# SRHT — THE CURRENT MATH, FOR THE NOTEBOOK
### Copy date: Aug 25, 2026 · engine v9.10.787 · every line below is verified

Rule for the old pages: anything marked ✗ gets crossed out; the ✓ line beside
it is what replaces it.

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## 1. RESONANCE  (which candidate is close to best)

✓  R = 1 − (V − Vmin) / (Vmax − Vmin + ε)
   V = this candidate's cost; Vmin/Vmax over the sibling set; ε > 0 tiny.
   R = 1 for the best cost, → 0 for the worst.

✓  Floor comes out by itself:  Rmin = ε / (s + ε),   s = Vmax − Vmin
   ✗ old hand-set floor 0.01 — cross out, never set a floor by hand again.
   ✗ old GS-based R — cross out (divide-by-zero at GS = 0).

## 2. SPIRAL RESONANCE — YOUR GOLDEN SPIRAL, CONFIRMED BETTER (use this)

✓  R_φ = φ^(−2θ/π)        θ = 2π·k/(n−1)     k = cost rank (0 = best of n)
   φ = (1+√5)/2 = 1.618…
   Same thing written your notebook's way:  R_φ = e^(θ·ln φ / (π/2)) run
   inward — resonance decays by one golden ratio every quarter turn.
   Best keeps R_φ = 1; the worst of the set keeps φ⁻⁴ ≈ 0.146.
   STATUS: beat the straight-line R on BOTH validity and cost,
   10 seeds × 300 = 3,000 instances (z = 3.67 and z = 3.05).
   In the engine: [config flag]=1 (value-form, partial gain; rank-form is the
   full gain and the upgrade path).

## 3. CONTRADICTION  (how much a candidate violates)

✓  χ = clip to [0,1] of ( v / vmax )        v = violation size
   ✗ old steps {1.0, 0.6, 0.1} — cross out (made an 81:1 cliff).

✓  BEST FORM — CAPX (won its 3,000-instance study):
   χ = dip / (repair the remaining future can actually reach)
   χ = 1 exactly when the dip can no longer be repaired → gate turns hard
   at true irreversibility, stays graded before it.
   WARNING (proved by breaking it): the repair bound must be SOUND — never
   overestimate reachable repair, or χ < 1 on branches that are already dead.

## 4. THE KERNEL

✓  g(R) = R² / (2 − R)         range (0,1], strictly increasing:
   g′(R) = R(4 − R)/(2 − R)² > 0.    Your first page's R² lives in the top.

## 5. THE SCORE  (one number per candidate)

✓  Ĉ = g(R) · (1 − χ)²         always in [0,1]
   Ĉ = 1 only at best-cost AND fully valid; Ĉ = 0 at full violation.
   ✗ the p in Φ = p×R²×(1−χ) — cross out: same p for all siblings at one
   depth, so it never changed a ranking (0 of 2,000). p stays only as the
   cross-depth priority.
   NOTE, honest: your linear (1−χ) tested equal-validity and CHEAPER on one
   family. The square is kept for the clean algebra below, not because it
   beat your original.

## 6. THE GATE  (one rule, not two)

✓  prune a branch  iff  Ĉ < θ         θ ≈ 10⁻³, or θ = β·(best Ĉ in set)
   ✗ old dual gate  X > θ·Xmax  OR  P < ε  — cross out, one floor covers
   both (proved; it prunes slightly MORE — superset, known and accepted).

## 7. DIAGNOSTIC ONLY

✓  X = χ·(1 − R)     read it, log it, NEVER gate on it.

## 8. THE CROSSOVER  (yours — the exact flip point)

✓  χ* = 1 − √( g(R_A) / g(R_B) )
   A valid-but-costlier branch beats an invalid-but-cheaper one exactly when
   the cheaper one's violation exceeds χ*. Costs close together → χ* small →
   even mild violations flip the order. Verified on 2,000 pairs, 0 misses.

## 9. THE TWO LAWS  (when the math matters, when it doesn't)

✓  LAW 1 (redundancy): if violation just tracks cost, sorting by Ĉ IS
   sorting by cost — the whole apparatus adds nothing. Know when to not
   use your own tool.
✓  LAW 2 (guidance): if violation carries information cost doesn't,
   Ĉ makes orderings NO cost-only sort of any width can make.

## 10. DEPTH  (live in the engine)

✓  Association spreads hop by hop, each hop worth 0.7^h of its score,
   h = 1…5, default 3.   Knob: [config flag].

## 11. KEEP-WHAT-WORKS  (live in the engine, flag [config flag])

✓  Answer judged GOOD  →  chain LTP at dopamine 0.95,
                          each cell's confidence + 0.05  (cap 5.0)
✓  Answer judged BAD   →  each chain synapse weight × 0.90,
                          each cell's confidence × 0.97
   Small on purpose: one click nudges, repetition sculpts, decay forgets.

## 12. CROSSED OUT FOR GOOD  (keep the list — it's part of the math)

✗  cos(χπ/2) "Born rule" second gate — classical taper, double-counted χ.
✗  τ scaling — a positive constant never changes a sort.
✗  Discrete χ steps — replaced by the continuous χ and the crossover.
✗  Bridge damper  mean(Φ_A, Φ_B) / (1 + |χ_A − χ_B|)  — REFUTED
   (your own margin note called it: "too blunt"). For two constraints
   use the joint gate  g(R)·(1−χ_A)²·(1−χ_B)²  or Deb's rule
   (feasible-first); the set-level Bundle(top-k) is still open.

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Every ✓ above has a proof or a pre-registered experiment behind it in
[local path]✗ has the test that
killed it in the same folder. Nothing on this page is old.