Connascence-matrix analysis — measured 2026-07-26

⚠️ SUPERSEDED MATHS — read foundational-values-and-creed.md §6 and §11 instead. This evidence file endorses the three-term composition (base + 0.5×fo + 0.25×so) and quotes a "~6%" residual; the adopted design is two-term damped (residual 4.70%) and rejects second-order outright. Its "49/49" mismatch count is now stated as 42/42 off-diagonal, and "48 of 49 symmetric" as one asymmetric pair of 21 (47/49 cells). Kept verbatim as the measurement record.

Source: https://patterns.sddevelopment.be/practices/ammerse_impact_analysis/. Coefficients now available (previously not in-repo). All results computed, not asserted.

1. The truncation is SOUND — the design's central maths question is answered favourably

First-order matrix M (7×7, rows = impacting value, cols = impacted).

Dominant |eigenvalue| λ = 2.3350 (power iteration, 3000 iters)
Max abs row sum (upper bound) = 4.0000
AMMERSE applies divisor 6 in the formula
Effective per-step gain λ/6 = 0.3892 < 1 ⇒ the Neumann series converges
Truncation error after two terms is governed by (λ/6)³ ≈ 0.0589 (~6%)

base + 0.5×first_order + 0.25×second_order is a legitimate limited-horizon truncation with ~6% residual. "Good enough for government work" is quantitatively defensible here. This retires the convergence objection.

2. The second-order matrix is NOT derivable from the first-order one

The article states second-order values are obtained "by multiplying the first-order impact matrix with itself." Tested and false. Hypotheses tried, mismatches out of 49 cells:

Hypothesis	Mismatches
`M×M` (as published claim)	49/49
`M×M` with diagonal zeroed	42/49
indirect paths only, `Σ_{k≠i,j} M[i][k]·M[k][j]`	42/49
`M + 0.5·(M×M)`, diagonal zeroed	40/49
`1.75 × M`	32/49 (best fit, still wrong)
first-order with diagonal zeroed	42/49

The published normalisation is also not raw / max|raw|: for the Agile row that gives [0, −0.58, 0.29, 0.12, −0.88, 0.58, 0.50] against the published [0, −0.69, 0.25, 0.05, −1.00, 0.82, 0.56].

Consequence: the second-order matrix is independently authored judgement, not a computed ripple. Two design implications:

The 0.25 × second_order term is a second independent input, so it needs its own provenance record — it cannot inherit the first-order matrix's.
The composition invariant ("authored delta is the only persistable term; the matrix supplies derived ripple") holds for first-order and does not hold for second-order.

Recommendation: use first-order only. The existing in-repo tactic already calls first-order "the standard path" and second-order "high-stakes only". That guidance is now backed by a measurement rather than a preference — and first-order alone carries the sound λ/6 = 0.389 gain.

3. One probable typo in the published first-order matrix

The first-order matrix is symmetric in 48 of 49 cells. The single exception:

Maintainable → Extensible = +0.75
Extensible → Maintainable = −0.75

The published second-order matrix is fully symmetric (0 asymmetric pairs). Given the surrounding symmetry, this is very likely a sign error worth reporting upstream. It matters because it is the one cell that would make the matrix directional, and any consumer must decide which sign to trust.

4. The matrix is symmetric — which vindicates the "correlation" framing for the matrix

An earlier review argued the authored number is a gain, not a correlation, because correlation is symmetric while causal links are directed. Measured: the connascence matrix is symmetric (48/49, see §3), i.e. correlation-shaped, not gain-shaped.

So both framings are right, in different places:

Artefact	Shape	Right name
Value↔value connascence matrix	symmetric	correlation-like — the operator's framing is correct here
Node→node `impacts` edge	directed	gain / elasticity — directed, so not a correlation
Artefact→value `delta`	directed, intervention effect	effect of adopting, neither of the above

The earlier "it's a gain not a correlation" correction applies to the edges, not to the matrix. Withdraw it for the matrix.

5. The absolutism problem — diagnosed, and it is the renormalisation

Operator's concern: iterating past tertiary effects over-amplifies the signal and converges on 1s, 0s and −1s; a dampening effect (lowering each tier's relative weight) was attempted, but the maths was unverified.

Verdict: the concern is real and correctly observed, the dampening is sound and in fact conservative, but the cause is NOT the iteration. It is the per-tier max-renormalisation.

5.1 With per-tier renormalisation: absolutism is provable, not incidental

Renormalising after each matrix application is power iteration. M^k·v / ‖M^k·v‖ converges to the dominant eigenvector regardless of the starting vector. So every artefact's higher-order profile converges to the same pattern, and the base vector's identity is destroyed.

Demonstrated with three deliberately dissimilar starting artefacts, renormalised each tier:

Artefact	tier 3	tier 5
rigour-heavy	`[1.0, 0.37, −0.06, 0.07, 0.27, −0.30, 0.61]`	`[1.0, 0.47, −0.36, 0.03, 0.43, −0.57, 0.63]`
velocity-heavy	`[−0.62, 1.0, 0.52, 0.35, 0.89, 0.71, −0.82]`	`[−0.61, 1.0, 0.43, 0.32, 0.95, 0.56, −0.82]`
`safe_to_fail`	`[0.05, 0.72, 0.66, 0.25, 0.02, 1.0, −0.13]`	`[−0.46, 1.0, 0.66, 0.36, 0.74, 0.86, −0.76]`

All three migrate toward the same attractor — the dominant eigenvector of M:

Agile −0.55 · Minimal +1.00 · Maintainable +0.35 · Environmental +0.30 · Reachable +0.97 · Solvable +0.46 · Extensible −0.77

Under renormalisation, every artefact eventually "says" Minimal-and-Reachable-maximal, Agile-and-Extensible-negative. That is the absolutism, and values piling onto ±1 and 0 is its signature. Note the published normalised second-order matrix contains cells at exactly −1 and +1 — the fingerprint of max-normalisation.

5.2 Without renormalisation, the dampening already solves it — with room to spare

Per-tier contribution (1/2)^(k−1) · (M^(k−1)·b) / (N−1)^(k−1), so the decay ratio is

r = 0.5 × λ/(N−1) = 0.5 × 0.3892 = 0.1946

Measured contribution norms for a real artefact (safe_to_fail):

Tier	Contribution norm	Cumulative
1 (base)	1.2971	1.2971
2	0.1215	1.3135
3	0.0123	1.3234
4	0.0014	1.3239
5	0.0002	1.3239

Tail beyond tier 3 = 0.91% of base. Stopping at tertiary is not a compromise; it is already past the point of measurable difference.
Tail beyond tier 2 = 4.70%. Two tiers would also be defensible.
Even undamped it converges — ratio 0.3892, tail beyond tier 3 ≈ 9.6%. The divergence fear was unfounded in both cases; saturation came entirely from renormalising.

5.3 The fix: one final scaling, never per-tier

Since |base| ≤ 1 element-wise, the composed value is bounded by the series sum S = 1/(1−r) = 1.2416. So scale once, at the end, by (1−r) = 0.8054:

composed(v) = (1−r) × [ base(v) + Σ_{k≥2} (1/2)^(k−1) · (M^(k−1)·base)(v) / (N−1)^(k−1) ]

Verified on three real corpus artefacts: every output lands inside [−1, 1], and because every artefact is multiplied by the same constant, all relative ordering is preserved exactly.

Artefact	base max\|·\|	output max\|·\|	in range
`safe_to_fail`	0.90	0.740	✓
`avoid_gold_plating`	0.70	0.640	✓
`AMMERSE_impact_analysis`	0.60	0.438	✓

Generic over N: r = 0.5 × λ/(N−1) and λ ≤ N−1 (§1), so r ≤ 0.5 for any admissible consumer basis, giving S ≤ 2 and a final scale never below 0.5. The rule holds for any value system, and both λ and N are computable from the supplied matrix.

5.4 Consequences for the design

Drop per-tier renormalisation. Keep the dampening. The dampening was the right instinct; the renormalisation was silently undoing it.
This is a second, independent reason to use first-order only. The published second-order matrix has already been max-normalised, so consuming it as-is imports the very step that causes absolutism.
Truncation depth becomes a free choice rather than a defensive one: tertiary leaves <1%, secondary <5%. Pick on taste, and record the number.
r is a useful reportable diagnostic. As r → 0.5 the basis is approaching total agreement and the higher tiers stop adding information — the same degeneracy warning as §1, from the other direction.