; two are deliberate observational traps.
THE 15-EXAMPLE SUITE
Exact class versus the echo and versus a spectral-radius-only reading. Filter by outcome.
03
Audit of the original claims
The phase-inletting and projected-stability statements of the original manuscript (its §5 and §4) were checked against their text. They hold exactly under explicit conditions; the counterexamples locate the boundaries of the mechanisms.
THEOREM A · PHASE INLETTING
Conjugation preserves the class. ρ(RθT) is independent of θ exactly for T ∈ ℂ ∪ ℂ̄ (scaled isometries). supU ρ(UT) = σmax(T).
Counterexample to the unrestricted claim: T = [[1,1],[1,1]] with U = diag(1, −1) gives ρ(T) = 2 but ρ(UT) = 0; U and T do not commute.
THEOREM B · PROJECTED GROWTH
T4.1: with relative error η the orbit stays in an invariant cone; asymptotically limsup |w|/|u| ≤ η/(|λ|−|μ|−η). T4.2: with absolute error h, escape is guaranteed for |u₀| > h/(|λ|−1), and this radius is sharp. Direction is not preserved.
CHECKS RUN ON THE REFINED STATEMENTS
Counts are from the recorded audit; a violation would be a counterexample to the refined statement within the generated cases.
04
Frozen admissibility rules AR-1, AR-2, AR-3
Protocol: freeze and hash the rules → predict (rule code must not call the exact classifier; source-checked) → hash the predictions → compare with exact computation. Outcomes: AGREE, DISAGREE, ABSTAIN, UNCONFIRMABLE.
OUTCOMES BY RULE
AR-3 BY PROJECTION FAMILY
The 92 disagreements are all in REL_QUANT (the k-bit relative quantiser): the frozen clause was false as stated.
AR-3 → AR-3.1 · FALSIFIED, REFINED, CONFIRMED
Refinement ηeig = κ∞(V)·|λ|·2−k, confirmed on fresh seed.
AR-2 · BOUNDARY SENSITIVITY
Bauer–Fike margin. Sharpness is measured by the ratio of the smallest destabilising perturbation to the certified margin.
05
PRD-1 · projection route dependence
Spectral frame T = diag(λ, μ), |λ| > 1, |λ| > |μ|. Four nonlinear projections: dead-zone (DZ), clipping (CLIP), lattice rounding (LAT), state gate (GATE), plus two linear controls. R is the structure-aware rule; C is a classical comparator that knows only the projection's error bound (escape radius), plus the exact classifier on ΠT for linear projections.
COVERAGE BY FAMILY · RULE R
Bars sum to each family's case count (denominators on the right). Gold segments are finite-horizon-consistent only (200 exact steps); no asymptotic certificate.
R VERSUS C
Nonlinear cases, n = . C is given an error bound only; a classical analyst given each projection's definition could match R's coverage.
ROUTE DEPENDENCE
The unprojected classifier is wrong about the projected regime in the cases below. A flip is a case where the projected regime is not divergent though the unprojected classifier says divergent.
PER-FAMILY TABLE
EVIDENCE LEVELS USED FOR EVERY VERDICT
06
Projection explorer
Choose T = diag(λ, μ), a projection and a starting state. The panel evaluates rule R's closed-form condition (a port of the recorded rule) and, for comparison, iterates the projected map for 200 steps in floating point. This is an illustration; the recorded results use exact rational arithmetic.
07
Linear control and the comparator amendment
The first run did not meet its own success criterion. The failure was in the comparator mapping, not in rule R; the correction is labelled post-hoc and the first-run numbers are kept.
CONTROL CONTRADICTIONS · BEFORE / AFTER
WHAT HAPPENED
08
Independent verification
A separately written checker (independent_check_prd1.py, seed ) does not use the original rule source and draws a distinct sample. It is an independent implementation, not an independently conceived theory and not a proof checker; it re-checks 200 exact steps.
2,400 FRESH CASES
FRESH-SEED RE-RUN OF PRD-1
Seed , same families, with the amended comparator mapping applied.
09
What is established, what is not, and where ESC-I sits
DEMONSTRATED
An exact, reproducible classification of real 2×2 echoes; the gap between exact class and finite-horizon reading is quantified by two identified traps.
Corrected, proved conditions replacing two unrestricted claims of the original manuscript.
Admissibility rules can be written, frozen, falsified, repaired and re-confirmed under an explicit protocol.
A projection-aware rule predicts the regime of Π∘T for four projection families in the spectral frame, with no observed contradiction.
NOT DEMONSTRATED
That the algebra of complex numbers emerges uniquely from UNNS generative admissibility without its defining structure (J² = −I) being assumed.
That the rules exceed classical dynamical-systems theory, or that R beats an equally informed classical analyst.
That finite-horizon verdicts are asymptotic theorems.
Any correspondence with STRUC-I ladder admissibility or STRUC-PERC-I connectivity; the open percolation-to-admissibility conjecture is not touched.
Independent formal proof checking of the theorems.
THREE INSTRUMENTS, THREE PREDICATES
Structurally related vocabulary, mathematically distinct objects. No cross-instrument equivalence is established; ρ means spectral radius here and structural pressure in STRUC-I.
STRUC-I v1.0.4
Ordered numerical ladders under perturbation.
inv(Pε; L) ≤ ν(Vε(L))
Does the perturbed ordering stay within its vulnerability budget?
STRUC-PERC-I v2.5.0
Full pairwise vulnerability graphs.
(i,j) ∈ Eκ ⇔ |Δi − Δj| ≤ ε(κ)
Does connectivity persist across the scale range?
ESC-I
Real recursive states, projected.
xn+1 = Π(Txn)
Can the orbit regime be classified or certified?
A rigorous map from ESC-I trajectories to STRUC-I ladders, and a test of whether projection regimes track ladder admissibility or percolation, would be a new research question; it is not a result of this study.
10
Provenance · hashes, seeds, timestamps
Hashes are SHA-256 of the frozen rule text or prediction file recorded in each report. Timestamps are UTC from the reports.
ESC-I sits beside this material: the foundation documents frame the program's admissibility and regime theory, while ESC-I asks how an exact real recursion and its projection behave.
INDEX Browse the archive by title
Grouping is by title only, for navigation; it is not a claim about each document's findings. The title list is the one used on the SCR dashboard; this page does not re-inspect the archive.
unns.tech · UNNS Substrate Research Program · 2026 Every figure on this page is read from the recorded ESC-I reports or stated in the synthesis manuscript; nothing is estimated. The explorer is an illustration.